Mathematical models of biological waste treatment processes for the design of aeration tanks

Water Research eel. 12, pp. 491 to 501 00,13-1354/TS/07(ll.(b191 $(12.00/0 ~Pergamon press lad.. 1978. Printed in Great Britain. MATHEMATICAL MODELS OF BIOLOGICAL WASTE TREATMENT PROCESSES FOR THE DESIGN OF AERATION TANKS V. A. VAVILIN and V. B. VASILYEV Water Problems Institute, U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. (Received 10 June 1977; in revised form 19 October 1977) Alaetract--The paper discusses the application of formal mathematical models of biological treatment in designing aeration tanks. It deals with a complete mixing aeration tank, an ideal plug-flow tank, and a many-step aeration tank. Selection is substantiated of the simplest non-linear Monod's model for biochemical oxidation. A new model is advanced taking into consideration, among others the effect of pollutant sorption by activated sludge. The model permits calculating the optimal volumes of the aeration tank and stabilization tank in the contact stabilization system. b D x D L F x F L i K Ki K L l L Lo L v L, L f Ll L , Lm~, Lint. L, Pe z Pe x q r l r,. TI') T(.) T~ T~ NOMENCLATURE decay activated sludge coefficient, dispersion cooflicient for activated sludge, T~ dispersion coefficient for organic matter, reaction term for microorganisms, reaction term for substratum, T (') p$ ordinal step number of a many-step aeration tank, function making the distinction between an n- T~I step aeration tank and an ideal plug-flow aeration tank, sorption coefficient, u inhibitory coefficient, V substratum concentration at which the growth rate is one-half of the maximum, V tank length V, summary concentration of organic matter, X concentration of organic matter on the inlet of -Y the aeration tank-sedimentation tank system, concentration of poorly oxidized organic matter, X, effluent organic matter concentration, influent organic matter concentration, X '~ organic matter concentration of the i-th step of the n-step-aeration tank, X o organic matter concentration at which I /F L is minimum, X, substrate concentrations at which dependence X" T,.(L,) has local maximum or minimum, X*' soluble pollutant concentration, X Peeler's number for substratum, Peclet's number for activated sludge, AX sewage flow, recirculation coefficient, Y nominal detention time. W contact time of contact stabilization system, z stabilization time of contact stabilization system, mixed liquor retention time of the complete mixing aeration tank, ~. mixed liquor retention time of the/-th step of 2 the n-step-aeration tank, 2" total mixed liquor retention time of the n-step 2 aeration tank, 20 mixed liquor retention time of the plug-flow 2 aeration tank, contact time of the contact stabilization system 491 where the aeration and stabilization tanks are complete mixing tanks, stabilization time of the contact stabilization system, where the aeration and stabilization tanks arc complete mixing tanks, contact time of the contact stabilization system where the aeration and stabilization tanks are plug-flow tanks, stabilization time of the contact stabilization system, where aeration and stabilization tanks arc plug-flow tanks, flow velocity summary volume of the contact stabilization system, aeration tank volume, stabilization tank volume, summary activated sludge concentration, mean level of activated sludge in the aeration tank, activated sludge concentration on the exit of the aeration tank, total suspended solids concentration without storage substrate, activated sludge concentration on the inlet of the aeration tank, return activated sludge concentration, sorbed pollutant concentration, equilibrium storage substrate concentration, activated sludge concentration on the exit of the stabilization tank, change of the activated sludge concentration in the aeration tank caused by decaying processes, microorganisms yield constant, waste activated sludge coefficient, dimensionless distance from the inlet of the aeration tank, g of sludge to g BeD ratio, maximum growth rate of microorganisms, dimensionless variable which is equal to X' /X% equilbrium value of 2, 2-value on the exit of the aeration tank, 2-value on the inlet of the aeration tank, 2-value on the exit of the stabilization tank. The respective dimeasionalities are given in the text. 492 V.A. VAVILIN and V. B. VASn YEV INTRODUCTION The following set of differential equations is com- monly used in designing the aeration tank in the activated sludge system (Fig. 1) saturated with oxygen (Lee & Takamatsu, 1976): 1 d2X dX - - -+ tFx(X,L) = O Pe x dz z dz 1 d2L dL Pe L dz 2 dz iFL(X, L) = O, ( l) the boundary conditions being as follows: [ dX JPe x dz = X - X o z =0) 1 dL (1.1) I-[~eL-dT= L -- Lo, f !dX | Pe x dz 0 z = l{ !dL! _ (1.2) Pet. dz O, where X is the summary activated sludge concentra- tion, L is the summary concentration of organic pollutant, X o and L o are the same on the inlet of the aeration tank, Fx(X, L) and FL(X, L) are the reaction terms for microorganisms and substratum, res- pectively, z is the dimensionless distance from the inlet, l = T/u is the mixed liquor retention time (T is tank length and u is flow velocity), Pe x = ul/D x and Pel. = ul/Dt- are Peclet's numbers (D x and Dt- are dispersion coefficients for microorganisms and sub- stratum, respectively). The mixed liquor retention time is calculated for two limiting cases: complete mixing (Pex, Pe L --. O) L o - L e T~m = FL(L,, Lo ' Xo ), (2.1) plug flow (Pex, Pet. - , ~) Lo Ty = FL(L-~o ' Xo ) (2.2) L. Sedimentation \ L t LoF----------~Le~ \ / / I qlr+w) I X " qr ] qW Fig. 1. Aeration tank-sedimentat ion tank system. where L e is the substratum concentration on the exit of the aeration tank. During biological waste treatment processes the mixing regime can be practically realised in a rather good degree, on the contrary to the ideal plug flow regime. The degree of axial mixing is rather high in real plug-flow aeration tanks, Pe equalling 2-10, i.e. these real tanks differ substantially from ideal ones (Khudenko & Shpirt, 1973). Ideal plug-flow aeration can practically be pro- duced in a many step aeration tank (Milbury et al., 1965; and Khudenko & Kanenko, 1974), which is a vessel divided by baffles with holes into several sections. The baffles prevent back water flow, and Peclet's number increases for the whole tank. If every step of the aeration provides complete mixing, the total mixed liquor retention time is ~ Li 1 -- Li T = ,.~"(~-') (2.3) i=1 where i is the ordinal number of the step and n is the total number of steps. The optimal technological scheme of a biological treatment plant should decrease to a minimum the ratio of the total volume of treatment facilities, V, to the sewage flow, q, for the given requirements to the quality of purified water. It goes to decreasing the mixed liquor retention time = V/q(1 + r), where r is the recirculation coefficient. FORMAL BIOCHEMICAL OXIDATION MODELS OF THE MONOD'S MODEL TYPE It is obvious from formulas (2.1) and (2.2) that mixed liquor retention time depends on the function FL(X, L), i.e. on the kinetics of sewage organic matter consumption and activated sludge growth. According to Gaudy et al., 1967, pure culture growth models can be used to describe dynamically these two variables, Monod's model (Monod, 1950) providing for the best coincidence with experimental results. In conformity with this model, dX _ #, XL dt K L + L (3) dL /~ XL dt Y(Kt. + L)" where /,,, is the maximal specific growth rate of microorganism, K L is the saturation constant, which is equal to the substratum concentration at which the growth rate is half of the maximal rate, and Y is the microorganism yield constant. The processes with activated sludge are also described by modified Monod's models, in particular, by Herbert's model (Herbert, 1961), which takes into Mathematical models of biological waste treatment processes for the design of aeration tanks 493 account sludge autooxidation: dX I~XL bX dt K L + L (4) dL #.,X L dt Y(K L + L)' where b is the auto-oxidation rate constant. With inhibiting sewage, the kinetics of biooxidation is better simulated by Haldane's model (Haldane, 1931): d X Iz.X L dt K L + L + L2/Kt (5) dL /a XL dt Y(K L + L + L2/Ki) where K~ is the inhibition constant. It is clearly seen that for K i --, co system (5) changes to system (3). Other varieties of Monod's model are also known. (Ierusalimsky, 1967). The constants of Monod's model, which were found experimentally, are given in Table 1. It is noteworthy that complex biological and physico-chemical processes occur in the activated sludge system. Models of the Monod's model type are invariably formal and take account of only the most general phenomena, such as growth and decay of microorganisms, saturation of the growth rate on the substratum, and inhibition by high substratum concentration and by metabolic products. Theoretical and experimental studies of processes involving activated sludge (Grieves et al., 1964; and Fan et al., 1970) indicate that the hydrodynamic regime of flow in an aeration tank (regime of mixing) greatly affects the treatment rate. Complete mixing aeration tanks can be used to advantage under some condition while plug-flow aeration tanks, under the others. As mentioned earlier, ideal plug-flow aeration can be provided to some or other extent in a many step aeration tank. Hence the design of such a tank appears to be of interest. Several authors tackled the problem (Erickson & Fan, 1968; Zaprudsky & Gyunter, 1973; Vavilin & Vasilyev, 