CURRENT RESEARCH Role of Diffusion in Biological Waste Treatment Wladimir Gulevich,' Charles E. Renn, and Jon C. Liebman Department of Environmental Engineering Science, The Johns Hopkins University, Baltimore, Md. 2121 8
The uptake rate of nutrients bq microbiota has been described b> different authors either in terms of rates of biochemical reactions (chemical kinetics approach) or, rather recently. in terms of diffusional mass transfer t o biologically active surfaces. To determine whether the nutrient uptake rate was dependent on the external velocity field, the microorganisms were grown on a rotating disk which would present mathematically the simplest surface from the standpoint of mass transfer, and Nhich was used successfully in other studies of heterogeneous chemical kinetics. In the laminar flow region, the rate of glucose uptake was found to be influenced by the velocity field near the active biological surface. This finding implies that the over-all transformation is diffusion dependent, and that the diffusional mechanism must be considered when design of waste treatment facilities is to be accomplished. The rotating disk rnethod was well suited to the study of nutrient uptake rates of biological slimes.
I
nterest in the prediction of waste purification rates developed rather early in this century, as evidenced by the publications of Theriault (1918, 1927) and Streeter and Phelps (1925). The early writers indicated the resemblance between a first-order kinetic reaction rate and the rate of satisfaction of biochemical oxygen demand, but were careful to refrain from imputing a kinetic mechanism to the biological purification phenomenon. However, a kinetic explanation of the obserbed rates proved very attractive in its mathematical and logical siniplicitq. and these rates were explained by subsequent authors in terms of kinetic mechanisms of various orders of reaction and increasing mathematical complexity. Recently. scheme5 have appeared (Young and Clark, 1965; 1965) which use the Michaelis and ReVelle, Lqnn. o r d., Menton enzqmatic reaction simplification, leading t o a secondorder reaction rate expression. In the past decade, however, several authors, mainly at Rice Universit) (Swilley and Atkinson, 1963; Swilley, Bryant, et d., 1964; Atkinson. Busch, et al., 1963; Busch, Grady, et d., 1962) have described nutrient uptake rates in terms of a mass I Present address, Sanitary Engineering Department, Preventive Medicine Division, Walter Reed Army Institute of Research, Washington, D . C. 20012
transfer mechanism. Their order-of-magnitude calculations and experimental studies showed that diffusional control of the over-all transformation rate should receive closer scrutiny. These studies are of great importance since they indicate that nutrient uptake rates might be improved by physical manipulation of the waste treatment processes; a n improvement which could not be achieved if the rates were controlled by reaction kinetics. In view of their findings, this work is a n attempt to design a n experiment which would determine whether or not diffusional control is possible, and whether further elaboration of the mass transfer model proposed by Swilley and Atkinson (1963) would produce fruitful results.
