5-Civil - IJCSEIERD - MATHEMATICAL - Sohail Ayub - Paid

International Journal of Civil, Structural, Environmental and Infrastructure Engineering Research and Development (IJCSEIERD) ISSN 2249-6866 Vol.2, Issue 3, Sep 2012 52-65 Š TJPRC Pvt. Ltd.,

MATHEMATICAL MODELLING IN ANAEROBIC PROCESS 1

PUSHPENDRA KUMAR SHARMA, 2NASIM AHMAD KHAN & 3SOHAIL AYUB 1

Assistant Professor, Deptt. of Env.Engg.,HCST, Mathura,UP, India 2

3

Ex-Professor, Deptt. of Civil Engg., AMU, Aligarh, UP, India

Associate Professor, Deptt. of Civil Engg., AMU, Aligarh, UP, India

ABSTRACT Dynamic models are used to describe the operation of biological wastewater treatment process and to establish the most effective control strategy. These models still require further development since most biological processes are not yet fully understood with respect to interaction of process variables.

The optimum performance of any biological wastewater treatment process is to produce a specific effluent with a minimum consumption of energy, materials, etc. therefore the advantage of mathematical modeling of anaerobic process for wastewater treatment stems from the improvement in process performance and potential savings in construction and operating costs. An attempt has been made for the formulation of models for anaerobic process in general and review of various other models available for UASB process.

KEYWORDS: Mathematical Modelling, Anaerobic Process,Waste Water Treatment.

INTRODUCTION Biological processes for wastewater treatment have been used for many decades yet there is a continuous need for a better understanding of their mechanisms. In this respect mathematical modeling of biological wastewater treatment offers a number of potential benefits. An important potential of modeling is related to the design aspects of the process Steady-state models are normally used for the design of various biological processes and they can also be used for prediction of their performance.

Both aerobic and anaerobic biological processes have been modeled but in this model our discussion will be limited to only anaerobic one. As any biological system, a generalized anaerobic system can be visualized as a physico-chemical system interacting with a biochemical or biological one (Fig 1). The physico-chemical system consists of the phases: the gas, the liquid and the biologically inert fraction of solids. The biochemical system is made of the microbial cells and of the related exoenzymes which act as a mass and energy transfer unit.

53

Mathematical Modelling in Anaerobic Process

GAS PHASE

Biochem system

LIQUID

INERT SOLID PHASE

ORGANICS

SUBSTANCE AS SUSPNDED SOLID

INORGNICS PRECIPITATES CARRIERS AND MEDIA

Fig.1 schematic representation of the interactions between the physico-chemical and the biochemical components in anaerobic system.

FORMULATION OF MODELS

Kinetics of the anaerobic digestion process is not a particular one and the universal model for biological treatment process can basically be applied. Anaerobic digestion has two phases, acidogenic and methanogenic phase as stated before, but common type of models is employed to describe both of them. Material balance and rate equation of reaction are considered to formulate kinetic equations.

MATERIAL BALANCE

Fig 2 shows the schematic diagram of a conventional anaerobic digester which is mixed completely and fed continuously without solid recycle. Material balance for the substrate gives

ds  (Q ( S1 − S 2 )  = dt  V − ry 

(1)

Gas

54

Pushpendra Kumar Sharma, Nasim Ahmad Khan & Sohail Ayub

Q S1 V

Q S2 Q= V/ſ ſ =V/Q = 1/D Sx

Fig: 2 Schematic Digram Of A Completely Mixed Anaerobic Digester Where, S

=

Substrate concentration in the reactor,

Q

=

Flow rate.

S1

=

Influent substrate concentration,

S2

=

Effluent substrate concentration (S2-S) ,

V

=

Reactor Volume.

rv

=

Substrate utilization rate per unit volume, and

t

=

time.

As only the steady state conditions is considered here,

ds =0 dt Therefore,

ry = ( S1 − S 2 ) D

(2)

Where, D=

Q/V = dilution rate (reciprocal of hydraulic retention time, θ).