1976). Because the models of biooxidation kinetics are nonlinear, it is rather difficult to derive a simple formula in a general form for determining the necessary number of steps and the optimal relation- ship of thier volumes. It can be done, however, by simplifying to some extent models (3) to (5). As shown below, the accuracy of the calculations deteriorates negligibly. We use BischotFs method (Bischoff, 1966) to illustrate the calculations. Let us consider the dependence of the reciprocal of sewage consumption rate on the concentration of sewage organic matter 1/FL(L ). In the general case this dependence must have a minimum at some value of L,,, because when there is a wealth of substratum the rate of the process is limited by the concentration Of microorganisms, then the rate increases with biomass growth and, finally, drops following substratum disappearance (Fig. 2). 0 L e L m L o L Fig. 2. Dependence of the pollutant consumption reciprocal rate upon pollutant concentration. According to (2.1) to (2.3), the time of retention in a mixing tank is numerically equal to the area of a triangle with the sides L o - L, and 1/FL(Le), that in a plug flow tank, to the area bounded by the curve I/FL(L ) and by the lines L = L o and L = L , and the total time of retention in a many step tank, to the sum of areas n rectangles. Proceeding from the minimal area limitation we conclude that the relationship among Lo, L,, and L, determines the type of aeration tank. Table 1. Kinetic coefficients for activated sludge process Waste water composition r, KL, 8 O f sludge b, Experimental g I- 1 g BOD h- 1 technique Reference Synthetic wastes Glucose Domestic wastes Glucose Peptone 0.13 0.5-0.6 0.15 0.05 0.25 0.04 0.6 BOD 5 0.075--0.125 0.6 0.006 COD 0.028 0.67 0.003 COD 0.355 0.42 0.0036 BOD 5 0.065 0.43 BOD s Gyunter et al., 1967 Gaudy et aL, 1967and Srnivasaraghavan & Gaudy, 1974 Bcnedek & Horvath, 1967 Stack & Conway, 1959 Garrett & Sawyer, 1951 494 V.A. VAVILIN and V. B. VASILYEV Thus, if L e >I L,,, a mixing tank is more advantage- ous, if L,, t> L o, an ideal plug-flow tank is better, and if L e ~< L m ~ Lo, a layout whose first stop is a com- plete mixing tank and second step is an ideal plug- flow tank provides the best results (Fig. 2). The analytical expression for 1/FL(L ) can be obtained from models (3) to (5), assuming two postulations for the processes in the treatment system. First, the return activated sludge concentration, X , is independent of the activated sludge concentra- tion on the exit of the aeration tank, X,. Second, sludge autooxidation in the tank is negligible, because we are only concerned with the content of sewage organic matter on the exit of the tank, rather than with the overall balance of sludge biomass in the system. That the first postulation is valid follows from the theory of designing a secondary clarifier (Dick, 1972), which indicates that X depends solely on the area of clarifier cross-section, recirculation coefficient, r, and sludge settling capacity. The concentration X should be stabilized in the activated sludge system (Srniva- saraghaven & Gaudy, 1974), which can be done by changing W, because in this case, given the steady treatment, X and L, will vary negligibly. On the contrary, attempts to control the system by main- taining the X, :X, ratio constant lead to great fluctua- tions of X° and L . To prove the second postulation we compare the change in the sludge concentration by decay, AX, with the mean level of activated sludge in the aeration tank, X. Obviously: AX btX --~- =-~ = b~. The detention time does not generally surpass 10 h, and, if we use the data from Table 1, AX/~ = 5 % at worst and AX/~ = 3 % at best. Therefore, we can assume for the processes in the aeration tank, that dX - - ~ y dL or, in the integral form, X =X o+ Y(L o -L ) where the concentrations X o and L o can be con- sidered parameters, because rX,. Lf X o = ~-~r, Lo= l + r" Hence, for Monod's model, 1 1 Y(K L + L) F,(L) (dL/dt) #,.[X o + Y(L o - L)]L' (6.1) while under the inhibiting action of sewage 1 _ Y(Kt, + L + L2/K~) (6.2) F,(L) #mL[Xo + Y(L o - L)] Differentiating (6.1) and (6.2) and setting the derivative to zero, we show easily that in the first case L =K, (~1 + X°+YL° 1), (7.1) m . yKl~ while in the second case YK L L,. = y +(X ° + YLo)/K i x (x// l+ (X°+ YL°)(Y +(X°-YLo) /K i )1 ) KLy2 - - _ . (7.2) It ought to be noted that for the characteristic meanings of K L and Y given in the Table, and for the influent concentration of the activated sludge X o = 2 g 1- l Lm ' calculated on (7.1) formula, is about 0.3 gl-1. Therefore, situation, when L e/> L m ~ 0.3g 1-1 is practically impossible. At the same time complete mixing aeration tank upon these condition works unstable. During the treatment of domestic sewage, the organic matter content of which is generally low (L o ~ 0.2 g 1-t BeDs), Lm may surpass L o. A many step aeration tank is optimal in this case. Assuming that the activated sludge concentration maintains approximately the same level, close to X o, during the treatment process, we prove easily that the total detention time will be the least if, in the i-th step, Y f.