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

C H A P T E R 4

Mathematical Models

This chapter describes m a t h e m a t i c a l 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:

9 Kinetics of reaction (including inhibition kinetics)

9 Mass transfer from gas to liquid

9 Mass transfer from liquid (substrate or nutrient) to surface of the micro- organism

9 Heat transfer

9 Mixing and agitation

9 Type of reactor

9 Physical properties of the m e d i u m

9 Physical properties of the microorganism

9 Mode of operation of the reactor

Modeling the pollutant transport process requires knowledge of the following subjects:

9 Diffusion coefficients of the pollutant in soil, liquid, or atmosphere

9 M e d i u m conditions (temperature, pH, etc.)

9 Flow and turbulence

9 Concentration of the c o n t a m i n a n t

9 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 R e a c t o r

t - ~ S(t) dS

d So rs

C o n t i n u o u s l y Stirred T a n k R e a c t o r

(4-1)

V (So - S)

M o n o d C h e m o s t a t

The Monod Chemostat is an extension of the continuously stirred tank reac- tor (CSTR)model, which considers both substrate utilization and the cell growth

The cell balance is

where D - V/qo If the number of cells entering the reactor is approximately

0, then

The substrate balance is given by

where kt is the specific growth rate and is given by the Monod equation,

~tmaxS

K s + S

This leads to equations for exit substrate and cell concentrations, respectively,

C S T R w i t h Recycle

Plug Flow R e a c t o r

_

V _ rls(t) dSm (4-10)

qo dSo - r s

Mathematical Models 41

Fed B a t c h 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 concentra- tion and reactor volume to vary w i t h time

S e q u e n t i a l B a t c h R e a c t o r

A sequential batch reactor operates in the fed batch mode; both concentra- tion and reactor volume vary w i t h time

E x t e n d e d Fed B a t c h

In this m e t h o d 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 F i l t e r

The performance of a biotower (a tall biotrickling filter w i t h 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, m g / m 2 m i n

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 ( - ~ D A s H K s m / u g S S i )

organic compound concentration

D = mass diffusivity in biofilm, m 2 / m i n

As = biofilm surface area per unit volume of packing material, m 2 / m 3

H = height of the biofilter, m

m = t h e r m o d y n a m i c distribution coefficient

ug = superficial gas velocity, m / m i n

= biofilm thickness, m

Si - influent concentration, m o l / m 3

42 Biotreatment of Industrial Effluents

FIGURE 4-1 Three-phase system

Ks = M i c h a e l i s M e n t e n c o n s t a n t , m o l / m 3

and ~ = c o r r e l a t i o n c o n s t a n t s

A d e t a i l e d m o d e l for t h e bed can be w r i t t e n f r o m the basic m a s s b a l a n c e

e q u a t i o n s Steady state m a s s transfer c o m b i n e d w i t h r e a c t i o n s in t h e gas, liquid, and b i o m a s s p h a s e s 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 t h e gas phase S u b s t r a t e flux r e s u l t i n g f r o m c o n v e c t i o n in t h e x

d i r e c t i o n is negligible w h e n c o m p a r e d w i t h diffusion Inside t h e liquid and

t h e biofilm, liquid transfer in t h e z d i r e c t i o n is neglected

Vg-~z -Dg-d~x2 = 0 for - x 3 < x < 0 (4-12)

Vl-~z - Dl-d~x2 - Xl,max Ks + ~ = 0 for 0 < x _< Xl (4-13)

- D b - ~ z 2 - D b ~ x 2 -Xb~tmaXKs + ~ = 0 f o r x l < x < x2 (4-14)

w i t h b o u n d a r y c o n d i t i o n s ,

S(x = 0, z) = So,1 for 0 < z < L S(x, z = 0 ) = 0 f o r 0 < x < x 2 S(x, z = 0) = So, ~ for x3 < x < 0

dSx=x2/dx = 0 for 0 < z < L

1 a n d g r e p r e s e n t liquid and gas, respectively Dg, D1, and D b are t h e diffusion coefficients in t h e gas, liquid, and b i o f i l m phases, respectively

M a t h e m a t i c a l Models 43

Reaction Kinetics

The simple kinetic equation for a single substrate reaction is

where n is the order of the reaction and k is the rate constant, which is a function of temperature and could be of the form