Nutrient
Uptake Model and Tlieoretical Requirements
Swilley, Bryant, et al. (1964) described a reasonable pathway for nutrients from the bulk of the solution to the organism and their transformation t o bacterial waste. They postulated a series of mechanisms: diffusion from the bulk of the liquid phase to the active surface through a thin boundary layer, adsorption o r surface reaction (transformation) at the bacterial surface, transport through the membranes to the interior of the cell, a volume reaction within the cell, and reverse pathways for metabolic wastes. With each of these steps, one may associate a time-dependent rate, the absolute magnitude of which is largely dependent on the rate of the particular step, but also dependent on the rates of the previous and subsequent steps. For example, if the rate of the surface reaction were significantly lower than the diffusion rates and rates of volume reaction within a cell, a pile-up of nutrient would occur at the surface, and the over-all transformation rate would adjust itself to this slowest step. If the over-all process is studied, the variables of significance will be the ones describing the surface reaction, with all other variables being hidden. A review of the significant variables shows that while the local concentration of the nutrient species plays a role in each of the steps, the velocity field outside the cell mass is of interest only in the first step. Thus, if a n experimental study were to show a dependence of the over-all process rate on the velocity of the liquid, it would indicate that the diffusion step influences this process rate and that a certain amount o f irnprovement might be expected if the velocity field were changed. I t is of interest to consider the first two steps. If the surface transformation (adsorption or surface reaction) rate is much higher than the diffusion rate, the nutrient arriving :it the
Volume 1, Number 2, February 1968
113
surface will be used up as it arrives, so that the local nutrient concentration on the liquid side of the interface will be zero for all practical purposes. If, on the other hand, the rate of surface reaction is appreciably lower than the rate of diffusion, n o concentration gradient will be established in the liquid phase, and the local concentration a t the surface will be the same as in the bulk of the liquid. The case when both rates are of the same magnitude is much more difficult to resolve mathematically, since the local surface concentration would be neither zero nor the bulk value. The absolute value of the local surface concentration would depend o n the relative magnitudes of the diffusional and transformation rates. Since the latter is the most general case (the other two being limiting solutions), preferably any experimental apparatus should be capable of determining the local surface concentrations, either by experimental means or by calculation. Otherwise, no meaningful estimates of the theoretical diffusional flux may be made. This implies that the momentum and mass-transfer equations (Navier-Stokes and diffusional partial differential equations) should be solvable with a minimum of simplifying assumptions. The surface which satisfies most of these requirements is a thin disk rotating about its axis in a large body of fluid. The momentum transfer (Navier-Stokes) equation for the rotating disk was first solved by von K B d n (1921), with subsequent additions and corrections by Goldstein (1935) and Sparrow and Gregg (1959). The velocity pattern of the fluid in the vicinity of the disk is shown in Figure 1. The final solution of the equation for the axial velocity, V,, was o b tained in terms of two infinite series-ne valid in the vicinity of the disk and the other a n asymptotic expression valid sufficiently far from the disk surface.
+ (1/3)Z3+
(Vt)z,o = d ~ [ - 0 . 5 1 0 2 3 Z 2
where v
(V,),,,
=
.I
rotational velocity, second-'
Z
zd&,
distance from the disk surface in the axial direction (dimensionless)
+ 2.112e-a-88147z 1 . 1 2 3 2 ~ ' . ~+ ~ 0.67056t-2-65341z ~~" + . . . . . . . . .) =
(1)
kinematic viscosity, sq. cm./second
w = =
. ...
d&-0.88447
(2)
layer remains constant across the whole disk (Frank-Kamenetskii, 1967). This property of the disk is valuable when boundary conditions for the mass transfer equation are considered. The mass transfer equation, in the absence of volume reactions, is written in the usual cylindrical coordinates as:
where c = point concentration of nutrient (grams per cc.) DAB = diffusivity of nutrient A through solvent B (sq. cm. per second). The solution of this equation for the disk was thoroughly discussed by Levich (1942, 1944a, 1944b, 1948, 1962). For the most general case, the boundary conditions are: In the bulk of the fluid nutrient c = c,asz+
where c, is the bulk concentration of nutrient (grams per cc.) At the transforming surface
(i:)
DAB where k ,
=
(Vz),-
-0.51023z2d~
=
-0.88447dz
(3) (4)
An inspection of these expressions reveals that the axial velocity is independent of either the radial or angular directions. The boundary layer thickness is usually defined as that distance from the surface at which the velocity reaches some preselected high percentage of the velocity at great distance from the surface. Thus, the rotating disk may be termed uniformly accessible, since the thickness of the boundary
k,c," as z
-+
0
of the nutrient at the surface (grams per cc.)
cs = concentration
m
=
empirical order of surface transformation (dimensionless)
For the disk (a uniformly accessible surface), cs remains constant. The second boundary condition can be simplified considerably if the rate of chemical reaction or adsorption is significantly higher than the rate of diffusional mass transfer. Under such conditions, the surface concentration becomes: cS = O a s z + O
For this simple boundary condition the solution of the partial differential equation assumes a simple form (Levich, 1962);
s:
exp(
c = (c=) =
= 2-0
surface transformation rate constant (cm. 3n1-2 second-')
Both series converge rapidly, so that it is jaastifiable (Levich, 1%2) to retain only first terms, giving: (V,),,,
a
~~
sd"
exp(
L
DAB
Vz(Y..>.
i~ V A Y ) f i Yj d f
DAB
On substitution of the appropriate expressions for V,, the above expression simplifies to (7) and
r is the gamma function.
