A complete graphical solution is obtained for the completely mixed biofilm-activated sludge reactor (hybrid reactor). The solution consists of a series of curves deduced from the principal equations of the hybrid system after converting them in dimensionless form. The curves estimate the basic parameters of the hybrid system such as suspended biomass concentra- tion, sludge residence time, wasted mass of sludge, and food to biomass ratio. All of these parameters can be expressed as functions of hydraulic retention time, influent substrate concentration, substrate concentration in the bulk, stagnant liquid layer thickness, and the minimum substrate concentration which can maintain the biofilm growth in addition to the basic kinetics of the activated sludge process in which all these variables are expressed in a dimensionless form. Compared to other solutions of such system these curves are simple, easy to use, and provide an accurate tool for analyzing such system based on fundamental principles. Further, these curves may be used as a quick tool to get the effect of variables change on the other parameters and the whole system. Keywords Activated sludge · Biofilm carrier · Graphical solution · Hybrid reactor · Steady state · Wastewater treatment −1 Abbreviations K Specific decay rate ( T ) −1 −3 a Specific surface area of biofilm reactor ( L ) K Monod half-velocity coefficient ( M L ) s s −1 b Sum of specific decay and shear loss rates ( T ) L Thickness of the stagnant layer (L) −1 * b Specific shear loss rates ( T ) L Dimensionless Thickness of the stagnant D Molecular diffusion coefficient in biofilm layer = L (KX )∕(K D )(D /D ) f f s f f w 2 −1 3 −1 (L T ) Q Flow rate into the reactor = (L T ) 2 −1 D Molecular diffusion coefficient in water ( L T ) Q X Dail y mass of the wasted sludge (M/T) w w u f Ratio between flux into actual and deep biofilm (Q X ) Dimensionless daily mass of the wasted sludge w u −1 −3 F/M Food to the biomass ratio (T ) S Effluent substrate concentration ( M L ) * * F/M Dimensionless f ood to the biomass ratio = (F/M) S Dimensionless effluent substrate (1/K) concentration = S/Ks −2 −1 J Substrate flux into biofilm (ML T ) S Dimensionless minimum substrate concentra- min tion that can sustain biofilm = 1/(YK/b −1) J Dimensionless substrate flux = J/ (K KX D ) s f f −3 S Influent substrate concentration ( M L ) o s K Maximum specific rate of substrate utilization S Dimensionless influent substrate −1 (T ) concentration = S /K o s S Minimum substrate concentration at diffusion −3 layer (M L ) * Moharram Fouad S Dimensionless minimum substrate concentra- m123f12317@yahoo.com tion at diffusion layer = S /K s s Renu Bhargava −3 S Subs trate concentration within biofilm ( M L ) moharramf2001@yahoo.com S Dimensionless substrate concentration within Department of Civil Engineering, University of Mansoura, biofilm = S /K Mansoura, Egypt f s 2 θ Empty retention time in the reactor (T) Department of Civil Engineering, IIT, Roorkee, Roorkee 247667, India Vol.:(0123456789) 1 3 62 Page 2 of 10 Applied Water Science (2018) 8:62 θ Dimensionless empty retention time in the reac- in evaluating this value, which is required in any analytical tor = a θ (KX D )∕(K ) solution (Fouad and Renu 2005c). Only knowledge of hydrau- f f s SRT Sludg e residence time of the whole (T) lic retention time (HRT), influent substrate concentration (S ), ave o SRT Dimensionless sludge