M i c h a e l i s M e n t e n E q u a t i o n

The rate equation for enzyme catalyzed reactions is given by the Michaelis Menten equation

r s = K m + S

The rate equation changes in the presence of inhibitors The most important

of these are as follows

rs Km (1 q- I / K i ) q- S

U n c o m p e t i t i v e i n h i b i t o r

Vma X S

rs = Km q- S (1 q - I / K i ) (4-19)

N o n c o m p e t i t i v e i n h i b i t o r

Vma X S

S u b s t r a t e i n h i b i t i o n ( u n c o m p e t i t i v e )

VmaxS

P r o d u c t i n h i b i t i o n ( u n c o m p e t i t i v e )

VmaxS

rs = (Km q- S)(~ Jr-(S o - S ) / K i )

44 Biotreatment of Industrial Effluents

M o n o d E q u a t i o n

T h e r e are several forms of modified M o n o d equation; the basic one is:

~max S

Rapid growth

~max S

KsSo + Ki + S Teisser model

[-t [-tmax ( 1 - - e -s/Ks) (4-25)

Moser model

1 +KsS -~

account

~,~max S

K I X + S

Substrate and product inhibition models are also possible T h e y are similar to the e n z y m e rate equations If two substrates are rate limiting,

t h e n the M o n o d equation becomes:

~tmaxS1 82

Ks1 + $1 Ks~ + $2 Growth rate equation with maintenance of the cells

~[max S

KG+S

Oxygen Transfer Rates

T h e oxygen transfer rate per u n i t liquid v o l u m e is kLa (C* L - C L ) T h e t e r m

w i t h i n parentheses is the driving force, kL is the mass transfer coefficient, and a is the interfacial area per u n i t liquid volume

Mathematical Models 45

M a s s T r a n s f e r a n d D i f f u s i o n C o e f f i c i e n t s

There are several equations available for estimating the mass transfer and diffusion coefficients

Mass transfer coefficient from fluid to solid or gas to liquid for particles less than 0.60 m m

where kr is in m/s, diameter in m, density in kg/m 3, viscosity in kg/m.s, and g in m/s 2

Gas to liquid mass transfer for bubble swarms when d < 2.5 m m

7~/1/3 ~'/1/3 (4-31) Nsh kLdp/Do2 - 0.3 l~,sc ~'Gr

Gas to liquid mass transfer for bubble swarms when d > 2.5 m m

Nsh krdp/Do2 - 0.3-~,Sc ~'Gr

Mass transfer into a free liquid surface or into a falling film

where ~ is the surface renewal time, which is the stream depth per average velocity Another equation is,

Mass transfer correlation for an agitated aerated vessel under turbulent

12 ~T 1/37~T 3/4 (4-36) Xsh 0 Sc ~'Re

(In this equation, we use the impeller diameter as the characteristic length for NRe)

Mass transfer correlation for agitated vessel under turbulent condi-

Taking the power input by the agitator into consideration

9(~7M1/4MB/41~T1/4 ( )1/4 NSh K L D T / D 0 2 0 Sc ~'Re ~'p D4/VDi (4-37)

46 B i o t r e a t m e n t o f I n d u s t r i a l E f f l u e n t s

(In this equation, the characteristic length of Xsh is the tank diameter and the characteristic length of NRe is the impeller diameter.)

Interfacial area per unit liquid volume (a)

Equations are found in the literature for calculating eg, bubble v o l u m e to reactor v o l u m e (which generally varies between 8 and 30%)

Diffusivities of small solutes in aqueous solutions with molecular weights less than 1000 (Geankoplis, 2003)

9.96 x 10-16T

~tB A Semiempirical equation of Polson for calculating diffusivities with molecular weights above 1000,

9.40 x 10-15T

~tBlW A

Wilke-Chang equation for calculating diffusivities for dilute solutes in liquids

1.173 x 10 -16 ((pMA) 1/2 T

gB v~

Mass transfer correlations for packed beds

J D - 1.09NR2e/3/r

JD 0.25NR 0"31/E

JD 0.4548NR0"4069/~

NRe = 0.0016 to 55, Nsc = 165 to 70,600 (4-43) NRe = 55 to 1500, Nsc = 165 to 10,690 (4-44)

where JD - kcNs2cl31 v'

v' = e m p t y tube velocity of the gas

Guedes de Carvalho et al (1991) give an empirical correlation for mass transfer to or from the spheres in a fluidized bed reactor