* n- - i+ l i - I n n - i J T!") = J~/L o Le - ~/L o L e #mXo (8) and y Corresponding to this case dependence between reciprocal rate pollutant consumption and meaning of mixed liquor retention time for the different types of aeration tanks is given in Fig. 3. Calculations have been done under following meanings of constants and concentration: Pm =0" lh t ;K L=0.04g l - tBOD;Y=0.6g of sludge/g BeD; L o =0.2g I -~BOD;X = 2gI -1: L e = 0.01 gl - t BeD. It is easy to check, that in this case L m = 0.338 g 1- 1 BeD. Let us estimate the influence of disregard to the increase of activated sludge on accuracy of the many step aeration tank calculation. Because a many step aeration tank approaches a plug-flow tank in its characteristics, we use (2.2) in our estimation, namely, calculate. Mathematical models of biological waste treatment processes for the design of aeration tanks 495 T I0 .2 ,,d \ Tom = 2.7h Tot. = O.9h T (5~ 1.06h T- ' - - - - 0 0.1 0.2 L, g/L BOD Fig. 3. For calculation of the many step aeration tank. Explanations are in the text. T,+(X = Xo) - ~,t T,: Lo Lo f Y(K~ + L)dL _ f Y(K1, + L)dL #,,,LX o J/~.,L[X o + Y(L o - L)] - -Lm Lo : to f Y(KK, + L)dL #,~L[X o + Y(L o - L)]" (lO) After necessary transformation, it is easy to show that relative error in definition of the mixed liquor retention time will not be more than YLo/X o value, which approximately coincide with Y(L o - L , ) /X o- activated sludge increase value. For the example, examined above, YLo/X " -6~. Thus, (8) and (9) formulas will be fair if the condition YLo/X o ¢ 1 is executed. Let us introduce the value T ("~ - TpflX = Xo) fa t = T :(X = x o) = KI.{n(~/Lo/L , - 1) - In LolL,} L o -L ,+ K LInLo/L" (11) which makes the distinction between a many step tank and an ideal plug-flow tank. When substitute in (11) formula numerical meanings for n = 1,2,3... shall resultj ~ = 2,fl 2) = 0.5, j~a~ = 0.27,j~4) = 0.185, fls) = 0.14, f16) = 0.117. It is easily seen that as few as 4 or 5 steps arc sufficient for the process to be as productive as that in an ideal plug-flow tank. The case of waste treatment with inhibiting action, when kinetic of the process corresponds to Haldane model, is of particular interest. The thing is that in these conditions it is necessary to secure stabilized work of the aeration tank side by side with the intensification of the process. At the same meaning of the mixed liquor retention time in the complete mixing aeration tank concentration of the pollution can settle low (L~ g L0), or high (L e ~ Lo). Under the influence of different destabili- zing factors (fluctuations of q and L0) the system may go over out of the condition with good degree of treatment into the contrary condition with practically absent treatment effect. It is possible that in some cases concentration L,, calculated by the (7.2) formula, can correspond to the field when the system work is unstable. Consequently, calculating of the aeration tank for the toxic waste treatment, it is necessary to know the limits of pollutant concentration variations in the aeration tank, within which the work of treatment system will be stabilized. It is easy to understand this reasoning if applying it to the dependence of the mixed liquor retention time for the complete mixing aeration tank upon substrate concentration. From formulas (2.1), (6.2) ensure that: T~.,(L.) = Y(L° - L')(KL + L. + L~/K,) #=L.[X o + Y(L o - Le) ] (12) Curves (12) calculated at different constants K L and K I are shown in Fig. 4. It is seen that in definite meanings of these con- stants function T(L,) has two extremes, moreover, they become more pronounced when values of both constants reduce, and on the contrary, when K L and K l increase, T(L,) function monotonously decreases. For comparison, monotonous dependence T(L,) is given in Fig. 4 for Monod's model (point line). Consider the diagram of dependence (12), calcu- lated at K L = 0.05 g l - t ; Ki = 0.1 gl - t (Fig. 4, solid line). Let us mark through Lmi n and Lm. x pollutant concentration meanings at which dependence (12) has accordingly conditional minimum and maximum. It is seen, that if actual mixed liquor retention time is within limits To,(/~,) ~< T ~ T~=(/~,x), then three aeration tank stationary work regimes with concen- trations of pollutant L~)), Le~2), Le°), which" correspond to the cross points of T~=-const. straight line with To,(Le) curve, will be possible. However, stability analysis of conditions (1), (2), (3) (according to Liapunov) show us, that intermediate state, situated on the segment of the curve where the derivative is [dT(Le)/dLe] > 0, is unstabilize. Thus, only two work regimes of the system with low value 496 V.A. V^V]HN and V. B. VASILYEV • "0 I h -~ -- KL=O.O5g lL BOD,K, =OJ g / t BOD I ; I t.',,," . - - - - -KL=O. Iq / t . BOD,K~=O. Ig / t . BOD | ~ / y =0.6gss/gBOD . . . . Kc ' ,O .OrSg/ tBOD,K , ' ,O.07g/LBOD I I, 11 xo=2g/ t . . . . . . KL=O.Ig/LBOD,K : ~ ~ ',\ Lo:o.6g/LBoo II, , \ . - - - - -~.