4 EXPERIMENTAL DISK,
VARIABLE SPEED MOTOR
:RATOR
I
d
NUTRIENT SOLUTl O N
Figure 1. Velocity distributions on the rotating disk
Figure 2.
Growing apparatus
The integral expressions have been tabulated by Abramowitz (1 95 I). Since the distribution of nutrient concentration is of less interest than the magnitude of the mass transfer rate, the value of diffusional flux is of great interest:
Figure 3. Testing apparatus
with the rotating disk apparatus. The value of the rotating disk in studies of heterogeneous chemical reactions has been well substantiated by many other authors, the most recent being Riddiford (1 966). Description of Experiments
where E l = diffusional flux (grams per sq. cm. per second). When the indicated differentiation is performed and the results are algebraically simplified, the result is
where K1 is a parameter dependent on the properties of the nutrient and the solvent. The expression for difusional flux in the more general case when both the surface transformation rate and the mass transfer rate are of the same order of magnitude is more complicated and usually involves a solution of two simultaneous equations:
where
cy
=
DAB 1.61166/(,
(-y>
I'"'
d;;
Dqfi
This involves knowledge of kinetic or adsorption constants in addition to the properties of the nutrient. If one plotted for a particular concentration the diffusional flux, calculated from Equation 9 with appropriately transformed axes, a linear relationship between flux and square root rotational velocity would result. If such a linear relationship were found experimentally, the argument for complete diffusional control of the over-all transformation would be supported. On the other hand, any curvature shown by the experimentally determined points would argue for a mixed control, where both kinetic and mass transfer rates are of equal importance. Such a clear cut separation of mechanisms is easily obtained
Two separate experimental setups were employed: one for growing the bacterial films on the experimental surface and the other for testing. The former was extremely primitive. As shown in Figure 2, a n earthenware vessel provided with a fritted glass aerator was used to contain the nutrient solution. The Lucite disk used in the testing was held just above the liquid level, so that the foam generated by the aerator could impinge on the disk. The disk was kept in this position until the desired thickness of the bacterial film was built up and was ready t o be used in the experiment. The second setup was almost equally uncomplicated. A 4-liter glass desiccator was used to contain the glucose solution, the uptake of which was studied. The disk, covered with biological slime, was immersed in the glucose solution and was rotated with a variable speed motor suitably geared down to the desired velocities. A solid state motor controller, rather insensitive to line voltage fluctuations, was used to set the motor at the desired velocity settings. This apparatus is shown in Figure 3. Small amounts of C1'-glucose, tagged at the initial Cgroup, were added to the glucose test solution to act as a radioactive tracer. Samples of the solution were withdrawn as the test progressed and prepared for radio-counting in either a IiqLiid-scintillation counter or in a Geiger-Muller counter. Care was exercised in making the experimental disk. The body of the disk was made of l,',ti-inchLucite and was machined to close tolerances so that the disk would be circular. It was mounted on a stainless steel shaft which was provided with two roller bearings. The whole assembly was made true, SO that no wobble could be detected at speeds far exceeding the ones used for experimentation. After several unsuccessful trials to grow even, adhering Volume 2, Number 2, February 1968 115
bacterial slimes which could withstand higher rotational velocities, the following procedure was adopted. A balanced synthetic nutrient was used throughout the study. It consisted of, in grams, milk peptone, 249; meat peptone, 167; urea, 43.2; and distilled water, 8 liters. Five milliliters of this stock solution produce 1 liter of nutrient with an approximate biochemical oxygen demand (BOD) of 200 mg. per liter. The strength of the nutrient was adjusted so that the desired thickness of slime could be grown after 3 days of residence in the growing vessel. The experimental disk was kept just above the nutrient level, so that the foam formed during aeration would collect on the underside of the disk. When the Lucite disk was immersed in the solution, slimes failed to grow. The nutrient solution was seeded anew prior to each experiment with several trickling filter stones obtained from the local sewage treatment plant. Failure to reseed invariably produced profuse floc within the body of the nutrient with only flaky and uneven slimes on the surface of the disk. Microscopic examinations of the surface cultures indicated that the young slimes were morphologically different from slimes grown in old solutions. Soon after reseeding, the slimes consisted of various microorganisms, mostly bacteria and fungi. Initially, the most common organism was a bacillus growing in branching chains and having a central spore. As the aerated culture aged, there was a