residence time of the stagnant liquid layer thickness (L), the minimum substrate con- whole system = (SRT)(k ) centration which can maintain the biofilm growth (S ), and d min V Total reactor volume (L ) the desired value of substrate concentration in the bulk (S) −3 X Biomass concentration (ML ) permit computation of suspended biomass concentration (X) X Biomass concentration = (X/a) (K)∕(K X D ) and food to biomass ratio (F/M) for that system. Further, the s f f −3 X Biofilm microbial density (M L ) additional knowledge of fraction YK/K permits computation f d Y Yield coefficient (M /M ) of sludge residence time (SRT) and wasted mass of sludge x s α Product coefficient in factor f expression (Q X ). Computation the value of J in this solution is not w u β Exponentional coefficient in factor f expression required, but J can be determined easily if it is needed in other applications. Introduction Development of the graphical solution Activated sludge process (ASP) reinforced by biofilm car - The process diagram of aerobic hybrid system is shown in rier to get aerobic hybrid system has become a major concern Fig. 1a while Fig. 1b defines its schematic, which is used for (Gautam et al. 2013). Today this technique can eliminate most the present study. The system is assumed to run under a steady problems of ASP like unavailability of large space require- sate condition with a rate limiting substrate concentration ment to construct new plants or extension of other existing through the reactor and has same kinetics for both cultures plants and complaints such as low resistance, low stability, (suspended and attached). For this system, the graphical solu- and low efficiency (Lessel 1994; Randall 1996; Muller 1998; tion for X, SRT, Q X , and F/M can be deduced. w u Wanner et al. 1998). Despite of widespread use of this system, the design criteria for most of its parameters are still unclear. Biomass concentration The analysis of this system seems very complicated due to difficulty of the biofilm analysis, differentiation between the The substrate balance for the hybrid system can be written as suspended and the attached culture behavior, and complexity follows, of the combined system (Islam et al. 2013; Goswami et al. dS KS 2017). Describing this system using any steady state model = Q(S − S)− aVJ − VX o (1) dt K + S based on Monod kinetics, often yields, a set of algebraic equa- tions not having an explicit solution even for biofilm reactor where dS/dt = the growth rate of substrate concentration in only (Kim and Suidan 1989). At present, hybrid reactors are the bulk, Q = flow rate, a = specific surface area of the bio- normally designed using experimental results for a recom- film, V = reactor volume, K = maximum specific rate of sub- mended ratio of the biofilm carrier based on field experience, strate utilization, and K = Monod half-velocity coefficient. some iteration steps do not include all of its parameters (Lee At steady state dS/dt vanish and the resulting equation after 1992) or the system is simplified as a two separate reactors run rearranging the terms is: in series (Gebara 1999). These methods of application neither (S − S) K + S XK o s give precise results nor a good estimation for system param- = − J (2) a a S eters (Goswami et al. 2017; Sarkar and Mazumder 2017). The present study has focused on deducing an easy graphical solu- where θ = empty retention time in