Specific substrate utilization rate, rx, is defined as follows: rx

=

rv/X

(3)

55

Mathematical Modelling in Anaerobic Process

Where, X

=

biomass concentration in the reactor.

Eqn. (2) is written as rx= (S1-S2)D/X

(4)

Material balance for biomass gives the following equation: dX dt

=

-QX/V + µ

X

(5)

Where, µ = specific biomass growth rate.

Biomass in the influent is ignored here. Under the steady state conditions, eqn. (5) yields, µ

=D=

1/θ

(6)

The growth of micro-organisms is proportional to the substrate utilization, and biomass decay or endogenous metabolism should also be taken into account to describe the specific growth rate as follows: µ

= YG. rx - b

(7)

Where, YG b

= growth yield coefficient , and = specific biomass decay coefficient.

Combining eqn. (6) and (7) we get,

D = YG rx − b or

rx =

D b + YG YG

(8)

(9)

These coefficients can be evaluated by making laboratory –scale experiments. Chemo stat type reactor is usually operated under various retention times, in other words various dilution rates, Influent substitute concentrations, S1, is controlled and effluent substrate concentration, S2, and biomass concentration, X, are measured. Specific substrate utilization rates are calculated according to Eqn. (4) and plotted against dilution rate, D. The slope and intercept of the regression line for these plots give YG and b according to eqn. (9).

Pushpendra Kumar Sharma, Nasim Ahmad Khan & Sohail Ayub

56

RATE EQUATION The substrate utilization rate depends on the substrate concentration and several types of equations have been proposed to describe the relationship. The most popular is the Monod-type equation which is written as follows: rx = S/(Ks + S).rx, max

(10)

Where, rx, max = maximum specific substrate utilization rate, Ks = Saturation constant for substrate, and S = Substrate concentration in the reactor (S1- S2) The value of rx is half of rx, max when substrate concentration is Ks. This equation is quite similar to the equation proposed by Monod, which describes the relationship between the growth rate and the substrate concentration. One of the following three types of linear plot can be used to evaluate rx, max and Ks using experimental data of rx and S.

1 Ks 1 1 = + rx rx ,max S rx ,max rx = rx ,max − Ks

(11)

rx S

(12)

S S Ks = + rx rx ,max rx ,max

(13)

These are called Liveweaver-Burk plot, Eadie plot and Hofstee plot, respectively. Maximum specific biomass growth rate, µ µ

max =

YG rx,max – b

max is

obtained by substituting rx,max for rx in eqn. (7) as follows.

(14)

A critical retention time, Өc, is defined as the retention time below which substrate utilization does not occur because of the Washout of the micro-organisms. By using Eqn. (6), θc can be represented as follows:

Qc =

1

µ max

(15)

Using equations (4), (8) and (9), the substrate and the biomass concentrations in the effluent under a given dilution rate are expressed as follows:

57

Mathematical Modelling in Anaerobic Process

S2 =

Ks ( D + b) YG rx ,max − ( D + b)

(16)

X=

YG ( S1 − S 2 ) D+b

(17)

Graef and Andrews (1974) developed a dynamic model of the biochemical process linked to the physico-chemical one. This model involves intricate interplays between physical, chemical and biological factors. It therefore exemplifies a synthesis of classical engineering skills with basic biological knowledge and it has been used as a basis for the majority of anaerobic mathematical modeling studies published in the last decade. The key factors of the above dynamic model are: 1.

An inhibition function to relate volatile acids concentration and specific growth rate for methane (Acetoclassic) bacteria.

2.

Consideration of the unionized (acetic) acid as the growth limiting substrate and inhabiting agent.

3.

Consideration of a toxicity effect which increases the decay coefficient Kd proportionally to the concentration of the toxicant.

4.

Consideration of the interactions which occur in and between the liquid, gas and biological phases of the digester.

The model is related to the anaerobic degradation of sewage sludge but as hydrolysis and

acidification were assumed not to be rate-limiting, a microbial population only is considered i.e. the methanogenic acetoclasts which produce most of the methane in conventional digesters.