~. I\ i \ .~-~ ~. \ ~\ ',,, 'X / . / "\ . 1 I I t L[" Lm,On ~ L,~, 02 L~o, Oh,,,_, 04 L e g/k BOD Fig. 4. Dependence of the mixed liquor retention time for the complete mixing tank upon the effluent concentration L, in the case of high inhibiting influence. Explanations are in the text, of effluent pollutant concentration L = L~ ~ and with high concentration -L e =tt3~-e, can be actually realised. It is necessary to note, that when effluent concentra- tion of activated sludge X o = 0, the dependence (12) has only single extremum in point Lm~" = x/KLK~ (Spicer, 1955). To the right of this point stationary states of the system are unstabilized, so in this field of L e meanings, complete washing away of the activated sludge culture out of the aeration tank takes place. When X o is different from zero, attempts to obtain analytical expressions for the extreme points Lmi ~ and Lm, ` fails. In this case analysis of aeration tank work in view of stability can be checked easily with the help of function diagram T~,,(Le). Let us consider, for example, the calculation of the aeration tank under following constants and para- meters meanings: /~r~ = 0.1 h - t ; Y = 0.6g of sludge/g BeD; K~. = 0.07 gl - t BeD; K~ = 0.1 gl - t BeD; L 0 = 0.6g 1-1BOD;L ,=0.01g l - IBOD;X o =2g1-1 . According to (7.2) formula, concentration wdl be L m = 0.08 g l - t BeD. I.e. in this case, when L~ ,~ L ,~ L o and the many steps scheme of the aeration tank with the pollutant concentration in first stage L t = L= must be optimal. Let's investigate now the first stage work stability of the aeration tank in the Le = L m = 0.08gl- point, making a diagram of T,.(L ) function. Accord- ing to the diagram (see Fig. 5, solid line) this point generally is in the stabilized segment of the curve. However, if mixed liquor retention time or concen- tration L o diverge from the average level only at 5 ~, what takes place rather often in actual conditions, influent pollutant concentration in the first step increases sharply, according to the level L e = 0,314 g 1- t BeD (AT) and L = 0,36 g l - 1 BeD (ALe). This situation is illustrated with the dotted lines in Fig. 5. In order to guarantee the stabilized work of the aeration tank first stage, it is necessary that the mixed liquor retention time in it will be no less than T m (Lmax) value, moreover T~M (Lm,~) value ought to be determined according to the diagram of T~m(L~) curve, calculated not for the average meaning of concentration Lo, but for the maximum possible Lmax" If we assume that amplitude of concentration oscillation L o does not exceed 10~, then, according to the graphics shown in Fig. 5 with the dotted lines, we shall get T 1 = 4, 16h, L 1 = 0.05 gl-1 BeD for the first step, For the calculations of the next steps one can use (8), (9) formulas, substituting, accordingly, meanings L~ and Xt = X o + (L o - Lt), instead of concentra- tions L 0 and X o and diminishing n value on 1 as well. This procedure is quite admissible, because while substrate concentrations are low, the inhibiting effect and sludge increase maybe neglected. For example, for the three steps scheme of the aeration tank under constants and parameters given Mathematical models of biological waste treatment processes for the design of aeration tanks 497 "1 _ .m 4 Tc~(Lm) 3 - L o • 0 .6g / t BOD . . . . Lo= 0 .63g/ t BOD 'Lo= 0 .66g/L BOD L. . . . . . . . . . . . . j . - • / . j ~-\ \"~''-"'" ~" ~- I'" "-'-- ---- . " 4 /(t "0 .07 0/L BOO ' ]~ =O. Ig / t BOD J '1 ,to =2g/ t ss I i I I 1 J I , I J L L m 0.1 0.2 0.3 0.4 L~ O/L BOD Fig. 5. For calculation of many step aeration tank for toxic waste treatment. Explanations are in the text. above, we shall get T~ ) = 4.16 h, Y L -- 0.29 h, = 0.25 h, T TM = T~ 3) + T~ 3) + T(33) = 4.7 h. In order to get the same treatment effect in com- plete mixing tank, the mixed liquor retention time must be equal to 12.2 h. Along with the high oxidation rate in the aeration tank-sedimentation tank system, good sedimenta- tion properties of the activated sludge should be provided. Hence in designing an aeration tank, such an internal characteristic is prcselccted as sludge age, which affects the constants of the formal models of the Monod's