noticeable shift in microbiological composition of the slimes: The fungi completely disappeared, and the dominant bacillus diminished in numbers, so that when the same culture was aerated longer than about 2 weeks the slimes were composed almost exclusively of streptococci. This shift in cultural characteristics was probably due to a selection caused by unfavorable conditions to aerophilic surface organisms which occurred in the growing vessel. The thickness of the slime was not considered critical from the standpoint of the nutrient uptake, since both diffusion and subsequent chemical reaction or adsorption steps are surface-dependent mechanisms, rather than volume-dependent. Qualitatively, however, there was an upper limit to slime thickness, since at higher rotational velocities there was a tendency to erode the surface. Such uneven, pitted surfaces would be completely unusable in the study. After the desired thickness of biological slime had been built up on the surface of the disk, the disk was transferred to the testing apparatus, so that the uptake of nutrients under controlled conditions could be studied. Prior to transfer, the slimes were starved for 24 hours to improve the nutrient uptake rate during the test run. Glucose was selected as a test nutrient primarily for practical reasons : Its metabolic pathways and biochemical behavior have been well studied, and its liquid diffusivity value is available in the literature. This permitted the comparison of experimental data with the theoretical equations without having to determine the diffusivity, which is a rather involved study in itself. To reduce the amount of analytical work and especially the time-consuming wet analysis, tracer techniques were re116 Environmental Science and Technolog?
lied upon mainly. If the amounts of intermediate tagged metabolic products diffusing back into the bulk of substrate which might be formed were were negligible, and if the C1400? to be driven off by acidification of the scintillation sample, the tracer techniques would give sufficiently accurate results. These assumptions were borne out by the experimental results. To assure that aerobic conditions prevail, especially at higher initial concentrations of glucose (100 mg. per liter). the test solution was prepared with well-aerated BOD dilution water which contained the requisite amounts of inorganic salts but was not seeded with sewage organisms. Pilot runs were performed to determine the dissolved oxygen (D.O.) concentration during the 2-hour runs. At no time did the final D.O. concentrations fall below 3 mg. per liter when the initial concentration was above 7 mg. per liter. The maintenance of this D.O. concentration was due to the considerable re-aeration action of the rotating disk. The concentration of 3 mg. per liter is normally well above the threshold value at which nutrient uptake may become dependent on the oxygen available rather than nutrient concentration. The test nutrient concentration varied between 33.3 and 100 mg. per liter of glucose. These limits were selected because below the lower limit, diffusional fluxes would be small, so that experimental error might mask the uptake; while above the upper limit the nutrient concentration may cease to be a limiting factor, and the fluxes may become functions of reaeration rates. Prior to the starting of a run, the prepared test solution was sampled to establish the initial tracer concentration. In each experimental run, the disk was immersed in the test nutrient solution to a depth of about one-half inch below the liquid surface. The upper limit of rotational velocities (about 100 r.p.m.) was dictated by rather severe vortex formation. Above this velocity, the upper surface of the disk became dry and cavitation started at the disk edges. Since this would indicate turbulent regime, such runs would be totally unusable for theoretical analysis. In addition, the associated Reynolds number (about 20,000) was sufficiently high for the purposes of this study. The selected rotational velocity was kept constant during the run; no variation in speed due to power surges was detected. Samples of the test nutrient solutions were withdrawn at various intervals. These intervals were in most cases kept uniformly at 10 minutes to facilitate subsequent statistical treatment of the results. The total duration of the single run was usually 2 hours. Longer experimental runs were not justified, since biological seeding of the test solution may reach such proportion that the uptake may occur not only on the disk surface but also in the body of the liquid. In addition, it was desirable to avoid large changes in concentration in order to permit the use of the statistical smoothing procedure described later. The samples were analyzed by tracer techniques. Two difyerent counting systems were employed in the determination of the activity-a Packard TRICARB liquid scintillation spectrometer with Bruno-Christian scintillator and a Nuclear-
Table I. Raw Data for Diffusional Fluxes Date July 17, 1966 Run No. 717 initial concentration of glucose: Rotational velocity of the disk: Radioactivity counting time: Temperature:
100 mg. per Lirer 10 revolutions per minute 40 minutes per sample 15" C.