the reactor. tion, which can be used for design and control of this system. Rewriting (2) in dimensionless parameters for S, S , J, and In this technique, the basic parameters of the system could θ yields, be computed from its basic equations (substrate and biomass ∗ ∗ (S − S ) balance). The relationship between variables and dimension- S ∗ o X (3) = − J ∗ ∗ less parameters are drawn in the form of a set of curves. From 1 + S √ √ these curves, the hybrid system can be designed and controlled * * where X = (X/a) (K)∕(K X D ) , J = J/ (K KX D ) , s f f s f f accurately based on fundamental principles and according to * * S = S/K , S = S /K , and θ = a θ (KX D )∕(K ) s o s f f s the kinetics of its cultures (attached and suspended). Further, In which D = molecular diffusion coefficient in biofilm, these curves are easy to use and can be considered a good tool and X = biofilm microbial density. The following solution to approve the performance and the characteristics of such of J has been developed by Suidan et al. (1989). system. In addition these curves has the value of substrate flux into biofilm (J ) in implicit form, so many steps has been saved 1 3 Applied Water Science (2018) 8:62 Page 3 of 10 62 Fig. 1 Aerobic hybrid process in (A) a Basic component of aerobic Bio-film hybrid process. b Schematic of aerobic hybrid process carrier out return waste Settling tank Reactor air Completely (B) mixed Q,X, o S, o (Q -Q ), Xe, S o w Settling Hybrid tank reactor Q X w u Sáez and Rittmann (1988, 1992) presented the following S = dimensionless minimum substrate concentration that min √ set of parametric equations, which give a unique value of J can sustain biofilm, and L = L (KX )∕(K D ) (D /D ) in f s f f w * ∗ for each value of S > S . which D = molecular diffusion coefficient in water. min This model is the most widely accepted model, used for ∗ ∗ ∗ ∗ S = S + J L (4) biofilm application. It is simplified in one-dimension and con- ∗ ∗ sider a single substrate as a limiting factor. J = fJ (5) deep After combining Eqs. (4–6) and (3), X can be expressed as * ∗ ∗ * * a function of (S , S , S , L ,θ ). Figure 2 represents the rela- o min * * * ∗ s ∗ tion between X and ( S −S )/θ for a fixed values of S = 0.10 f = tanh α − 1 (5a) o min S * * ∗ * min with L = 0.0 and L = 1.0 and S = 1.0 with L = 0.0 and min * * L = 1.0. Each chart contains different values of S ranging ∗ from 0.10 to 100.0 and from 1.0 to 100.0 for S = 0.10 and min α= 1.5557 − 0.4110 tanh log S (5b) min S = 1.0 respectively. min β= 0.5035 − 0.0257 tanh log S (5c) min Wasted mass of sludge ∗ ∗ ∗ 0.5 J =[2(S - ln (1 + S ))] (6) deep s s At steady state, biomass balance for hybrid system is: where S = dimensionless minimum substrate concentration dX YKS s s = VX − K + YaVJ − Q X (7) d w u at diffusion layer, f = ratio between flux into actual and deep dt K + S b s t biofilm, α and β are product and exponentional coefficients where dX/dt = changing rate of biomass concentration in the respectively in factor f expression (Eq. 4), and J = dimen- deep reactor, Y = yield coefficient, b = specific shear loss rates, sionless substrate flux into deep biofilm, b = sum of specific decay and shear loss rates, Q X = mass t w u 1 3 62 Page 4 of 10 Applied Water Science (2018) 8:62 4 4 (A) (C) 10 10 X*= (X/a) (K/Ks Xf Df) 1/2 X*= (X/a) (K/Ks Xf Df) 1/2 (HRT) * = a (HRT) (K Xf Df/Ks) ½ (HRT) * = a (HRT) (K Xf Df/Ks) ½ 3 3 S* = S/Ks