Biotreatment of industrial effluents CHAPTER 4 – mathematical models

Biotreatment of industrial effluents CHAPTER 4 – mathematical models Biotreatment of industrial effluents CHAPTER 4 – mathematical models Biotreatment of industrial effluents CHAPTER 4 – mathematical models Biotreatment of industrial effluents CHAPTER 4 – mathematical models Biotreatment of industrial effluents CHAPTER 4 – mathematical models Biotreatment of industrial effluents CHAPTER 4 – mathematical models

CHAPTER Mathematical Models This chapter describes mathematical models for the design of basic batch and continuous reactors; the design of aerobic activated-sludge process; the mass transfer and diffusion correlations needed for design; diffusion through landfill; and diffusion of airborne pollutants In order to develop a reactor model, the following information is required: Kinetics of reaction (including inhibition kinetics) Mass transfer from gas to liquid Mass transfer from liquid (substrate or nutrient) to surface of the microorganism Heat transfer Mixing and agitation Type of reactor Physical properties of the m e d i u m Physical properties of the microorganism Mode of operation of the reactor Modeling the pollutant transport process requires knowledge of the following subjects: 9 9 Diffusion coefficients of the pollutant in soil, liquid, or atmosphere Medium conditions (temperature, pH, etc.) Flow and turbulence Concentration of the contaminant Physical properties of the m e d i u m and the pollutant Basic Reactor Models The basic design parameters for various types of reactors are as follows; 39 40 Biotreatment of Industrial Effluents Batch Reactor ~ S(t) dS dSo rs t- (4-1) C o n t i n u o u s l y Stirred T a n k Reactor V 0- qo = (So - S) (4-2) rs Monod Chemostat The Monod Chemostat is an extension of the continuously stirred tank reactor (CSTR)model, which considers both substrate utilization and the cell growth The cell balance is -0 X(~t-D)+DXo (4-3) where D - V/qo If the number of cells entering the reactor is approximately 0, then D =, (4-4) D (So - S ) = ~t X / Y (4-5) The substrate balance is given by where kt is the specific growth rate and is given by the Monod equation, ~t = ~tmaxS Ks+S (4-6) This leads to equations for exit substrate and cell concentrations, respectively, S = DKs/~tmax - D X Y(So- maxDKS-D ) (4-7) (4-8) C S T R w i t h Recycle D(1 - f ) = ~t (4-9) Plug Flow Reactor _ _ V _ rls(t) qo dSo dSm -rs (4-10) Mathematical Models 41 Fed Batch R e a c t o r In this mode of operation there is no outflow, but after the initial reactor charge, nutrient(s) addition is intermittent, causing the substrate concentration and reactor volume to vary with time dS dV V - ~ + S d-( = qoSo - rs V (4-111 S e q u e n t i a l Batch R e a c t o r A sequential batch reactor operates in the fed batch mode; both concentration and reactor volume vary with time E x t e n d e d Fed Batch In this method the feed to the reactor is constant, leading to a constant substrate concentration inside the reactor (i.e., dS/dt - 0) D e s i g n of B i o t r i c k l i n g Filter The performance of a biotower (a tall biotrickling filter with well-structured packing that uses a modular plastic media, leading to high porosity) is given by the following correlation: Effluent to influent substrate biological _kH/qn oxygen demand (BOD)(mg/L) = e where Q = hydraulic loading rate, mg/m2.min k = treatability constant, a function of wastewater and m e d i u m characteristics (per minute), = 0.01 to 0.1 (0.06 for modular plastic media) H = bed height, m n = 0.5 for municipal waste and modular plastic media (an empirical constant) For a trickle bed air biofilter, a performance equation similar to the previous one can be written as Effluent to influent volatile o~ exp (-~DAsHKsm/ugSSi) organic compound concentration D = As = H = m= ug = = Si - mass diffusivity in biofilm, m / m i n biofilm surface area per unit volume of packing material, m / m height of the biofilter, m thermodynamic distribution coefficient superficial gas velocity, m / m i n biofilm thickness, m influent concentration, m o l / m 42 Biotreatment of Industrial Effluents FIGURE 4-1 Three-phase system Ks = Michaelis M e n t e n constant, m o l / m and ~ = correlation c o n s t a n t s A detailed m o d e l for the bed can be w r i t t e n from the