fermentation in fed-batch reactors—application to the sewage sludge anaerobic digestion

Pergamon Chemical Ea~ineeria~ Scwnce, Vol. 50, No. 13, pp. 2117-2126, 1995 Copyright C.~ 1995 Elsevier Science Ltd

Printed in Great Britain. A l l rights rcacrved 0009-2509/95 $9.50 + 0.00

0009-2sog(gs)00oss-s

FERMENTATION IN FED-BATCH REACTORS--APPLICATION TO THE SEWAGE SLUDGE

ANAEROBIC DIGESTION

R. FONT and J. M. LOPEZ CABANES Departamento de lngenieria Quimica, Universidad de Alicante, Apartado 99, Alicante, Spain

(Received 23 March 1994)

Al~traet--Fed-batch reactors, with cyclical feeds and extractions, are used in fermentations. The behaviour of this type of reactor under quasi-steady state is discussed. With fermentations, the washout conditions are calculated for cases of homogeneous fermentation and for heterogeneous fermentation. In the case of heterogeneous fermentation (fermentation of particulate substrate), the residence time distribution of the solids has been taken into account. Fed-batch reactors can also be used as differential reactors. The performance equations deduced are used for correlating the experimental data obtained in the anaerobic digestion of sewage sludge.

INTRODUCTION

In an ideal fed-batch reactor, a volume is rapidly introduced in the reactor. The mixture is stirred and the reaction takes place for a period of time. After- wards, a volume, equal to that introduced, is extracted from the reactor. The reactant medium is mixed with a new addition of feed in the following cycle with the same reaction time as in the previous one. After sev- eral cycles, and when the reactant concentrations are the same for all the volumes fed and the operating conditions are kept constant, the concentration of the products in the extracted volume can reach constant values. Under these conditions, the system is at quasi-steady state.

It can be tested that the fed-batch reactor is equiva- lent to the recycle reactor (reactor plug flow reactor with recirculation). It can also be deduced that the behaviour of this reactor is intermediate between the batch reactor (when there is no volume fed or extrac- ted) or the continuous stirred flow reactor (when the ratio between volume fed and extracted is large).

Fed-batch fermentors have been used by some re- searchers (Hashimoto et al., 1981, 1982; Jain and Kumar, 1986). The simulation of the dynamical be- haviour of the fed-batch reactor and the steady-state multiplicity and stability for different cases can be observed elsewhere (Ausikaitis and Engel, 1974; Svobodova et al., 1983, Zubickova et al., 1987).

In this paper, the solutions at quasi-steady state for homogeneous and heterogeneous fermentation are obtained. The relations deduced have been applied to the experimental data obtained in the anaerobic fer- mentation of sewage sludge at laboratory scale. These relations can also be applied to industrial fed-batch fermentors.

F U N D A M E N T A L S O F THE FED-BATCH REACTOR

The different steps of a fed-batch reactor can be observed in Fig. 1.

In the addition step, a volume Vo is introduced and mixed with the reactant medium. The reaction takes place for a time t,, and then a volume Vo is extracted. The ratio "R" is defined as the volume extracted divided by the remaining volume in the reactor, so the volume of the reactant mixture is Vo(l + R). For a reactant A, whose concentration in the inlet flow to the system is CAo, when the quasi-steady state is reached, the concentration C,~i, after the mixture of the new volume Vo and the remaining reactant medium equals

CAo + RCA/ c~i. - (1)

R + !

where (?As" is the concentration of A at the end of the reaction period and is also the concentration in the volume Vo extracted, which leaves the system.

On considering the kinetic expression for the batch reactor, it can be written that

f c... = ic.0 + Rc,, )/IR + ll dCA

t , = . ( 2 ) J (-'A ! - - rA

The mean residence time of the mixture in the system can be calculated as follows: when a volume Vo

REACTION STEP

VO ADDIT ION ~ _ A [ N _~,_._._ C A F

c 'o ,~ ( R . l , V o /

V 0 STEP

Vo I_1 CAF

R: RECYCLE RATIO

Fig. 1. Steps of a fed-batch fermentor.