S* = S/Ks 10 10 So* = So/Ks So* = So/Ks L* = 0.0 L* =1.0 S*min = 0.10 S*min = 0.10 2 2 10 10 S*=0.4 S*=0.4 X* X* 1 1 10 10 S*=0.1 S*=0.1 0.2 0 0 10 10 1.0 2.0 5.0 25.0 10.0 50.0 X*=0.59 0.2 0.5 2.0 10.0 5.0 0.5 1.0 25.0 100.0 50.0100.0 -1 -1 10 10 -1 0 1 2 3 -1 0 1 2 3 0.346 0.346 10 10 10 10 10 10 10 10 10 10 X*=0.0 (So*-S*)/HRT* (So*-S*)/HRT* (B) (D) X*= (X/a) (K/Ks Xf Df) 1/2 (HRT) * = a (HRT) (K Xf Df/Ks) ½ X*= (X/a) (K/Ks Xf Df) 1/2 S* = S/Ks (HRT) * = a (HRT) (K Xf Df/Ks) ½ So* = So/Ks 3 S* = S/Ks L* = 0.0 So* = So/Ks S*min = 1.0 L* =1.0 S*min = 1.0 X* X* S*=1.0 2.0 4.0 10 50 S*=1.0 2.0 4.0 10 25 100 3.0 5.0 -1 5.0 25 3.0 100 -1 0 1 2 3 -1 10 10 10 10 10 10 -1 0 1 2 3 10 10 10 10 10 (So*-S*)/HRT* (So*-S* )/HRT* * * ∗ * ∗ * ∗ * ∗ * ∗ Fig. 2 X as a function of S and ( S −S)/HRT at a L = 0.0, S = 0.10, b L = 0.0, S = 1.0, c L = 1.0, S = 0.10 and d L = 1.0, S = 1.0 o min min min min * * of wasted sludge, and k = specific decay rate. At steady state Substituting X from (3) the final equation for (Q X ) d w u dX/dt will vanish then combining (2) and (7) and substitut- can be obtained as follow. ing b = k + b (Rittmann 1982a) the resulting equation is ∗ ∗ t d s ∗ ∗ (S − S ) YK 1 + S 1 + S 1 o ∗ obtained as: (Q X ) = − + J − − 1 w u ∗ ∗ ∗ K S S S min (11) Q X = YQ(S − S)− VXK − YaVJ (8) w u o d After combining (4–6) and (11) the term (Q X ) can be w u * ∗ ∗ * * expressed as a function of S , S ,S , L , θ , YK/K . Fig- o min By rearranging (8), the following equation can be ures 3 and 4 represent the relation between (Q X ) and ( S w u obtained * * ∗ * * −S )/θ for a specific set of S L , and S which are used min Y(S − S) K K XK YK earlier for computation X , but for two different values of Q X = − − J (9) w u aV k a k a b YK/K , 50 and 500. d d t d YK Making substitution for as (1∕S + 1) (Suidan et al. min b Sludge residence time 1989; Rittmann 1982a) and transferring (9) in dimensionless By definition, SRT for the combined biomass (suspended form yield. and fixed) is given by the following relationship after ∗ ∗ (S − S ) neglecting the biomass in the final effluent. YK 1 ∗ ∗ ∗ (Q X ) = − X − J + 1 (10) w u k S d M + M min Total active biomass sus. bio. SRT = = (12) ave √ (M ) +(M ) Total wasted sludge w sus. w bio. * K where (Q X ) = Q X ∕ (K KX D ). w u w u s f f aV k where M = the active mass of biofilm which is given by bio 1 3 Applied Water Science (2018) 8:62 Page 5 of 10 62 5 5 10 10 (A) (C) L* = 0.0 L* = 1.0 S*min = 0.1 S*min = 0.1 4 4 Y K /Kd = 50.0 Y K /Kd = 50.0 10 10 S* = S/Ks S* = S/Ks So* = So/Ks So* = So/Ks 3 HRT* = a(HRT)(K Xf Df/Ks)½ HRT* = a(HRT)(K Xf Df/Ks)½ (QwXu)*= (Qw Xu)(K/Kd)/[(aV) (QwXu)*= (Qw Xu)(K/Kd)//[(aV) (K Xf Df Ks)½] (K Xf Df Ks)½] (QwXu)* (QwXu)* 1. S*=0.100 2 1. S*=0.100 2 10 10 2. S*=0.200 2. S*=0.200 3. S*=0.500 3. S*=0.500 (QwXu)* =15.0 (QwXu)* =13.0 4. S*=1.000 4. S*=1.000 1 5. S*=2.000 5. S*=2.000 6. S*=5.000 6. S*=5.000 7. S*=10.00 7. S*=10.00 0 0 10 8. S*=25.00 10 8. S*=25.00 2 3 4 5 6 7 8 9 10 9. S*=50.00 9. S*=50.00 3 4 56 7 9 8 10. S*=100.0 10. S*=100.0 -1 -1 -2 -1 0 1 2 3 -2 -1 0 1 2 3 0.346 0.346 10 10 10 10 10 10 10 10 10 10 10 10 (So*-S*)/HRT* (So*-S*)/HRT* (D) (B) L* = 1.0 L* = 0.0 S*min = 1.0 S*min = 1.0 Y K /Kd = 50.0 Y K /Kd = 50.0 S* = S/Ks S* = S/Ks So* = So/Ks So* = So/Ks 3 10 HRT* = a(HRT)(K Xf Df/Ks)½ 10 HRT* = a(HRT)(K Xf Df/Ks)½ (QwXu)*= (Qw Xu)(K/Kd)/(aV) (QwXu)*= (Qw Xu)(K/Kd)/(aV) (QwXu)* (QwXu)* 1. S*=1.00 1. S*=1.00 