N~h [4 + 0.576N~e 0"78 + 1.28N~e + 0.141 (dp/Dr) N~e2] 0"5 (4-46) where N' - Sh v~kr, DT/edp

g~e - ~/~gmf DT/ed p

Mathematical Models 47

Activated Sludge Process

Most of the biomass occurs as flocs or aggregates, and the concentration

of biomass is referred to as mixed liquid suspended solids (MLSS, ~1500

to 3500 rag/L) Sludge loading (kg BOD/mg/day, ~0.4 to 1.2) is the ratio of biodegradable organic material to active biomass (Forster, 1990)

flow rate x BOD

Organic loading rate (kg MLSS.day/OLR, kg BOD) =

BOD (g/L) x flowrate(m3/day) BOD

For high rate, conventional, and extended aeration systems, the OLR varies between 0.5 and 5, 0.25 and 0.45, and 0.05 and 0.2, respectively Sludge age (s)

MLSS (g/L) x aeration tank volume (m 3) sludge wasted (kg/day)+ [flow rate (m3/day) x effective solids (g/L)]

(4-49)

M i n i m u m sludge age compatible with nitrification (t*)

1/[0.18 - 0.15 (7.2 - pH)] e 0"12(T-15) (4-50)

Daily carbonaceous oxygen requirement (kg/day)

Air requirement for extended aeration

BOD + 4.34NH - 2.85NT + (0.024 MLSS)Vr x 1.07 T-2~ (4-52)

Air requirement for extended aeration and burning only carbonaceous material at 20~

The mass balance equation for biomass production in a completely stirred, tank-activated sludge reactor is (see Fig 4-2)"

Biomass in the influent + biomass produced

= biomass in the effluent + sludge wasted

dX

q o X o q- V 77- - (qo - q w ) X e -ff q w X w

48 Biotreatment of Industrial Effluents

Treated effluent CZ> S, X I Settling tank I

"J O

V

t i t s , x

Air

~, ~:w Underflow q~ S~ X w Recycle of concentrated sludge

Sludge wasted

FIGURE 4-2 Activated sludge process

B l o m a s s p r o d u c t i o n rate

When the biomass amount in the inflow and outflow liquid streams is negligible, then Xo and Xe - O Equation (4-54)becomes

Mass balance e q u a t i o n for substrate utilization

Substrate c o n s u m p t i o n rate

dS F mS I X

dt - LKs + S Y (4-58)

Since degradation is taking place only in the aeration tank,

Substituting Eqs (4-58) and (4-59) into Eq (4-57)

~tmS qo Y(So - S)

Cell residence t i m e (sludge age)

Mathematical Models 49

o r

1 [.tmS

Oc Ks + S

H y d r a u l i c r e s i d e n c e t i m e

Combining Eqs (4-56) and (4-60), i.e., cell and substrate utilization balances,

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

(1 +keOc) O

V o l u m e t r i c l o a d i n g rate is the ratio of the mass of BOD in the influent

to the volume of reactor

Food to m a s s ratio is the ratio of mass of BOD removed to the biomass

in the reactor

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 (So - S) + (1 - a)Ks In Si/S

where ~ is the recycle ratio and Si is the substrate concentration after mixing the feed with the recycled sludge

(So + aS)

(1 + a )

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

where S and So = the soluble food entering and leaving the system,

respectively

k = the first order rate constant

0 = the hydraulic residence time

If n reactors are arranged in series then,

So (1 + ke/n) n

where Sn is the concentration of the soluble food leaving reactor n

(4-71)

Transport in Soils

When liquid organic pollutants are released into the soil, they can become physically bound w i t h i n the soil phase, as well as at the pore spaces that sep- arate 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 f t E2Df

RK ~ + (1 - ~)KpPs

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 c o n t a m i n a n t concentration

in the fluid and solid phases

Df diffusion coefficient of the c o n t a m i n a n t in the fluid phase

e = bed porosity

The a m o u n t of c o n t a m i n a n t remaining in a spherical particle at any

t i m e t is obtained by integrating the diffusion equation as follows:

M ( t ) _ 6 ~ 1 [ (Deff/RK)n2~2t]

n = l where Mo is the initial a m o u n t of c o n t a m i n a n t and R is the radius of the particle Except for early times, the second and subsequent terms in the