model type (Stens¢l & Shell, 1974). The sludge age influences the best relationship between the volumes of the aeration tank and sedimentation tank (Vavilin & Vasilyev, 1977a). A FORMAL MODEL OF B IOLOGICAL WASTE TREATMENT TAK ING ACCOUNT OF SORPT ION Models of the Monod's model type do not take account of the fact that sorption occurs in the acti- vated sludge system simultaneously with bio-oxida- tiom and the rate of sewage organic matter sorption may sometimes materially surpass that of bio-oxida- tion. In consequence, on the surface of the activated sludge floes some surplus food forms--the storage substrate. The storage substrate forms both on the surface of the floes (adsorption) and within the floes (absorption). We advanced (Vavilin & Vasilyev, 1977b) a new formal model of biological treatment, which takes account of sorption and can be applied in designing the contact stabilization system (Fig. 6). The stabili- zation tank is actually the first stop of the aeration tank, which does not receive sewage flow and where the substratum sorbed by sludge is after-oxidized. The new model is written as the following set of differential equations: dL d--[ = - ~KL(X*" - X') dX~- = KL(X" - X') - & X ~ dt Y dg ac dt = #MX'C (13) where L is the pollutant content of the solution, X~ is the sorbed pollutant concentration (storage sub- strat¢), X =c is the total suspended solids concentration without storage substrat¢, X =" is the equilibrium storage substrate concentration, K is the sorption constant, and j8 is the g:g BOD ratio. Set (13) assumes that the bio-oxidation rate, #,/Y, is maximal for X = > 0 (flocculas of activated sludge hold some sorbed substratum). 498 V.A. VAV=LIN and V. B. VASILYEV Sedimentation L, Lo,~ol-7,~ L,,~,\ \ / ¢-.j qll+r} x~ ° Aerahon ~ 1 q(I-w) StaOili\zotion q(r+w} L,,),,~Le,;%X, I I x, q, ~ qw Fig. 6. Contact stabilization system. L~ The quantity of storage substrate, which can be taken by the mass unit of the activated sludge is not a constant value in general and depends mainly on activated sludge characteristics (waste composition and concentration). However, in the first approaching X ~' ~ X*' can be taken. In this case, while analysis system (13) it is convenient to use the new sizeless variable 2 = X~/X ~. It is easy to show that now system (I 3) will transform dL - f lKLX=¢(2 * - 2) dt where dX ac dt =/~'X~¢ (14) d2 dT = KL(2* - 4) - #, , /Y - It"J. X 5. 4* = - - = const. X as Determining unambiguously the constants of the model requires a series of experimental dependences, X'( t ) and L(t), obtained for the different initial values of X o and L o. The constants ktr, and Y appear to have the same limits as the respective Monod's constants. The constant 4" is known (Blackwell, 1971) to reach unity in some cases. To analyze the model, we selected the following values: It,. = 0.06 h - t , K = 12 1 g- BOD/h, 4" = 0.4, Y = 0.6 g of sludge/g BOD and = 1 g BOD/g of sludge. Model (14) is in good agree- ment with experimental results (Weston, 1961) as to the kinetics of consumption of the pollutant by activated sludge, taking into account that the sewage contains poorly oxidized substances (L D = 0.01 g1-1 BOD) and assuming that L = L ~ + L(t). In the subsequent text, for simplicity, we omit the superscripts of the variable X, i.e. we assume that X ~ is X . Model (14) can be used to compare the biological treatment system without a stabilization tank with that separately stabilizing the activated sludge. We determine the efficiency of a treatment system as a function of the outlet pollutant concentration. By efficiency it is meant the ratio of the total volume of the system, V, to the sewage flow, q. Thus, for the aeration tank V/q = (l + r) T (L ) (15.1) and for the contact stabilization system V/q = (1 + r )T (L ) + rT~(Le) (15.2) where T¢ is the contact time, T is the stabilization time, and r is the return activated sludge coefficient. We assume that L < L I- Let the aeration tank and stabilization tank be complete mixing tanks. Then the steady process of biological treatment is described by a set of equations of material balance (Fan et al., 1970). According to model (l 4), the set of equations for the flow diagram in Fig. 1 is as follows: L I - L - T , , f lKL X (2* - ). ) =0 l+r rX r (1 + r)(1 + 2) X + T I t , X =0 (16) KLe(2* - ).) - It, 2 - It, ./Y = O. The last equation in the system takes a simple form, because without a stabilization tank 2 o = 2 e. The fulfilment of this condition means that the oxidation in the secondary clarifier is slowed down. Because X is the total concentration of the sorbed and activated part of the sludge supplied to the recirculation system from the secondaD clarifier, we have for the outlet to the aeration tank that rX r (1 + r )X o = 1 + 2" For 4 < 0 model (14) ceases to work because the bio-oxidation starts to be entirely limited by diffusion of the pollutant to the flocculas of activated sludge, which takes place when L ~ L* = I t , /YK2* . In this case, the following set of equations is valid: L I - L - ~ , , f lK2*X L , =0 l+r (17) rX,. X o+ T YK2*X L e =0. l+r The set of equations for the flow diagram in Fig. 6 is as follows: L~ _ L - T~mf lKL~X, (2* - 2 )=0 l+r r_X.