2117

2118

is fed to the reactor, a fraction 1/(1 + R) of this vol- ume leaves the system after a time t,; in the reactor, a fraction R/(I + R) of the volume fed Vo remains inside; after another period of time t,, a fraction 1/(1 + R) of the remaining R/(I + R) leaves the system; consequently, the fraction [(R/(I + R))x (1/(1 + R)] has been inside the system for a time 2t,; in the same way, it can be deduced that a fraction [(R/(1 + R)] 2 [1/(1 + R)] has been inside the system for a time 3t, and so on. The residence time of the mixture inside the system, therefore, is

1 R t, + 2t, ~ = 1 +

( R ' ~ 2 1 + \~-~--R} ~ 3t, + . . . . t,(R + 1). (3)

In the previous expression, it has been considered that R/(1 + R) is less than 1.

From eqs (2) and (3)

f (C4o + RCA t )!(R * 1 ) d C A r = ( R + l) - - - (4)

dC~, - - r A "

Equation (4) is identical to the equation deduced for recycle reactors (Levenspiel, 1979). This means that some equations and considerations deduced for one type of reactor can be used for the other.

FED-BATCH REACTOR AS DIFFERENTIAL REACTOR

When the ratio R is large, the difference between the concentrations CAi, and C A f is small. Consequently, the batch reactor (in the system described) can be considered as differential. Therefore, if R is large, from eq. (4), it can be deduced that

((CAo + RCAf)/(1 + R)) - CAy r = E1 + R] (5)

- r* (C*)

where - r* is the reaction rate approximately corres- ponding to a mean concentration C,~, which equals

((CAo + RCA.f)/(I + R)) + C A f c~=

2

CAo + 2RCAs + CAf

2(1 + R)

Taking into account eqs (5) and (6), it is deduced that

CAO -- C A f CAO -- C A r - r* (C*) = (7)

r tp(1 + R)

The difference CAo -- CAI can be obtained without very accurate analytical methods and, therefore, using fed-batch reactors, it is possible to determine reaction rates for different values of concentrations.

R. FONT and J. M. LOPEZ CABANES

substrate A + ceils ~ products + more ceils.

It is considered that the maintenance process, in which the cell must consume substrate to survive, is slow in comparison with the fermentation, as occurs with the anaerobic fermentation of different sub- strates.

If CA is the concentration of substrate A (kg/mS), Cx is the microorganism concentration and G, is the product concentration (kg/mS), on considering the yield coefficients Yx (kg microorganisms generated/kg of substrate A consumed) and Yv (kg product for- med/kg of substrate consumed), it can be deduced that

Cx - Cxo = Yx(CAo - CA) (8)

Ce - Cro = Ye(CA0 - CA) (9)

where CAo, Cxo and Cpo are the initial concentrations of substrate, ceils and product, respectively, in the feed to the reactor.

Some of the kinetic models applied for fermentation processes use the concept of specific growth rate of microorganisms #, which is equal to

rx # = - - ( 1 0 )

Cx

where rx is the reaction rate corresponding to the generation of microorganisms per unit of reactor volume.

For a batch fermentor, eq. (10) can be written as

1 dCx # = - - ill)

Cx dt

For a continuous reactor, rx must be deduced from the corresponding material balance applied to the fermentor.

The specific growth rate of microorganisms can be related to the substrate concentration CA. One of the expressions used in the Chen and Hashimoto model equation

C A / C A o

# = tt,. K + (l - K)CA/CAo" (12)

For any type of reactor, taking into account eqs (8) and (9), the following is deduced:

(6) r X rp

- rA = ~ = r p (13)

From eqs (11)-(13), when there are no cells in the feed (Cxo = 0), then

rx pCx" 1 - rA = rx Yx ~x [Yx(Ca° CAI")'I

/~. Gz/Go × K + (1 - K)CAt/CAo" (14)

Substituting eq. (14) into eq. (2), and integrating

t,/~,, In R + ! ! + R CA//CAo = + K In (151 R (R + I)CAI/C~o

FUNDAMENTALS OF THE FERMENTATION

For a fermentation process, the following scheme can be written:

Fermentation in fed-batch reactors

Equation (15) is the performance expression for fed-batch reactors, with the Chen and Hashimoto kinetic law, when there are no microorganisms in the feed.