Time after Start, Min.
Total Tracer Count, Counts
Counting Rate, Counts/Min.
0 10 20 30 40 50 60 70 80 90 100 110 120
79,960 80,138 79,008 SO, 228 78,378 78,559 76,633 76,698 76,452 17,443 74,514 75,150 73,646
1998 2003 1975 2005 1959 1963 1961 1917 1911 1936 1863 1879 1841
Chicago Geiger-Muller counter with a 1.4-mg. per sq. cm. mica window which was moderately sensitive t o carbon-14 tracer. All samples were sufficiently active to produce counting rates which had standard deviations of about 0.5 %. To prevent undue adsorption of active species on the laboratory glassware, it was treated with a hydrophobic silicone compound. Experiiiwntul Results
The raw data, a sample of which is shown in Table I, consisting of the initial glucose concentration, rotational velocity of the disk, and the time series of the tracer activities, were used to calculate the diffusional fluxes. This was accomplished by obtaining zero time intercept and slope by least squares analysis. In effect, this procedure gave the average uptake rate for the run. An example of such least-squares fit is shown in Figure 4. Three different nutrient concentration levels were investigated. To compare the different runs, the diffusional mass transfer rate obtained was transformed in such a manner that it would be brought to a common basis. This artificial variable was called specific flux and was obtained from the time series in the following manner:
where
= = AD = c, =
At = Ar = =
VT =
specific flux, cm. per hour diffusional flux, grams per sq. cm. per hour area of the disk, 78.5 sq. cm. actual initial concentration of the nutrient, grams per cc. time interval between two successive samplings or the nutrient substrate, hours fraction removed during time period, dimensionless, Ac/c, total volume of solution, 3000 cc.
The results (Table I1 and Figure 5 ) , though rather widely scattered, indicate a possibility of some slight curvature. The data were subjected t o statistical analysis and some orderof-magnitude estimates of the kinetic rate constants of Equations 10 and 11 were obtained. Since the equations involve trial and error solutions and rather cumbersome least-square curve fitting, the calculations were performed on it digital computer. The complete program used is given by Gulevich (1967) together with some intermediate calculations. The statistical response surface was rather flat with the limited number of data points, so that the determined constants are of rather limited accuracy. The curve offering the best fit is shown in Figure 5 by a dashed line. The determined constants were approximately k,
=
5
m
=
1'
x
lo-: (grams/cm.)o.j second
(2
Discussion of' Results The experimental results may be summarized as follows. The rate of nutrient uptake by biological slimes from liquid media is diffusion dependent. Since only one nutrient has been investigated, this general statement should be qualified somewhat. The above statement obviously depends on the chemical species involved and the relative magnitudes of their diffusivity and the rate constant for the biochemical reaction according t o which these materials are utilized by the organisms. If the dimensionless reaction number, defined as
is very miich greater than unity (as it is for glucose), the diffusion process is the controlling mechanism. However, conceivably, for some nutrient species the reaction number may
Time
Figure 4.