__ _ X + T~,. i t , ,X = 0 l+r -2 + T~,.[K(2* - 2e)L e - I tm/r - M,,2o] = 0 X = Xr(l + Y2y(l + ).), T~,. = V) , /p . (18) Mathematical models of biological waste treatment processes for the design of aeration tanks 499 The equation for X s takes into account that there is an increment of the activated sludge mass in the stabilization tank. L e being small, there mainly occurs oxidation of the sorbcd pollutant in the stabilization tank until 4 = 4 s = 0. Hence in calculating the stabilization time, the following equation of material balance of the variable was considered valid: 4 - ~,, Y = 0. The sets of equations (16) to (18) permit determining the efficiency of treatment depending on L~ by formulas (15.1) and (15.2). Let the aeration tank and stabilization tank be ideal plug-flow tanks, then the steady biological treatment is described by the relevant set of differ- ential equations: dX d---r = TPflz'~X dL = - --Trl3KXL(4* - 4 ) d - - ; dX d-~-- = Tp/[KL(4* - 2) - /~ .2 - ~.lr] (19) where z is the dimensionless distance from the inlet, with z = 0 on the inlet of the aeration tank (X(0) = X 0, L(0) = L 0, 4(0) = 40) and z = 1 on the outlet of the aeration tank (X(1) = X=, L(I) = L,, and 2(1) = 4). The following conditions should be added to (19): aeration tank without a stabilization tank: 2(0) = 4(1) = 4~, X(0) = rX (1 + r)(l + 4 3 1 L(0) = =~" (20.1) l+r aeration tank with a stabilization tank : rX( l + Y4) 4(0) = 0, 40) = 4,, x (0) = (1 + r)( l + 4j' L I (20.2) L(0) = 1 + r" In describing the processes in the stabilization tank for small Le, the equation d4/dt = - #, JY is suffi- cient. Hence fo d2 Y4 7" ; , = = (21) According to (20.1) and (20.2), the initial condition for the inlet of the aeration tank depgnds on the outlet value of the variable 4, but is not related to X and L, on the exit of the aeration tank. Hence, with the present formulation of the task, the following can be done in particular calculations: adopt some fixed J~ - 2 o g 3 - - i 3 i 4 L o 0.025 0.05 E f f luent organic mat ter concent ra t ion , g/t BaD Fig. 7. Dependence of treatment efficiency on the pollutant content ofeffiuent water, L,: (1) plug-flow aeration tank with- out stabilization tank: (2) plug-flow aeration tank with stabilization tank: (3) mixing aeration tank without stabili- zation tank ; (4) mixing aeration tank with stabilization tank L r = 0.4 g I- l BOD. value of 4=4 from the range 0 0 corresponds to unstable steady states of the system. To treat the sewage thoroughly (L, = 0.005 g 1-t BOD) in a plug-flow tank, it is necessary that V/q > (V/q)m x (see Fig. 7, curve 1). For a certain value of V/q there are three possible steady states, the inter- mediate of them being unstable. In fact, there are always fluctuations of the concen- tration and amount of pollutant in the sewage under treatment. Because (V/q),,,~, is a function of Lo, the 500 V.A. VAVXtaN and V. B. VASILYEV stable work of the aeration tank without a stabiliza- tion tank requires that for every possible change of q and L o For the contact stabilization system the ratio falls rapidly with the increasing L,. Hence the fluctilations of L~. and q will not materially affect the quality of the effluent water. Moreover, there is a considerable gain in treatment efficiency, which is the more, the higher pollutant concentration. For example, in treating the sewage to 0.01 g 1-1 BOD, a plug-flow aeration tank with a stabilization tank will save 43 % of plant volume (L~. = 0.4 g l - 1 BOD). =- E :7 - o w c o 0 Effluent organic matter concentration, L o 0.025 0.05 g/ t BOD Fig. 8. Dependence of T" and T" in the contact stabilization system on the pollutant content of effluent water, L e. The dependence of the contact time in an aeration tank (T~) and the stabilization time (T,) on the pollut- ant content of the effluent water is given in Fig. 8. The figure shows that T+/T~ and, accordingly, V/V increase with the pollutant content of the effluent water. In practice, T~ is selected greater than T~ (Eckenfelder & O'Connor, 1961). Indeed, for a plug- flow tank, this occurs near L~ = L ~, i.e. is valid including the low pollutant content of the effluent water. In a mixing tank, however, the stabilization time is greater than the contact time for relatively high L e. CONCLUSION The above formal models can be further elaborated concerning a more sophisticated mechanism of bio- chemical oxidation of an organic pollutant by a microbial aggregate (Matson & Characklis, 1976), the formation of activated sludge flocculas from dispersed particles (Parker et aL, 1972), the more complete trophic chain substratum-bacteria-protozoa (Curds, 1971), and other aspects. Pilot plants of biological treatment make it possible to decide whether or not a model can be applied to the particular problem by studying possible operation conditions. REFERENCES Andrews J. F. (1968) A mathematical model for the con- tinuous culture of microorganisms utilizing inhibitory substrates. Biotechn. Bioengng. 