From eq. (15), the fractional conversion XA of sub- strate can be expressed as

X A = 1 CAf C~o

1 = 1 - [ R -I l/r (16)

/ ~ - - ~ e x p ( / # , t , ) ] (R + 1 ) - R

The condition of loss of cells at quasi-steady state, termed washout, occurs when Cao/Caf equals 1, and in this case from eq. (16), it is deduced that

R + I t,#~, = In - - (17)

R

In order to obtain the production of cells or prod- ucts (values of Caf/C.4o less than 1), the following relation can be satisfied:

R + 1 t , ~ M > I n - - (18)

R

If R is specified, then

! R + I t, > - - In - - (19)

/z,. R

If t, is specified, then

R > [exp(g,~t,)] - 1" (20)

When t, tends to zero (behaviour of the fed-batch fermentor as a continuous stirred tank reactor), the equation that can be obtained either from eq. (16) when R tends to infinity or from a material balance applied to a continuous stirred tank reactor, is

X A = 1 - - C a r = 1 K (21) CAo ~,,,r -- 1 + K"

2119

is distributed in slabs with the same thickness. These slabs or layers can be fixed to an inert material in particles. The reaction rate is considered proportional to the concentration of microorganisms because these segregate the hydrolytic enzymes which attack the substrate surface. If the thickness of all the slabs or layers of substrate is constant, the substrate mass per unit of surface exposed to the enzyme reaction is also constant. Under this assumption, the reaction rate rp* of a particle per unit of time and per unit of surface can be expressed as

dM~, = _ kC~ (22) r~ = dt

where M~' is the substrate mass per unit of surface and the kinetic parameter k is the product of two other parameters: (a) the kinetic constant corresponding to the reaction between the enzyme concentration and the unit of surface, and (b) the ratio between microor- ganism concentration and the enzyme concentration inside the fermentor.

The cell concentration Cx, in accordance with the material balances applied to the fed-batch fermentor, varies between Cxi,, which equals [R/(R + 1)]Cxy and Cxf.

For a particle that has been inside the reactant mixture, integrating eq. (22), it can be deduced that

;o M* = M~o - k Cx dt (23)

where M*0 is the initial substrate mass per unit of surface and t* is the residence time of the particle considered inside the fermentor.

The fractional conversion of substrate (XAp) for a particle is

M~o - M~ (X Ap) ~" (24)

M?o

when M~o equals zero, (XAp) equals l, then rp* is nil. Some cases are discussed below in order to obtain

a general equation.

PARTICULATE SURSTRATE--REACTION PROPORTIONAL

ONLY TO THE MICROORGANISM CONCENTRATION IN

EACH PARTICLE

Most of the kinetic models developed are presented for solubilized substrates or substrates which solubil- ize quickly. In some fermentations, the slow step is the solubilization of the substrate, as occurs in the anaer- obic digestion of sewage sludge. In a previous paper, the kinetics for particulate substrates in CSTRs was discussed (Font and L6pez, 1990). One of the models considered was the shrinking core model (Levenspiel, 1979) in flat-shaped particles. This model is also de- veloped in this paper when using fed-batch fermen- tors. In this case, the consumption rate of substrate in a particle is proportional to the microorganisms con- centration and does not depend on the substrate amount. The reaction rate is nil when the substrate is depleted. This case can occur when the substrate mass

Fractional conversion equals I for all the particles In this case, the fractional conversion after the first

reaction stage of fed-batch fermentor equals 1 for all the particles. The substrate of the particles introduced in the first addition has reacted to depletion. Conse- quently,

fi M~o = k Cx dt. (251

The variation of Cx between Cxi,, which is equal to CxIR/(R + l), and C# during the reaction stage fol- lows an exponential law, as is shown below. On con- sidering the reaction stage, it can be written that

dCx 1" dC A'X

= Y~no - dt ] (26)