OTHER MODELS FOR UASB REACTOR Of the various high-rate anaerobic system developed so far UASB process is being widely used for the treatment of a number of industrial wastewaters. A review of literature reveals scanty information available regarding the modeling aspect of the process. Numerous models can be constructed for one reactor from which a few can be chosen on the basis of physical reality.

Heertjes and Van der Meer (1978) investigated the dynamic of liquid flow in an up-flow reactor used for anaerobic treatment of wastewater. The major points derived from this study were: 1.

In the up-flow 30 m3 volume reactor that was studied, 800-2000 mg/1. TOC wastewater, at organic loadings of about 5.5 kg TOC/m3 day and hydraulic loadings of 6-8 m3/m3, can be treated by anaerobic bacteria with efficiencies of about 95%

2.

In this process, for a sludge bed height of 2-3m, about 6m3/h. or 30m3/m2 day or 5m3/m3 day of gas (85% methane is produced).

Pushpendra Kumar Sharma, Nasim Ahmad Khan & Sohail Ayub

58

3.

Mechanical stirring is unnecessary because the gas produced causes sufficient mixing.

4.

It could be deduced that the reactor consists dynamically of three parts: the sludge bed sludge blanket and the settler.

5.

From the stimulus response experiments, performed under different conditions it was found that the sludge bed could be described as a combination of two ideal mixers. The sludge blanket could be described as one ideal mixer and the settler as plug flow region. Bypassing streams from the inlet to the blanket and in cases to the second ideal mixer of the sludge bed, plus return flows the sludge blanket to the second and first ideal mixer of the sludge bed complete the description.

6.

An increase of the gas production from 2.7 to 4.5 m3/h. caused no significant change in the fluid flow pattern.

7.

An increase from 1.2 to 2.2 m in the sludge bed height resulted in an improved efficiency, but also in an increased bypassing of the sludge bed.

8.

Based on the results of stimulus response experiments with laboratory, pilot and full-scale reactor. Van der Meer(1979) has reported that :

9.

Different flow rates of gas, caused by the different organic loadings and Purification efficiencies did not result in significantly altered fluid flow patterns.

10. A higher bed gives larger bypassing streams.

11. Change of settler did not strongly influence the fluid flow pattern in the reactor.

12. Due to accumulation of sludge in the settlers of both laboratory reactors and full-scale reactors, deal spaces were present. This sludge accumulated in the settler produced almost no gas because of the absence of substrate. This led to the conclusions that non-active sludge blocks the fluid flow. In the full-scale reactor, this accumulation was result of the increased mud concentration in the blanket; in the laboratory reactor the recirculation opening was too small, and therefore easily blocked.

13. Both the laboratory reactor and full-scale reactor contained two ideal mixers, which resulted from the bed heights in both reactors having the same small order of magnitude (0.4m). Pilot plant models contained three perfectly mixed regions as its bed height was larger (1.2 – 2.2 m).

14. Dead space region in the settler of pilot plant did not occur as large opening was provided in the bottom of the settler and sludge concentration in the blanket did not exceed 20 kg SS/m3.

15. High values of bypassing flow in the full-scale reactor (e.g. 86% of the total influent), were due to small height of the sludge bed.

59

Mathematical Modelling in Anaerobic Process

16. Increased height of the sludge bed may result in three effects : (i)

The increased thickening at the bottom.

(ii)

The greater stabilization of the channels used by the bypassing streams and

(iii)

The larger differences between the densities of the influent and the sludge mass.

Each of these effects can cause the larger bypassing flows, (Heertjes, P.M. 1978). Height of the

sludge bed should be restricted to a maximum (2-3m) to have reduced bypassing. Scaling-up of the reactor in the horizontal direction is quite feasible, and not restricted to any maximum, provided that the influent is well distributed over the bottom.

