preaerated in transit so that the biological oxygen demand (BOD) loading may be reduced before arrival a t waste treatment plants. Furthermore, pipeline fermentors which would be more efficient than conventional stirred-tank fermentors for certain insoluble-substrate utilizations (van Dedem and 1100Young, 1973), may be made possible with good aeration induced by surface baffles.
R e = Reynolds number in channel (= 4 r ~ p v / p ) Re’ = apparent Reynolds number in stirred cell, given as Re in channel to yield the same k~ values for mater Re, = Reynolds number of stirrer (= p N d z / p ) S = gas-liquid interfacial area in channel, cm* v = linear velocity of liquid, cm/sec V = volumetric flowrate of liquid, cm3/sec
Acknowledgments
u
Our thanks are due to Dr. G. D. Fulford and 11.Kilany for their assistance in the design of the flow channel and to the Department of the Environment, Canada, for financial support of the project. Part of the manuscript was prepared with assistance extended by Professor J. R. Bourne, Department of Industrial and Engineering Chemistry, Swiss Federal Institute of Technology, Zurich. Nomenclature
C = concentration of Tween 20, ppm C,, = C 0 2 concentration of inlet liquid, Jf Gout = C o r concentration of outlet liquid, M C* = saturation concentration of COZ, ;If d = diameter of a magnetic stirrer, em IZL = liquid-phase mass-transfer coefficient (water), cm/sec k ~ s =. liquid-phase mass-transfer coefficient (surfactant solution), cm/sec N = stirrer rotational speed, rpm Q = volumetric absorption rate, mol/sec TH = hydraulic radius, cm
GREEKLETTERS p p
= = =
static surface tension, dyn/cm density of liquid, g/cm3 viscosity of liquid, g/cm sec
literature Cited
Byers, C. H., King, C. J., AIChE J., 13, 628, 637 (1967). Burnett, Jr., J. C., Himmelblau, D. RI., AIChE J . , 16, 185 (1970). Cullen, E. J., Davidson, J. F., Chem. Eng. Sci., 6 , 49 (1956). Davies, J. T., AZChE J., 18, 169 (1972). Davies, J. T., Rideal, E. K., “Interfacial Phenomena,” 2nd ed, Academic Press, New York, N. Y., 1963, p 168. Fortescue, G. E., Pearson, J. R. A., Chem. Eng. Scz., 22, 1163 (1967). Goodridge, F., Gartside, G., Trans, Inst. Chem. Eng., 43, T62, T72 (1965). Hirose, T., Moo-Young, PI., Can. J. Chem. Eng., 47, 265 (1969). Hughes, R. R., Gilliland, E. R., Chem. Eng. Progr. Symp. Ser., S o . 16, 51, 108 (1955). Oliver, D. R., Atherinus, T. E., Chem. Eng. Sci., 23, 525 (1968). Treybal, R. E., “Mass Transfer Operations,” 2nded, McGraw-Hill, New York, N . Y., 1968, p 4.5. van Dedem, G., Moo-Young, M., Biotechnol. Bzoeng., 15, 419 (1973). RECEIVED for review May 17, 1972 ACCEPTED May 22, 1973
Mass Transfer to Naturally Flowing Streams V. M. Nadkarni* and T. W. F. Russell Department of Chemical Engineering, University of Delaware, Newark, Delaware 19711
It i s sometimes necessary for a process design engineer to estimate rates of oxygen transfer to naturally flowing streams when he i s concerned about disposal of waste waters. A typical correlation used to calculate the mass transfer coefficient i s the empirical one developed b y Downing and Knowles (1 964). In this short note a semiempirical correlaton i s presented which i s simpler in form than that ot Downing and Knowles and which i s superior in its agreement with published data.
I n rivers and streams the liquid Reynolds number is always quite high and for these turbulent flows the liquid-side mass transfer coefficient may be estimated with a relation of the following form ~
LL
(1)
= dDL/te
The characteristic length and velocity scales can be replaced by more easily measured parameters by considering their relationship to the rate of energy dissipation. Batchelor (1953) relates the basic length and velocity parameters to e as follows
i=
[pLa/E~1~4
where te is the average time of exposure of a liquid element a t the interface. An expression for the exposure time has been proposed by Banerjee, et al. (1968), which approximately represents the physical situation.
t,
=
(2)
i/lzil
where i and zi represent the average length and velocity parameters characterizing the turbulence structure, respectively. 414
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No.