2120 R. FONT and J. M. LOPF.Z CABANES

where no/(R + 1) is the concentration of particles in- troduced in the last addition with no-depleted sub- strate (no is the total concentration of particles: num- ber of particles per unit of fermentor volume) and S is the total surface of all the particles:

dCx I dt = Yxno ~ kCxS. (27)

Integrating eq. (27), and taking into account that, for the time t,, Cx varies between Cxi,, which equals CxrR/(R + 1), and Cx/, it can be deduced that

c,, dCx rxno S fo, ,ca+ l~cx, C'~- = ~ k dr. (28)

It can be observed that the variation of Cx between the corresponding boundaries follows an exponential law. Integrating eq. (28),

R 1 In = YxnokS-:---: t,. (29)

R + 1 / < + 1

On the other hand, the product YxnokS represents the specific growth rate of microorganisms that would be observed in a batch fermentor. This is deduced as follows.

In a batch fermentor, on considering that all the particles undergo the same process and the relation- ship between the substrate consumption and micro- organism formation, for eqs (I 1), (13) and (27),

1 dCx 1 dCa #-Cx dt =C~x rx d~-

Yx dM* YxnokCxS YxnokS. (30) = ~ no ~ ; - - S C x =

Using this concept of specific growth rate of micro- organisms, eq. (29) can be written as

In = / a ~ - - ~ t , = / t 1 R + I t, (31)

where/~ is the specific growth rate of microorganisms observed in a batch run.

Fractional conversion equals 0.5 for all the particles introduced in the last addition

In this case, it is considered that all the particles introduced in the first addition have a conversion 0.5 and the particles that have been inside the reactor for a time 2t, or a greater time, therefore, have a conver- sion equal to 1.

On considering the particles introduced in the last addition, it can be written that

k fi" Cx dt - M~o2 (32)

Taking into account that the substrates of the par- ticles corresponding to the last additions have been degraded, the variation of Cx can be related by the expression

dCx ( d e ) dt = Yx -~-~

= rxno ~-4~ +

\ R + I / _J

But YxnokS equals p and eq. (33) becomes

, R

R + I R + I - ~ - S

(33)

dCx R 2 (34)

Integrating eq. (34) between the limits t = 0 , Cxi. = CxIR/(R + l) and t = t,, Cx.r, it can be de- duced that

I n T = ~ 1 - Cx. (35)

On the other hand, rearranging eq. (34) and integ- rating between the limits corresponding to the reac- tion stage,

dCx= 1 - Cxdt. (36) C x , R ( R * 1 ) . ,

On considering eq. (32), eq. (36) becomes

CxI( I R + - \ ~ ] _]~- . (37)

From the relationship between microorganism formation and substrate degradation, when there are no microorganisms in the feed, it can be written that

Cx, = Yx(CAo- CA)= YxCaoXA. (38)

But

Cao = noM~oS. (39)

From eqs (37)-(39), it can be deduced that

YxnM~oX A ~ - - ~ S = tt l - ~ 2k

(40)

Rearranging eq. (40), and taking into account eq. (30), the following can be written:

XA =~(R+ 1) 1 - . (41)

Other cases If the numerical process is repeated when a time nt,

is necessary to deplete the substrate of a particle, the expressions deduced are

In R = ~ 1 - t, (42)

- - R n

Fermentation in fed-batch reactors

Eliminating the parameter n in eqs (42) and (43), it can be deduced that

- - R + 1 ( I n R / ( R + 1)) 2

1 R + 1-]" (44) X , - R In I /~t, - - - - In ~ /

Equation (44) is the performance equation for fed- batch fermentors in the case considered. Taking into account that t, = z/(R + 1), eq. (44) can also be writ- ten as

X--~A = (R + I)2(InR/(R + 1)) 2

[ ( ~ - I n 1 - ~ R + I

(r/tp) 2 (In t,/~) 2 (45)

/ ~ z [ - I n ( l - ~,,1 In z']]'7,/_ 1

In accordance with eq. (44), the washout of micro- organisms (when conversion of the substrate A equals zero) takes place when/a , equals In[R/(R + 1)], and consequently the conditions indicated by eqs (17)-(20) are applicable, substituting the term/am for homogene- ous fermentation by the constant/a for heterogeneous fermentation.