basic m a s s balance equations Steady state m a s s transfer c o m b i n e d w i t h reactions in the gas, liquid, and b i o m a s s phases are w r i t t e n as follows (see Fig 4-1) T h e r e is no r e a c t i o n in the gas phase Substrate flux resulting from c o n v e c t i o n in the x direction is negligible w h e n c o m p a r e d w i t h diffusion Inside the liquid and the biofilm, liquid transfer in the z direction is neglected dS dg S Vg-~z -Dg-d~x2 dS d2S Vl-~z - Dl-d~x2 - Xl,max =0 S Ks + ~ = d2S d2S S - D b - ~ z2 - D b ~ x -Xb~tmaXKs + ~= for - x < x < (4-12) for < x _< Xl (4-13) f o r x l < x < x2 (4-14) w i t h b o u n d a r y conditions, S(x = 0, z) = So,1 S(x, z = ) = S(x, z = 0) = dSx=x2/dx for < z < L for0 S, X "1Aeration tank tit qr, Xu I Settling tank I So, X~ V s,x Air qu Xw ~, ~:w q~ S~ Xw Underflow Recycle of concentrated sludge Sludge wasted FIGURE 4-2 Activated sludge process Blomass production rate d X - l KsgmS + Sl (4-55) When the biomass amount in the inflow and outflow liquid streams is negligible, then Xo and Xe - O Equation (4-54)becomes grnS Ks + S qwXw +ke VX = (4-56) Mass balance equation for substrate utilization qoSo - V(dS/dt) - (qo - qw)Se + qwSw (4-57) Substrate c o n s u m p t i o n rate dS dt - F mS I X LKs + S Y (4-58) Since degradation is taking place only in the aeration tank, S Se -~ Sw (4-59) Substituting Eqs (4-58) and (4-59) into Eq (4-57) ~tmS Ks + S = qo Y(So - S) VX (4-60) Cell residence time (sludge age) Oc = V X / q w X w 14-61) Mathematical Models 49 or [.tmS = Oc Ks + S (4-62) ke H y d r a u l i c residence t i m e (4-63) = V/qo Combining Eqs (4-56) and (4-60), i.e., cell and substrate utilization balances, (4-64) q w X w + ke - qoY(Sovx- S) Substituting Eqs (4-61) and (4-63) into (4-64) gives an equation for the amount of biomass in the exit stream of the activated sludge plant Y(So - S) Oc X - (4-65) (1 +keOc) O V o l u m e t r i c loading rate is the ratio of the mass of BOD in the influent to the volume of reactor (4-66) VL - qoSo/V Food to m a s s ratio is the ratio of mass of BOD removed to the biomass in the reactor (4-67) F / M = qo(So - S ) / V X If the aeration vessel is a plug flow type (complete mixing in the radial direction and no mixing in the direction of flow), then 0c = ~tm(So- S) (So - S) + (1 - a)Ks In Si/S -ke (4-68) where ~ is the recycle ratio and Si is the substrate concentration after mixing the feed with the recycled sludge Si (So + aS) (1 + a ) (4-69) Ponds and Lagoons In the case of facultative systems, complete mixing is assumed for the liquid portion of the reactor The solids that fall to the bottom are not resuspended; hence the balance considers only the soluble BOD This soluble food is 50 Biotreatment of Industrial Effluents assumed to be distributed uniformly, and if the rate is assumed to be first order, then S So = + k0 where S and (4-70) So = the soluble food entering and leaving the system, respectively k = the first order rate constant = the hydraulic residence time If n reactors are arranged in series then, Sn So where Sn is (1 + (4-71) ke/n) n the concentration of the soluble food leaving reactor n Transport in Soils When liquid organic pollutants are released into the soil, they can become physically bound within the soil phase, as well as at the pore spaces that separate the soil particles from one another A single particle has an intraparticle porosity that characterizes the internal structure of the particle as well as an interparticle porosity that characterizes the packing of the particles An empirical relation that can be used to estimate effective diffusivity (Deft) of liquid through soil bed is (Middleman, 1998) D e ft RK = E2Df ~ + (1 - ~)KpPs (4-72) where RK = retardation of diffusion due to the absorption of the solute on the surface of the particle ps = mass density of the solid phase Kp - equilibrium constant relating the contaminant concentration in the fluid and solid phases Df diffusion coefficient of the contaminant in the fluid phase e = bed porosity The a m o u n t of contaminant remaining in a spherical particle at any time t is obtained by integrating the diffusion equation as follows: M(t)_ ~ Mo - ~ -~ [ (Deff/RK)n2~2t] exp - R2 (4-73) n=l where Mo is the initial a m o u n t of contaminant and R is the radius of the particle Except for early