2. S*=2.000 2. S*=2.000 3. S*=3.000 3. S*=3.000 1 4. S*=4.000 4. S*=4.000 5. S*=5.000 5. S*=5.000 6. S*=10.00 6. S*=10.00 0 7. S*=25.00 7. S*=25.00 1 1 10 8. S*=50.00 8. S*=50.00 2 3 7 9 9. S*=100.0 9. S*=100.0 4 6 5 -1 34 67 8 9 -1 -2 -1 0 1 2 3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 (So*-S*)/HRT* (So*-S*)/HRT* * * * * * ∗ ∗ ∗ ∗ Fig. 3 (Qw Xu) as a function of S and ( S −S)/HRT at Y K/K = 50.0, a L = 0.0, S = 0.10, b L = 0.0, S = 1.0, c L = 1.0, S = 0.10 and d o min min min L = 1.0, S = 1.0 min M = aV L X ∗ ∗ ∗ (13a) bio f f J (1∕S + 1)+ X ∗ min SRT = (16) where L = biofilm thickness which equals (JY)/(b X ) (Ritt- f t f ∗ ∗ ∗ ∗ ∗ YK/K (S − S )∕ − X − J (1∕S + 1) o min mann 1982a; Suidan et al. 1989) then * * where SRT = SRT K . Combining (3) and (16) the SRT M = aV J Y ∕b ave d (14b) bio t. may be written as: Substituting total wasted sludge from (8) yields, ∗ ∗ ∗ ∗ (S −S ) Y 1 1+ S 1+ S ∗ o aJV + XV J + 1 − + ∗ ∗ ∗ S S S min SRT = (14) SRT = ave (17) ∗ ∗ d ∗ ∗ (S −S ) YK 1+ S 1+ S 1 YQ(S − S)− VXK − aVJY ∗ o d − + J − − 1 ∗ ∗ ∗ ∗ K S S S min Multiplying both sides of (14) by K and the right hand side * * After substituting J from (4 to 6), SRT is plotted as a (numerator and denominator) by K/aV yields, * * function of S for a fixed value of S = 0.10 with L = 0.0 min YK XK * ∗ * * J + and L = 1.0 and S = 1.0 with L = 0.0 and L = 1.0 (Figs. 5, min b a * * SRT K = 6). Each chart contains different values of ( S −S )/θ rang- (15) ave d (S −S) o YK XK YK − − J ing from 0.01 to 1000. Also the curves are prepared for two K a a b d t values of YK/Kd (50 and 500). In these curves, it is better ∗ * * * Equation (15) can be expressed in a dimensionless form and easy to fix the values of ( S −S )/θ instead of S as in the as: previous curves (Figs. 3, 4). The use of the figures in design is exemplified in illustrative example at the end of this paper. 1 3 62 Page 6 of 10 Applied Water Science (2018) 8:62 (C) (A) L* = 1.0 L* = 0.0 S*min = 0.1 S*min = 0.1 Y K /Kd = 500.0 Y K /Kd = 500.0 S* = S/Ks S* = S/Ks So* = So/Ks So* = So/Ks 4 HRT* = a(HRT)(K Xf Df/Ks)½ HRT* = a(HRT)(K Xf Df/Ks)½ 10 10 (QwXu)*= (Qw Xu)(K/Kd)/[(aV) (QwXu)*= (Qw Xu)(K/Kd)/[(aV) (K Xf Df Ks)½] (K Xf Df Ks)½] (QwXu)* (QwXu)* 1. S*=1.00 1. S*=1.00 2. S*=2.000 2. S*=2.000 3. S*=3.000 3. S*=3.000 2 2 4. S*=4.000 4. S*=4.000 5. S*=5.000 5. S*=5.000 6. S*=10.00 6. S*=10.00 1 7. S*=25.00 1 1 7. S*=25.00 8. S*=50.00 8. S*=50.00 3 9. S*=100.0 9. S*=100.0 3 10 6 8 9 10 4 5 6 7 8 9 7 -2 -1 0 1 2 3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 (So*-S*)/HRT* (So*-S*)/HRT* (B) (D) L* = 1.0 L* = 0.0 S*min = 1.0 S*min = 1.0 5 5 Y K /Kd = 500.0 Y K /Kd = 500.0 S* = S/Ks S* = S/Ks So* = So/Ks So* = So/Ks 4 HRT* = a(HRT)(K Xf Df/Ks)½ HRT* = a(HRT)(K Xf Df/Ks)½ (QwXu)*= (Qw Xu)(K/Kd)/[(aV) (QwXu)*= (Qw Xu)(K/Kd)/[(aV) (K Xf Df Ks)½] (K Xf Df Ks)½] (QwXu)* (QwXu)* 1. S*=1.00 1. S*=1.00 2. S*=2.000 2. S*=2.000 3. S*=3.000 3. S*=3.000 2 4. S*=4.000 4. S*=4.000 5. S*=5.000 5. S*=5.000 6. S*=10.00 6. S*=10.00 1 7. S*=25.00 1 7. S*=25.00 10 10 8. S*=50.00 8. S*=50.00 9. S*=100.0 9. S*=100.0 6 8 6 9 7 9 7 8 -2 -1 0 1 2 3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 (So*-S*)/HRT* (So*-S*)/HRT* * * * ∗ * ∗ * ∗ Fig. 4 (Qw Xu) as a function of S and ( S −S)/HRT at Y K/K = 500.0, a L = 0.0, S = 0.10, b L = 0.0, S = 1.0, c L = 1.0, S = 0.10 d o d min min min * ∗ L = 1.0, S = 1.0 min * * where F/M = F/M (1/K). Substituting