Bhatia et. Al.; 1985, proposed a UASB reactor model for the steady state and transient behavior in microbial methanification. In the proposed model, Carbon flow inside the cell was discussed. Carbon in the substrate is utilized to produce energy (primarily in the form of adenosine triphosphate or ATP) or is used to build the infra structure of the cell for reproduction i.e. growth (fig. 3). Since the methane is known to be a by-product of the energy metabolic pathway, the higher the energy consumption, the higher the production of methane. From fig. 3 it is apparent that the methane production is proportional to ET and anything that increases ET ( i.e. EG or EM) will cause methane production rate to go up. Since most of the Carbon input in to the cell goes in to gas production most of the carbon would be accounted for by ET. When carbon is left unaccounted for can be construed to go in to waste products or new cellmass (this is usually 5-10%).

There is no direct data available in the literature to prove that methane production is either growth associated or non-growth associated. The only certainly that could be gleaned from the literature is that methane is a byproduct of the energy related pathway. The transient data already presented in the literature very strongly suggest that methane is non growth associated. Whether methane is growth or non growth associated, the efficiency of the proposed model is unaffected by this. This is because most of the substrate carbon goes in to gas production (90-95%), making the substrate consumption rate directly proportional to gas production rate and not to a weighted sum of growth rate and product generation rate, i.e. the model presented is generally applicable for both types of assiciatedness.

Bhatia et. Al. (1985), presented the following reaction(consumption) rate equation :

Where,

ri =

Ri Si (1 + K s i Si )[1 + ∑ j = 13 K lij S j ]

ri

= the rate of consumption of the ith substrate.

Ri = the maximum rate of consumption of the ith substrate.

Ksi = the substrate affinity coefficient of the ith substrate.(similar to the

(18)

Pushpendra Kumar Sharma, Nasim Ahmad Khan & Sohail Ayub

60

reciprocal of Ks in Michaelis-Mentin Eqn. Klij= Coefficient of inhibition depleting the inhibitory effect of Jth substrate on the consumption of the ith substrate.

CARBON

CH4

ENERGY (ET)

GROWTH (EG)

MAINTINANCE (EM)

Fig: 3 intercellular carbon flow

It should be noted that the amount of active cell-mass does not appear explicitly in Eqn. (8), but does so implicity in the term Ri : i.e. this term is a composite of the maximum specific consumption rate and the active biomass fraction consuming the ith substrate. Using the schematic diagram of the USAB (fig. 5) a description of the reactor dynamics was given by Bhatia et.al. (1985). They developed equations for the specific case of three acids (butyric, propionic and acetic) consumption in the reactor I as follows :

dS1 120 = Dl ( S10 − S1 ) − ri + r3 dt 88

(19)

dS 2 = Dll ( S 20 − S 2 ) − r2 dt

(20)

dS 3 = Dl ( S30 − S 3 ) − r3 dt

(21)

Where, S10 = the concentration of the ith substrate in the feed.

DL = the dilution rate in the reactor I i.e. the reactor that contains the biomass ; i=1, acetic acid ; i=2 propionic acid and i=3, butyric acid.

61

Mathematical Modelling in Anaerobic Process

In the equation (19) the last term on the right hand side is added because of the stoichiometery; i.e. 2 mole acetic acid, (molecular weight = 60) are produced per mole of butyric acid (molecular weight = 88).

Writing a mass balance for reactor II, we have,

ds′ i =

Du (Si – S′i)

(22)

dt Where, Du = dilution rate in reactor II,

Si = Inlet concentration of the ith substrate in the reactor II ( same as the exit concentration from reactor).

S′i = Exit concentration from the UASB system

EFFLUENT Si

REACTOR II

Si

REACTOR I

FEED S 10

Fig: 4 schematic diagram of reactor model.

Since there is a negligible amount, if any, of biomass present in reactor II, it can be safely assumed that there is no reaction occurring in the upper part. Thus the upper part has construed as a system adding a first-order lag to the transient response of the effluent from the biomass bed in the UASB system, which has no effect on the steady-state response.

62

Pushpendra Kumar Sharma, Nasim Ahmad Khan & Sohail Ayub

Bolle et. Al. (1986a) constructed a simple mathematical model as shown in fig. 5, based on the experience with a model with all dimensions 1/20th of a full-scale 800 m3 UASB plant. The UASB reactor can be divided in three compartments: the sludge bed, the sludge blanket and the internal settler.