The limit of the expression (45), when R tends to infinity or t, tends to zero, coincides with the equation corresponding to a continuous stirred reactor. This equation is

- - I XA = (46)

/~r[-- in(1 -- l/~u'r)]"

APPIJCATION TO SLUDGE FERMENTATION IN FED-BATCH REACTORS

In wastewater plants, the anaerobic digestion of sewage sludge is one of the alternatives for stabilizing the sludge, reducing the organic load and obtaining a residue with acceptable settling characteristics. As a consequence of the biogas formed, many wastewater plants use this biogas to maintain the fermentors at 30-35°C and, in addition, to generate electricity. Lit- erature on the anaerobic fermentation steps (hy- drolytic-fermentative, acetogenic and methanogenic) is extensive and several schemes are presented in the references outlined in this paper. In the anaerobic digestion of particulate matter, several researchers have deduced that the initial stage of solubilization is the slowest step (Kotze et al., 1968; Eastman and Fergurson, 1981; Pavlostathis and Gasset, 1986; Hobson, 1983; De Baere et al., 1984; Ghosh, 1984). Different kinetic models have been considered for the solubilization of particulate substrates in continuous reactors. Some authors (Eastman and Fergurson, 1981; Pavlostathis and Gasset, 1986; Gujer and Zehnder, 1983) assumed a first-order reaction and others a Monod or Chen and Hashimoto model (Gujer and Zehnder, 1983; Ghosh and Klass, 1978; Mosey, 1981; Pfeffer, 1981; Kennedy et al., 1987; Chen and Hashimoto, 1978, 1980; Hashimoto et al., 1981,

2121

1982). When the substrate is particulate and the slowest step is its solubilization, the kinetic expres- sions deduced for homogeneous systems in continu- ous fermentors or fed-batch fermentors cannot be applied (only for first-order reactions, does the mean conversion of a particulate matter equal the conver- sion in a homogeneous system for the same residence time).

The anaerobic digestion of sewage sludge was car- ried out in 1 I reactors. These reactors were slowly stirred in order to maintain constant operation condi- tions. The sewage sludge (primary sludge + biological sludge) was obtained from the wastewater treatment plant in Alicante (Spain). A scheme of the experi- mental apparatus used can be observed in Fig. 2. In the fed-batch runs, the experiments were made as follows: reactors of 1 1 volume were working as fed- batch digesters, with daily additions and extractions, using syringes. Runs were carried out at 32.5°C. After several cycles of addition, reaction and extraction, the biogas production reached a constant value (quasi- steady state). The pH values were between 6.8 and 7.2 without addition of any chemical reagent. The redox potential was around - 320 mV. The sewage sludge was kept frozen (it was tested that the batch fermenta- tion of the frozen sewage sludge was similar to the non-frozen sewage sludge).

The variables analysed were the following: (a) com- position of the gas by chromatography with a column molecular sieve (02, N2) and a Porapak Q column (N2 + 0 2, CO2, CH4, H2S); (b) solubilized COD, after having passed the sample through a glass fibre filter (Wathman GF/C); and (c) volatile acids (acetic, propionic, butyric, valeric, isovaleric) by GC with a Porapak Q column (prior to the analysis, oxalic acid was added to the samples and the chromatographic column was saturated with formic acid).

Experimental results Six runs at different residence times were carried

out. Table 1 shows the operating conditions and the

i \ i

l

Fig. 2. Experimental apparatus: (a) fermentor; (b) thermo- statized bath; (c) magnetic stirrer, (d) controller of agitation

time; (e) inverted bottle for gas collection.