X from (3) yields, Food/biomass ratio ∗ ∗ ∗ (S − S )∕ For the hybrid system F/M can be written as: o F∕M = ∗ ∗ ∗ ∗ ∗ ∗ J ∗(1∕S + 1) + [(S − S )∕ − J ](1∕S + 1) min o Q(S − S) (21) (18) M aVJY/b + VX * * After substituting J from (4 to 6), F/M can be * * * * ∗ ∗ ∗ expressed as a function of (S , S , S , L , and ( S −S )/θ . Multiplying both sides of (18) by 1/K and the right hand o min o Figure 7 represents the relation between F/M as a func- side (numerator and denominator) by 1/aV yields, * * * * ∗ ∗ tion of ( S −S )/θ for the same values of S L , and S as o min (S − S)∕a F 1 in previous curves (5,6) except that these curves are not (19) M K JYK/b + XK/a t dependent on YK/Kd. It must be noted that in all these curves the value of S must be greater than or at least Equation (19) can be converted in a dimensionless form * * ∗ ∗ equals S . When the value of S equals S the J will min min as: be zero and the role of biofilm in the reactor will vanish ∗ ∗ ∗ converting the system to a pure suspended reactor. At that (S − S )∕ ∗ o F∕M = (20) time the value of X, SRT, Q X , and F/M shall not depend w u ∗ ∗ J (1∕S + 1)+ X min on the value of S or L . min 1 3 Applied Water Science (2018) 8:62 Page 7 of 10 62 2 2 10 10 (So*-S*)/HRT* (So*-S*)/HRT* 0.5 =0.01 =0.01 2.0 (A) 0.1 (C) 0.51.0 1 1 10 10.0 10 5.0 20.0 0.1 0.346 1.0 50.0 0 0 0.346 10 10 2.0 5.0 20.0 SRT*= 0.3 100.0 10.0 SRT* SRT* 50.0 SRT*= 0.17 100.0 200.0 -1 -1 200.0 10 10 1000.0 L* = 1.0 L* = 0.0 1000.0 S*min= 0.10 S*min= 0.10 YK/Kd=50.0 YK/Kd=50.0 -2 -2 10 S* = S/Ks 10 S* = S/Ks So* = So/Ks So* = So/Ks SRT* =(SRT)( Kd) SRT* =(SRT)( Kd) HRT* = a(HRT)(K Xf Df/Ks)½ HRT* = a(HRT)(K Xf Df/Ks)½ -3 -3 10 10 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 S*=0.4 S*=0.4 S* S* 2 2 10 10 L* = 1.0 L* = 0.0 (So*-S*)/HRT* (So*-S*)/HRT* S*min= 1.0 S*min= 1.0 (B) =0.01 (D) =0.01 YK/Kd=500.0 YK/Kd=500.0 1 1 10 S* = S/Ks S* = S/Ks 0.1 So* = So/Ks 0.1 So* = So/Ks SRT* =(SRT)( Kd) SRT* =(SRT)( Kd) HRT* = a(HRT) 0 HRT* = a(HRT) 0 0.5 10 (K Xf Df/Ks)½ (K Xf Df/Ks)½ 1.0 SRT* SRT* 1.0 0.5 2.0 -1 -1 5.0 5.0 2.0 10.0 20.0 10.0 20.0 -2 -2 10 50.0 50.0 100.0 100.0 200.0 200.0 1000.0 1000.0 -3 -3 -1 0 1 2 3 4 -1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 S* S* * * ∗ * ∗ * ∗ * ∗ Fig. 5 SRT as a function of S and ( S −S)/HRT at Y K/K = 50.0, a L = 0.0, S = 0.10, b L = 0.0, S = 1.0, c L = 1.0, S = 0.10 and d o min min min * ∗ L = 1.0, S = 1.0 min * * * For the above variables, X = 0.0 at L = 0.0 and X = 0.59 Illustrative example * * at L = 1.0 from Fig. 2a and c respectively giving X = 0.501 at L = . 85 by using linear interpolation. Consider a true specific domestic wastewater having the From the definition of X , X = (0.501) (1.8)/ following kinetics (Heath et al. 1990; Lee 1992). (10)∕[(.01)(25)(.75)] = 0.124 mg/cm = 124.0 mg/L. S = 4 3 0 . 0 m g / L , S = 4 . 0 m g / L , K = . 0 1 m g / c m , o s −1 Further the value of J can deduced easily from (3) to be K = 10/day, Y = 0.45, K = 0.09 day , −1 −1 J = 0.346−(0.501)(0.4)/(1 + 0.4) = 0.203, so J = (0.203) b = 0.41 day , b = 0.32 day , S = 1.0 mg/L, t s min √ −2 −1 − 1 (.01)(10)(25)(.75) = . 278 mg cm day . L = 0 . 0 0 78 c m, H R T = 0 . 5 d a y, a = 1 . 