4, 1973
(3)
(4) (5) We thus have the following relation for the average liquidside mass transfer coefficient iL =
dDL(€/YL)1/4
(6)
I n the case of gravity-driven flows, a macroscopic energy
Table 1. Comparison of the Proposed Correlation (Eq 18) with Downing and Knowles Correlation. %-deviation Obsd no.
Ref
kLR,
cm/hr
ReL
kLP, cm/hr
kLP/iLR
x
100
%-deviation
kLm, cm/hr
kLDK/kLR
x
24.50 19.29 -21 3 38.10 60,430 26.60 23,48 -11 7 41.11 2 75 ,948 20.80 26.00 +25 0 44.33 3 82 ,932 23.16 4 -3 1 74 ,639 23.90 40.97 69.80 50.78 75,26 367,817 -27 0 5 6 50.30 49.79 60.33 -1 0 405 ,626 43.60 41.76 31.52 472 ,350 -4 0 7 53.80 45.98 -14 5 418,589 8 50.70 Yewdale, Beck (Owens, et al., 18.30 17.56 24.35 -4 0 98,948 9 23.10 23.17 41.68 98,517 1964) 10 +o 3 21.14 99 ,206 21,80 11 34.89 -3 0 12 33,72 32.20 280,836 47.13 +4 7 31.50 37.71 56.51 13 291 ,772 $20 0 14 290,185 31.20 36.16 51.64 +16 0 30,80 15 288 ,007 35.97 51.44 +16 0 48.80 16 (Gameson, et a/., 1956) 134,532 40,29 -17 4 36.80 17 40.50 43,44 143 , 389 36.74 +7 0 130,053 18 34.21 48.80 39,OO -20 0 40.50 133,315 19 34.18 40.31 -0 5 20 30.70 30.87 96 ,862 28.33 +o 5 25.50 81,445 21 36.03 32.43 +27 0 Ivel, Beck (Owens, et al., 56,232 22 5.42 7.10 5.91 -16 7 1964) 5.90 6.05 53,200 23 5.57 +2 5 24 8.70 6.26 44,070 -28 1 5,71 11.71 180 ,286 25 8.20 9.25 $42 8 26 6.74 -45 2 12.30 57 ,771 6.82 27 49,493 6.57 5.10 7.24 +28 7 5.74 28 48,343 3.3c 5.50 +74 0 29 12.03 11.00 180,720 9.75 +9 3 30 12.66 14.60 -22 0 200 ,676 18.70 158,851 31 3.21 8.24 +55 5 5.30 32 136,226 4.13 7.75 -4 4 8.10 &'L% = reported valu,3? of the mass transfer coefficient; &,p = mass transfer coefficient predicted by eq 18; ~ L =~ mass K efficient predicted by Downing and Knowles correlation (Downing and Knowles, 1964). 1
Black, Beck (Owens, et al., 1964)
0
balance relates the viscous dissipation of energy per unit mass, E , to the hydrodynamic parameters as follows E
=
S represents
[g/YL1141/oL[sDL11~4
(8)
For open-channel gravity-driven flow, Whitaker (1968) gives the following Impression for average liquid-phase velocity
oL
=
d[(1.082
x
1o-*)g sin e ~ ~ 4 ' 3 ] / n *
(9)
1%here 0 is the angle of inclination with the horizontal and the coefficient n i i a measure of roughness of the bottom surface. The hydraulic diametei, H L , is defined as
H L =
i.e
combining eq 8 and 12, we get
(7)
gSoL
where g is the acceleration due to gravity and slope.
E L --
100
+56 0 +55 0 +113 0 +71 0 $8 0 +20 0 -28 0 -6 0 +33 0 +80 0 +60 0 +46 0 +79 0 +66 0 +67 0 -25 0 -9 0 -30 0 -16 0 -8 0 f41 0 -24 0 -5 0 -34 0 $13 0 -44 0 +42 0 +67 0 -11 0 -32 0 -39 0 -49 0 transfer co-
4 X crowsectional area of liquid flow wetted perimeter ~~
(10)
I n the case of ordinary earth canals and rivers, the roughness coefficient n is equal to 0.04412 cmlis (Whitaker, 1968). Using this value of n, we get
JL =
0.6511/=(Re~O
7 5 / H ~Os) 1
(15)
Equation 15 is therefore valid for a cgs system of units. Mass transfer does not depend solely on how much energy is imparted to the system, but also on how that energy is distributed and ultimately dissipated. For this reason the effect of the geometry of the system on energy requirements needs to be considered. An empirical way of doing this is to modify eq 15 as follow
. HL = 4wh/(2h
+
W)
(11)
M here w is the average w d t h of the channel and h is the average channel depth. The slope is thus related to the phase velocity by the relation
S * D~~n*/(1.082 X 10-*)gH~*'~
(12)
The function @ accounts for the geometry of the system. The mass transfer data reported for four different natural streams (Gameson, e t al., 1956; Owens, et al., 1964) were used to find the functional form of a. The experimental values LL reported ranged from 5.10 to 69.88 cm/hr over a range of liquid Reynolds numbers from 44,000 to 475,010. The following correlation shown on Figure 1 predicts values of the mass Ind. Eng. Chem. Process Des. Develop., Vol. 12, No.