2122 R. FONT and J. M. LOPEZ CABANES

Table 1. Operating conditions and experimental results in the fed-batch runs

T = 32.5°C Initial total COD = 49.25 g O2/1 Initial solubilized COD = 10.63 g O2/1 of filtrate = 10.10 g O2/1 of sewage sludge (in this value, the fraction of solids that solubilizes for the first 5 d in a batch fermentor is also included)

Reaction time t, = I d

Residence time (d) 5 7 10 15 20 25 Recycle ratio 4 6 9 14 19 24 Period of time, for which the operating conditions were kept constant (d) 41 32 142 106 145 130

Biogas production (32.5~C, l atm, I of CH4/1 of sewage sludge fed) 1.85 8.66 10.0 11.7 11.5 12.0 1 of CH4/(I of sewage sludge d) 0.37 1.23 1.0 0.78 0.57 0.48 Mole fraction of CH,, 67-70 74-75 72-75 73-75 73-74 73-74 Mole fraction of CO2 30-32 23-25 25-27 25-26 24-26 24-26

Total COD of the effluent 43.34 26.90 22.82 20.84 19.20 21.52 Solub. COD of the effluent 7.40 5.92 4.01 3.20 2.63 2.02 COD corresponding to CH4 4.71 22.08 25.50 29.91 29.40 30.78 Error t in balance (%) + 2.4 + 0.5 + 1.9 - 3.0 + 1.3 - 6.2

T The error in the COD balance is calculated by the equation error = ((init. total COD - effluent total COD - COD of CH4)/initial total COD) x 100.

Table 2. Concentration (g/1 of sewage sludge) of volatile acids in the fed-batch runs

Residence time (d)

0 5 7 10 15 20 25

Acetic acid in 0.72 0.091 0.098 0.056 0.032 0.044 0.029 Propionic acid g/1 0.62 1.69 0.011 0.025 0.007 0.006 0.010 Butyric acid of 0.25 0.17 0.10 0.08 n.d. n.d. n.d. /-Butyric acid sewage 0.02 0.13 n.d. n.d. n.d. n.d. n.d. n-Valeric acid sludge 0.05 0.43 0.12 0.28 0.28 0.29 0.33 i-Valeric acid 0.03 0.03 n.d. n.d. n.d. n.d. n.d.

n.d. = not detected.

experimental results. The concent ra t ion of volatile acids is presented in Table 2. Figure 3 shows the methane volume produced (measured at the operat ing condi t ions 32.5~C and 1 atm) per sewage sludge liter at quasi-steady state (methane product ion cons tant between addit ions and extractions). The max imum methane product ion coincides with tha t obta ined from batch digesters (around 10-11 1 of CH4 per sludge liter). The values of the solubilized COD, total C O D and C O D corresponding to the CH4 evolved are plot ted vs residence time in Fig. 4. In Table 1, where a C O D balance is presented, it can be observed that the errors are small. Mean values of total C O D of sludge between the values determined analytically and those deduced from the C O D balance with the CH4 evolved were calculated and considered for cor- relating the experimental results (Table 3).

Discussion The total C O D of the sewage sludge used was

49.25 g/l. In the test performed with a residence time of 25 d, the value of total C O D corresponding to the sewage sludge digested was a round 20 g/l. There is, therefore, a var ia t ion of 29.25 g of COD/l , corres- ponding to the methane product ion. The initial solubilized COD, also including the C O D of the par-

l b

1 0

0 L I

0 1 0 2 0 3

I ~ E S I D E N C E T I M E ( D A Y S )

Fig. 3. Production of methane vs residence time.

ticulate mat ter tha t was solubilized quickly (for the first 10 d in a batch fe rmentor ) i s a round 10.10 g/l. At a residence time of 25 d, the value of solubilized C O D was 1.92. F rom these figures, it can be deduced tha t

4o I

5 0

~' 2o

1 0

/ / !

I I

l 0 2 0

Fermentation in fed-batch reactors 2123

Case I. In this case, the Chen and Hashimoto model has been considered. Consequently, for fed- batch reactors, eq. (16) relates the conversion of the reaction with the operating parameters R and t,.

Taking into account that the refractory coefficient R is the ratio between the non-degradable organic matter and the total organic matter, it can be written that

(COD)~ R* = - - (47)

(COD)o

where (COD)o~ is the corresponding value for reaction time infinity (all the degradable matter is depleted) and (COD)o is the initial concentration of organic matter in the feed. Consequently, the ratio CAI/CAo equals

CA___(I = (COD) - R*(COD)o (48)

CAo (COD)oi l - R*]

From eqs 06) and (48),

(COD) = R*(COD)o +

(1 - R*)(COD)o

[Rexp~mt,)/(R + 1)]l/r[R + 1] - R"

(49)

A Flexible Simplex Program was used to optimize the values of R. #m and K. The objective function O F is the sum of the squares of the absolute difference between the experimental values of C OD and the calculated ones. The optimum values are the follow- ing:/~, = 0.234 d - i, K = 0.26, R* = 0.375, variation coefficient = 2.4%.