In this model the main assumptions were: Both sludge bed and the sludge blanket are behaving like perfectly mixed reactors.

The settler volume can be described as a plug flow reactor.

The following mass balance can be derived from the tracer in the sludge bed and the sludge blanket.

Vbd

Vbd

dcbd = (1 − SFbd ).Qv.Cinf − (1 − SFbd ).QvCbd dt = (1 − SFbd ).Qv.(Cinf − Cbd )

(23)

dcb1 = (1 − SFbd ).Qv.Cbd + ( SFbd − SFb1 ).QvCinf − (1 − SFb1 ).Qv.Cbi dt GAS

(24)

EFFLUENT EFFLUENT

SETTLER Short circuiting Flow over sludge Blanket (SFbt) SLUDGE BLANKET Short circuiting SLUDGE BED Flow over sludge Bed (SFbd) DEAD SPACE

INFLUENT

INFLUENT Fig: 5 schematic representation of upflow reactor

Where , SF = Short circuiting flow as a fraction of the influent flow.

V = Volume of the reactor (m3)

63

Mathematical Modelling in Anaerobic Process

C = Concentration (kg/m3) θc = Volumetric flow (m3/hr) And subscripts, bd= bed, bl =blanket, Inf = influent.

RESULTS AND DISCUSSIONS The quantity of the fluid short-circuiting the sludge bed and sludge blanket is not known and cannot be measured directly, these short- circuiting flows are unknown, adjustable parameters included in the model. It can be concluded directly from the measurements whether it is necessary to include the liquid short-circuiting the sludge blanket as a parameter or not. When there is a dead space in a reactor, it is necessary to take in to account a third unknown, adjustable parameter. After the optimal parameter values have been calculated, the amount of active sludge bed is known and so are the volume of dead space and the short-circuiting flows. Determining the volume of dead space in the reactor in this manner is called the indirect method.

The main difference between the other models and the hydro-dynamical model proposed here in is the absence of back-mixing flows in the model described above. These back-mixing flows are closely related to the mechanism for transport of sludge in the reactor. From the results obtained during the study it could be concluded that the short circuiting flows over the sludge bed (SFbd) increases with increasing super-ficial as velocity. This effect is due to the decreased density of the sludge bed. The short-circuiting flow over the sludge blanket (SFb1), shows a maximum as a function of this could be due to High superficial gas velocities in the sludge blanket break up the shory-circuiting flow over the sludge blanket.

Low superficial gas velocities in the reactor cause the short-circuiting flow over the sludge bed and thus the short-circuiting flow over the sludge blanket, to be small. The results also indicate that the shortcircuiting flow over the sludge blanket decreases with increasing height of the sludge blanket.

The study also reveals that for normal superficial gas velocities (1-1.5 m/h) a height between 3.5 and 4 m is sufficient. This is in contradiction to the results of former investigators who found a minimum in the short-circuiting flow over the sludge bed as a function of the bed height for a height of approximately 2.5 m in the same system.

It was also found that the parameters of the fluid flow model can be expressed as follows:

SFbd = (- 0.25Hbd + 0.95) (0.42VSG+0.44) SFb1 = ( 0.16H2 b1- 1.24 Hb1 + 2.5) (-0.16V2 SG+1.6VSG).SFbd‌ Apart from the volumes of the different flow regions, two parameters are necessary and sufficient to describe the fluid flow in a well functioning UASB reactor i.e. the short-circuiting flow over the sludge bed and the short-circuiting flow over the sludge blanket.

Pushpendra Kumar Sharma, Nasim Ahmad Khan & Sohail Ayub

64

The results indicate that to prevent the UASB reactor from wash out, the settler efficiency has to be at least 99% because of the very slow growth rate of the methane bacteria. When the settler efficiency is over 99% the total sludge content of the reactor increases so the reactor is hardly ever in a steady state.

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