times, the second and subsequent terms in the MathematicalModels 51 infinite series can be neglected, which gives rise to a simple equation M(t) = 6Mo exp [ (Deff/RK)~2t] - R2 (4-74) This equation can also be used to estimate the time required for the concentration of the contaminant to reach half its initial value Municipal landfills require construction of a barrier that separates the contaminant storage region from the ground water supply The barrier could be a polymeric film with low permeability to toxins placed on top of a thick layer of clay Clay has a high resistance to hydraulic flow and hence prevents vertical flow of water due to gravity Nevertheless, diffusion of the toxins does take place across the barrier According to Fick's law, Flux through a polymeric film = (permeability of the film x concentration driving force)/thickness of the film Transport through the clay layer is assumed to be diffusion through a semi-infinite planar region with an initial concentration Co = and a constant concentration Cs at the landfill-clay boundary Then C(x, t ) = e r f c ( Cs B ) (4-751 2~/DABt where erfc is the complementary error function For example, if the thickness B = 10 m and DAB = 3.7x 10 -8 cm2/s, the time required for the lower surface of the liner to reach a value of 1% of the value of the upper surface will be 65,000 years If the liner thickness is m, then the time required becomes 650 years Suppose the landfill lies above an aquifer so that the water carries the pollutants away because of its flow The contaminant gets fully mixed with the aquifer flow Then the mass balance for the toxin will be Landfill flux x area of interface with the aquifer = contaminant concentration in the aquifer x flow rate of the aquifer (Middleman, 1998) Landfill flux at the interface - csC = ( - BY) - n ~ 2oo sin 11=1 where XB - DABt/B2 DAB-~y y=S dc I nn(y/B)nexp(-n2~2XB) (4-76) 52 Biotreatment of Industrial Effluents The boundary conditions for the diffusion are C=0 O 500 m Nighttime, x < 2000 m Nighttime, x > 2000 m a b c d 0.01082 0.04487 0.0049 0.01901 1.78 1.56 1.66 1.46 2.46 2.80 1.28 1.32 -0.56 -0.64 -0.6 -0.68 Mathematical Models MA MB NH Npe NRe Ns~ NT qo Q R R S So T U Umf V VA X Y E ~g ~S ~ma~, Ks 0c 53 molecular weight of solute molecular weight of solvent Ammonical nitrogen removed (kg/day) Peclet number Reynolds number Schmidt number Total nitrogen removed (kg/day) feed flow rate toxic release rate, g/s endogeneous respiration rate (3.9 mg O2/g MLSS/h) Gas constant substrate concentration in the exit stream substrate concentration in the exit stream temperature wind speed, m/s minimum fluidization velocity reactor or aeration tank volume (m3) molar volume of solute cell concentration yield coefficient bed porosity bubble volume to reactor liquid volume Monod specific growth rate viscosity of solvent Monod constants Association factor (1-2.6) residence time sludge age References Bailey, J E., and D F Ollis, 1977 Biochemical Engineering Fundamentals Tokyo: McGrawHill Kogakusha Tokyo de Carvalho G 1991 Trans Inst Chem Eng 69: 63-67 Forster, C., and D Wase 1990 Environmental Biotechnology Chichester, U.K.: Ellis Horwood 1990 Geankoplis, C J 2002 Transport Processes and Unit Operations New Delhi: Prentice Hall of India Kulov, P 1983 Chem Eng Comm., 21: 259-262 Kumar A., 1998 Estimating hazard distances from accidental releases Chem Eng August 121-124 Kumar A., 1999 Estimate dispersion for accidental release in rural areas Chem Eng July 91-94 Middleman, S., 1998 An Introduction to Mass Transfer and Heat Transfer, Principles of Analysis and Design New York: John Wiley Peavey, H S., D R Rowe, and G Tchobanoglous 1985 Environmental Engineering International Ed., New York: McGraw-Hill  ... 9(~7M1/4MB /41 ~T1 /4 ( Sc ~'Re ~'p D4/VDi )1 /4 (4- 37) where the power number Np is Np = P~ (pLN3D 5) (4- 38) 46 Biotreatment of Industrial Effluents (In this equation, the characteristic length of. .. 165 to 70,600 (4- 43) JD 0.25NR 0"31/E NRe = 55 to 1500, Nsc = 165 to 10,690 (4- 44) JD 0 .45 48NR0 "40 69/~ XRe 10tO10,000 (4- 45) where JD - kcNs2cl31v' v' = empty tube velocity of the gas Guedes... Jr-(S o - S)/Ki) (4- 21) 44 Biotreatment of Industrial Effluents Monod Equation There are several forms of modified Monod equation; the basic one is: ~max S ~ t - KsSo + S (4- 23) Rapid growth