10, 707. Benedek P. & Horvath I. (1967) A practical approach to activated sludge kinetics. Water Res. l, 663. Bishoff K. B. (1966) Optimal continuous fermentation reactor design. Can. ,L Chem~ Engn. 44, 281. Blackwell L. G. (1971) A theoretical and experimental evaluation of the transient response of the activated sludge process. Ph.D. thesis, Clemson Univ., Clemson, S.C. Curds C. R. (1971) A computer simulation study of predator- prey relationships in single stage continuous culture system. Water Res. 5, 1049. Dick R. I. & Young K. W. (1972) Analysis of thickening performance of final settling tanks. Prec. 27-th Ind. Waste Conf., Purdue. part 1, 33. Eckenfelder W. W. & O'Connor D. J. (1961) Biological Waste treatment. Pergamon Press, Oxford. Erickson L. E. & Fan L. T. (1968) Optimization of the hydraulic regime of activated sludge systems. J. War. Pollut. Control Fed. 40, 345. Fan L. T. et al. (1970) Effects of axial dispersion on the optimal design of the activated sludge process. Water Res. 4, 271. Garrett M. T. & Sawyer C. N. (1951) Kinetics of removal of soluble BeD by activated sludge process. Water Res. 4, 271. Gaudy A. F. et al. (1967) Kinetic behavior of heterogeneous populations in completely mixing reactor. Biotechn. Bioengng 9, 387. Grieves R. B. et al. (1964) A mixing model for activated sludge..L Wat. Pollut. Control Fed. 36, 619. Gyunter L. 1., Zezyulin D. M., Lensky B. P. & Yudina L. F. (1967) Treatment of metropolitan sewage waters in mixing aeration tanks. Vodosnabzhenie i santekhnika, No. 8, 3 (.in Russian). Haldane T. B. S. (1930) Enzymes, Longmans, London. Herbert D. (1961) A theoretical analysis of continuous culture systems. In Monograph. Symposium on Contin- uous Cultivation of Microorganisms, Sc.l, No. 12, 21. Ierusalimsky N. D., & Neronova N. M. (1965). A quantitative relationship between the concentration of metabolic products and the growth rate of microorganisms. Dok- lady AN SSSR, t, 161, No. 6, 1437 (in Russian). Khudenko B. M. & Shpirt E. A. (1973) Aerators for sewage water treatment (Aeratory dlya ochistki stocknykh vod). Moscow, Stroisdat. (in Russian). Khudenko B. M. & Kanenko G. K. (1974) Design and engineering of an aeration tank. Vodosnabzhenie i santckhnika, No. 11, 14. (in Russian/. Lee C. R. & Takamatsu T. (1976) Solution to the boundary- value problems on non-uniform axial mixing in tappered aeration. Water Res. 10, 861. Matson J. V. & Characklis W. G. (1976) Diffusion into microbial aggregates. Water Res. 10, 887. Milbury W. F. et al. (1965) Compartmentalization of aera- tion tanks../. Sanit. Engng Div. ASCE. 91, 45. Monad J. (1950) The technique of continuous culture theory and applications. Ann. Inst. Pasteur. 79, 390. Parker et al. (1972) Floc breakup in turbulent flocculation process. J. Sanit. Enflng Div. ASCE, 98, SA 1, 79. Mathematical models of biological waste treatment processes for the design of aeration tanks 501 Srnivasaraghavan R. & Gaudy A. F. (1974) Operational performance of activated sludge process with constant sludge feedback. Prec. 29-th Ind. Waste Conf., Purdue, part 2, 753. Stack V. T. & Conway R. A. (1959) Design data for com- pletely mixed activated sludge treatment. Sewage Ind. Waste 31, 1181. Stensel H. D. & Shell G. U (1974) Two methods of biological treatment design. J. Wat. Pollut. Control Fed. 46, 271. Spicer C. (1955) The theory of bacterial constant growth apparatus. Biometrics, 11,225. Vavilin V. A. & Vasilyev V. B. (1976) Design of the aeration tank-sedimentation tank system. Vodnye resursy, No. 2, 175 (in Russian). Vavilin V. A. & Vasilyev V. B. (1977a) Design technique for the aeration tank-sedimentation tank system taking account of the activated sludge age. Vodnye resursy, No. 1,166 (in Russian). Vavilin V. A. & Vasilyev V. B. (1977b) A mathematical model of biological treatment on the floccules of activated sludge. Doklady AN SSSR, t.233, No 5, 922 (in Russian). Weston R. F. (1961) Design of sludge reaction activated sludge systems. J. Wat. Pollut. Control Fed., 33, 748. Zaprudsky B. S. & Gyunter L. I. (1973) Optimal design of manystep aeration tanks. Mikrobiologicheskaya promy- shlennonst', No. 3, 14 (in Russian).