8 c m , 2 −1, 3 3 These results for X and J conform to Lee (1992), He D = 1.25 cm day X = 25 mg/cm , Q = 10,000 m /day, w f 2 −1 obtained the same results by analytical solution after five itera- and D = 0.75 cm day . tions. In additions, These results conform to Fouad and Renu Converting variables to dimensionless form: ∗ * ∗ (2005a, b, 2012). S = 4 30/ 10 = 4 3, S = 4 /1 0 = 0 .4 , S = 1 .0/ 10 = 0 . √ min Further the present technique can be used to get the values 1, L = . 0078 (10)(25)∕[(.01)(.75)] (.75/1.25) = . 85, of SRT, Q X , and F/M easily (as given below) which are not θ = (1.8)(0.5) (10)(25)(.75)∕(.01) = 123.24, w u available before. Repeating the same steps at L = . 85 and ∗ ∗ ∗ (S − S )∕q = (43 − 0.4)∕123.24 = 0.346. * * * YK/K = 50.0, the values of SRT , (Q X ) , and F/M can be d w u 1 3 62 Page 8 of 10 Applied Water Science (2018) 8:62 (C) (So*-S*)/HRT* (So*-S*)/HRT* 0.5 =0.01 =0.01 2.0 (A) 0.1 1.0 10.0 10 5.0 20.0 0.1 0.5 0 1.0 50.0 2.0 5.0 20.0 100.0 10.0 SRT* SRT* 50.0 200.0 100.0 -1 -1 200.0 10 L* = 1.0 1000.0 L* = 0.0 1000.0 S*min= 0.10 S*min= 0.10 YK/Kd=50.0 YK/Kd=50.0 -2 -2 S* = S/Ks S* = S/Ks So* = So/Ks So* = So/Ks SRT* =(SRT)( Kd) SRT* =(SRT)( Kd) HRT* = a(HRT)(K Xf Df/Ks)½ -3 HRT* = a(HRT)(K Xf Df/Ks)½ -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 S* S* (So*-S*)/HRT* (So*-S*)/HRT* 0.1 (B) =0.01 (D) =0.01 0.1 1.0 1.0 5.0 0 5.0 0 2.0 0.5 2.0 0.5 10.0 10.0 SRT* SRT* 20.0 20.0 -1 -1 50.0 50.0 10 100.0 L* = 0.0 100.0 L* = 1.0 200.0 200.0 S*min= 1.0 S*min= 1.0 YK/Kd=50.0 YK/Kd=50.0 1000.0 1000.0 -2 -2 10 S* = S/Ks 10 S* = S/Ks So* = So/Ks So* = So/Ks SRT* =(SRT)( Kd) SRT* =(SRT)( Kd) HRT* = a(HRT)(K Xf Df/Ks)½ HRT* = a(HRT)(K Xf Df/Ks)½ -3 -3 10 10 -1 0 1 2 3 4 -1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 S* S* * * * * * ∗ ∗ ∗ ∗ Fig. 6 SRT as a function of S and ( S −S)/HRT at Y K/K = 50.0, a L = 0.0, S = 0.10, b L = 0.0, S = 1.0, c L = 1.0, S = 0.10 and d o min min min L = 1.0, S = 1.0 min deduced to be 0.189, 14.7, 0.127 from which SRT, Q X , and The solution, which is presented by a series of curves, w u −1 F/M are 2.2 day, 1664.0 kg 1.26 day . can be used to estimate suspended biomass concentration, sludge residence time, wasted mass of sludge, and food to biomass ratio. The applicability of these curves has been Conclusion proven by compared with results of other researchers. The accuracy of this technique may be about 98% because of This paper presents a graphical solution for design and the inaccuracy in reading from the curves and use of the control of the biofilm-activated sludge reactor when it runs linear interpolation to get the final results. Equations 3, 11, under steady state and rate limiting substrate condition. 1 3 Applied Water Science (2018) 8:62 Page 9 of 10 62 0 0 10 10 (A) (C) HRT* = a(HRT)(K Xf Df/Ks)½ HRT* = a(HRT)(K Xf Df/Ks)½ F/M* =(F/M)(1/K) F/M* =(F/M)(1/K) 1000.0 So* = So/Ks 1000.0 So* = So/Ks 200.0 S* = S/Ks S* = S/Ks S*min= 0.10 S*min= 0.10 F/M*=.135 100.0 50.0 -1 L* = 0.0 -1 L* = 1.0 200.0 10 10 100.0 F/M*=.085 20.0 10.0 50.0 F/M* F/M* 20.0 5.0 2.0 10.0 5.0 -2 -2 0.346 1.0 10 10 0.5 0.346 2.0 1.0 0.1 0.5 0.1 (So*-S*)/HRT*=0.01 (So*-S*)/HRT*=0.01 -3 -3 10 10 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 S*=0.4 S*=0.4 S* S* 1000.0 (D) 200.0 100.0 1000.0 (B) Q X r u 200.0 20.0 50.0 50.0 100.0 5.0 10.0 -1 20.0 10.0 -1 1.0 2.0 5.0 F/M* 2.0 F/M* 0.1 1.0 -2 0.5 -2 L* = 1.0 L* = 0.0 S*min= 1.0 0.1 S*min= 1.0 S* = S/Ks S* = S/Ks So* = So/Ks So* = So/Ks (So*-S*)/HRT*=0.01 F/M* =(F/M)(1/K) (So*-S*)/HRT*=0.01 F/M* =(F/M)(1/K) HRT* = a(HRT)(K Xf Df/Ks)½ -3 HRT* = a(HRT)(K Xf Df/Ks)½ -3 -1 0 1 2 3 4 10 10 10 10 10 10 -1 0 1 2 3 