4, 1973 41 5
Equation 18 can be modified by changing the units and using eq 11 to a form more similar to that of Downing and Knowles
iL = 4
1 ~ 7%~ 0 yi
+ 2.4hlw)
(20)
Based on the data in Table I i t is clear that eq 18 or 20 allow one to estimate the mass transfer coefficient more accurately than that of Downing and Knowles. One can also be more confident that correlations presented here can be more safely extended to different situations since they have some basis in simple theory. Nomenclature
cross sectional area of flow, L* molecular diffusivity, L2/t Q = acceleration due to gravity, L/t2 go = gravitational constant, ML/(ft2) h = average depth of the liquid, L = hydraulic diameter, L = mass transfer coefficient, L/t 1 = average length scale of turbulence, L n = surface roughness coefficient, L1’6 Re, = Reynolds number of ith phase = H , ~ , / v ,dimen, sionless S = slope, dimensionless t, = exposure time, t Q = average velocity scale of turbulence, L/t u = average phase velocity, Lit w = average width of stream a t the waterline, L
A
D
Figure 1. Prediction of &L for natural streams using eq 18: A, Black, Beck (Owens, et a / . , 1964); 0,Yewdale, Beck (Owens, et a / . , 1964); E, Gameson’s data (Gameson, et a/., 1956); 0,Ivel, Beck (Owens, et a/., 1964)
transfer coefficient in reasonable agreement with the reported values for natural streams.
LL =
0 . 6 5 l d z ~ ( R e ~7sh0 O 8 6 7 / H ~947)1
(17)
For an air-water system, this relationship reduces to
LL
=
2.765 X 10-4(Re~07sh086’/H~l 947) YL =
(18)
0 . 0 1 cm2/sec
D L = 1.8 X
cm2/sec
Equation 18 gives the average liquid-side mass transfer coefficient in cm/’sec when h and H L are measured in cm. The Downing and Knowles correlation (Downing and Knowles, 1964) is presented in their paper as follows
K --
288~~,~h-~ . esp(-lOOh)]/[l ~{[l
+ 4 ~ - ’ . ~ ] )(19)
The constants in the above correlation are developed for the following set of units: h = average depth in m, w = average width in m, 8~ = mean velocity of water in m/sec, LL = mass transfer coefficient in cm/hr. d comparison of the predicted values of mass transfer coefficients, using eq 18 and 19 with the reported values, is summarized in Table I.
416 Ind.
Eng. Chem. Process Des. Develop., Vol. 12,
No. 4, 1973
=
=
SUBSCRIPTS G = gas phase L = liquid phase GREEKLETTERS = rate of dissipation of energy per unit mass, L2/t3 B p = viscosity, M / ( L t ) Y = kinematic viscosity, L2/t p = density,M/La literature Cited
Banerjee, S., Scott, D. S.,Rhodes, E., Ind. Eng. Chem., Fundam., 7, 22 (1968). Batchelor, G. K., “The Theory of Homogeneous Turbulence,” Cambridge University Press, England, 1953, Chapter VI. Downing, A . L., Knowles, G , Int. Conf. Water Pollut. 2nd, Adaan. Water Pollut. Res., 1, 268 (1964). Gameson, A. L. H., et aZ.,Water Sanzt. Eng., 6 , 52 (1956). Owens, >I., Edwards, R. W., Gibbs, J. W., Int. J . Azr Water Pollut., 8,469 (1964). Whitaker, S., “Introduction to Fluid llechanics,” Prentice-Hall, Englewood Cliffs, X. J., 1968. RECEIVED for review May 19, 1972 ACCEPTED May 7 , 1973