Example of data collected
Volume 2, Uumber 2 , Februar, 1968
117
5
4
-
I l l 1
-
---0
I I
t
I I
1
I
I
I
1
Specific Flux Calculated f o r Various K S and m i 1/2 from
1
1
1
1
0
EPS 9 and I O L i n e of Best F i t for Experimental Data E x p e r i m e n t a l Points
, L:
~~~~
~
~
.c
~~~~~~
Table 11. Calculated Diffusional Fluxes Rotational Velocity, Specific Flux, Run R.P.M. Cm./Hr. Initial Glucose Concn., 100 Mg./L.
5.
3
.-0
2
L
al 0
817 818 717 418 803 802 719 724 808 807 723
5 6 10 10 14 19 28 38 51 63 91
1.102 1.405 1.546 0.871 1.642 1.961 2.617 2.718 3.168 3.580 4.190
a In !
Revo!utions Per Minute
0 0
I
d
2
Figure 5. Comparison of experimental data with calculated values for various kinetic rate constants
Initial Glucose Concn., 33.3 Mg./L. 832 2 0.532 828 3 0.764 91 1 28 2.325 922 36 2.684 921 41 2.682 912 48 2.858 823 61 3.622 827 76 3.661 1010 86 3,877 83 1 95 4.905
qualification may not be a seriously limiting one for rather dilute domestic wastes and moderately polluted receiving waters, the over-all uptake rates of industrial wastes high in bacterial nutrients may no longer be predictable from nutrient concentration alone. For example, if the dissolved oxygen concentration falls below some value (variously reported, but in general below 1 mg. per liter) the over-all uptake may still be diffusion controlled, but dependent o n re-aeration rates, rather than nutrient concentration.
Initial Glucose Concn., 66.7 Mg./L. 0.560 1011 2 1012 6 1.628 1020 13 1.609 35 2.400 1021 1017 42 2.810 1018 46 2,962 1025 53 3.481 1026 72 3.794 1101 80 4.087 1102 98 4.541
The results of the present study agree with those of workers who have maintained that diffusional processes influence laboratory determinations of biochemical oxygen demand. The nutrient uptake rates are influenced by the velocity field of the substrate, not only in cases when the re-aeration rate is the controlling factor (such as in Warburg determinations), but also in the case of the standard bottle BOD test, where the available oxygen is in dissolved form. By necessity, in the latter case, the nutrient material must be very dilute so that the amount of dissolved oxygen is sufficient to satisfy the exerted demand and the concentration of dissolved oxygen does not fall to the level where it begins to control the oxidation rate. Because of this concentration limitation, the results of the study are directly applicable. The BOD test should be sensitive to the velocity field in the bottle, and the kinetic BOD constant would then be a value characteristic of the diffusionalflux under quiescent conditions. The difficulties connected with the reproducibility of such quiescent conditions help to explain the wide variations in the constant as reported by various authors even for identical substrates. This analysis of the BOD test is supported not only by the results of this study, but also by the data reported by Swilley, Bryant, et al. (1964) and Busch, Grady, et ul. (1962). When the biochemical oxygen demand test is made for the purpose of estimating the relative strength of a waste, the fact that the constant is velocity dependent is not fatal.
be small, sc that no diffusional gradient is established. Such might be the case for compounds of very low biodegradability or of some toxicity t o the organism. The application of the general statement is limited to dissolved substances only, since in dispersed media coagulation of colloids may, and normally does, occur. Such coagulation would drastically affect the diffusional fluxes so that the basic underlying theory would no longer hold. At higher nutrient concentration, the uptake mechanism by the slimes may change so that the nutrient concentration is no longer the limiting one, and the over-all uptake rate may depend on other substances which enter into the biochemical reaction. These other limiting substances may be the dissolved oxygen or even whole enzyme systems. While this last 118 Environmental Science and Technology
Applications
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