4 10 10 10 10 10 10 S* S* * * * * ∗ * ∗ * ∗ * ∗ Fig. 7 F/M as a function of S and (S −S)/HRT at a L = 0.0, S = 0.10, b L = 0.0, S = 1.0, c L = 1.0, S = 0.10 and d L = 1.0, S = 1.0 min min min min Gautam SK, Sharma D, Tripathi JK, Ahirwar S, Singh SK (2013) A 17, and 21 can also be used to determine the solution with study of the effectiveness of sewage treatment plants in Delhi accuracy reach to of 100%. region. Appl Water Sci 3(1):57–65 Gebara F (1999) Activated sludge biofilm wastewater treatment sys- Open Access This article is distributed under the terms of the Creative tem. Water Res 33:230–238 Commons Attribution 4.0 International License (http://creativecom- Goswami S, Sarkar S, Mazumder D (2017) A new approach for mons.org/licenses/by/4.0/), which permits unrestricted use, distribu- development of kinetics of wastewater treatment in aerobic tion, and reproduction in any medium, provided you give appropriate biofilm reactor. Appl Water Sci 7(5):2187–2193 credit to the original author(s) and the source, provide a link to the Heath MS, Wirtel SA, Rittmann BE (1990) Simple design of the Creative Commons license, and indicate if changes were made. biofilm processes using normalized loading curves. JWPCF 62(2):185–192 Islam MA, Amin MSA, Hoinkis J (2013) Optimal design of an activated sludge plant: theoretical analysis. Appl Water Sci References 3(2):375–386 Kim BR, Suidan MT (1989) Approximate algebraic solution for a Fouad M, Bhargava R (2005a) A simplified model for the steady- biofilm model with the Monod kinetic expression. Water Res state biofilm-activated sludge reactor. J Environ Manag 23(12):1491–1498 74(3):245–253 Lee CY (1992) Model for biological reactors having suspended and Fouad M, Bhargava R (2005b) Modified expressions for substrate attached growths. J Environ Eng 118(6):982–987 flux into biofilm. J Environ Eng Sci 4(6):441–449 Lessel TH (1994) Upgrading and nitrification by submerged biofilm Fouad M, Bhargava R (2005c) Mathematical model for the biofilm- reactors experimental from a large scale plant. Water Sci Technol activated sludge reactor. ASCE (EE) 131(4):557–562 29(10–11):167–174 Fouad M, Bhargava R (2012) Sludge age, stability, and safety factor Muller N (1998) Implementing biofilm carriers into the activated for the biofilm-activated sludge process reactor. Water Environ sludge process 15 years of experience. Water Sci Technol Res 84(6):506–513 37(9):167–174 1 3 62 Page 10 of 10 Applied Water Science (2018) 8:62 Randall CW (1996) Full-scale evaluation of an integrated fixed film- Suidan MT, Wang YT, Kim BR (1989) Performance evaluation of bio- media in activated sludge (IFAS) process for nitrogen removal. l fi m reactors using graphical techniques. Water Res 23(7):837–844 Water Sci Technol 33(12):155–162 Wanner J, Kucman K, Grau P (1998) Activated sludge process com- Rittmann BE (1982) Comparative performance of the biofilm reactor bined with biofilm cultivation. Water Res 22(2):207–215 types. Biotechnol Bioeng 24:1341–1370 Sáez PB, Rittmann BE (1988) Improved pseudo-analytical solution for Publisher’s Note Springer Nature remains neutral with regard to steady-state biofilm. Biotechnol Bioeng 3:379–385 jurisdictional claims in published maps and institutional affiliations. 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Applied Water Science – Springer Journals
Published: Apr 20, 2018
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