The values of the parameters obtained are similar to those found by Chen and Hashimoto (1978, 1980) on considering the O'Rourke (1968) data of anaerobic digestion of sewage sludge.

The Chen and Hashimoto and Monod equations can be applied to homogeneous systems, where the substrate is solubilized. Nevertheless, the slowest step of the majority fraction of the sewage sludge is the solubilization of the particulate substrate, and the kinetic expressions for homogeneous substrates such as those of Chen and Hashimoto and Monod must be considered as correlation equations.

The experimental results obtained in fed-batch fer- mentors seem to correlate satisfactorily to the model

;~ S l l } f NC[ T I M I ( D A Y S )

Fig. 4. Values of total COD (o), solubilized COD (,x) and COD corresponding to the methane evolved (o) vs residence

time.

for the test carried out with the greatest residence time (25 d), there is a variation of 59% of total COD, 43% corresponding to the particulate matter and the re- maining 16% corresponding to the solubilized or- ganic matter. The maximum biogas production was 1.23 m a biogas/(m fermentor d) for a residence time of 7 d. The production in the fermentors of the waste- water plants was usually less than the previous value due to high residence time design (more than 20 d). On the other hand, taking into account that the values of solubilized COD and the concentration of the acids decreased as the residence time increased, it can be deduced that the slowest stage in the anaerobic process is the solubilization of the particulate matter. Nevertheless, the remaining stages cannot be ignored, because there is a considerable concentration of solubilized organic matter.

Kinetics The kinetic models and schemes considered for

correlating the experimental results are the following: Case I: Overall behaviour of the sewage sludge. Case I1: A scheme of two reactions: particulate

substrate --* solubilized substrate ~ biogas.

Table 3. Experimental data considered for correlation in the fed-batch runs

Residence time (d)

0 5 7 10 15 20 25

Mean total value of COD' (g/l) 49.25 43.94 27.03 23.28 Solubilized COD (mg/1) 10.10 7.03 5.62 3.81 Particulate matter COD (total COD - solubilized COD) 39.15 36.90 21.41 19.47

20.09 19.52 19.99 3.04 2.50 1.92

17.05 17.02 18.07

*Mean total value of COD = [total COD + 49.25 - (CODk-~]/2.

2124 R. FONT and J. M.

developed previously for particulate substrates, tak- ing into account a constant growth rate of microor- ganisms to the depletion of the substrate. The par- ticles or layers of substrate are considered slabs with the same thickness. This expression, in accordance with eqs (45) and (48), is

(COD) = (COD)o - (1 - R*)(COD)0

×[ --(z/t ,)2( In t,/,) 2 ].

l /~,ln ( 1 . l l n ~ ) ] (50)

The optimum values obtained in the correlation of the experimental values by a Flexible Simplex Pro- gram are the following: /~ = 0.223 d - l , R * = 0.328, VC = 3.6%.

It can be deduced that both correlations, consider- ing solubilized substrate or particulate substrate, lead to similar results in the parameters of t~ or/~,, and R*. These results are in agreement with those presented in a previous paper with CSTR fermentors, where it was deduced that the experimental data that are corre- lated satisfactorily by the Chert and Hasimoto model with a value of K close to 0.25-0.3 could also be correlated satisfactorily with the particle shrinkage model in flat-shaped particles or layer with the same thickness.

Case II. In accordance with the characteristics of the sewage sludge used and the experimental results, the following scheme is proposed:

Initial reaction 1 solubilized solid ) organic - - organic matter matter

Initial solubilized organic matter (including that proceeding from the solid fraction that is solubilized quickly for the first 10 d in a batch run

reaction 2 ) biogas

Reaction 1 is the solubilization of the particulate matter. The experimental values of the particulate COD considered are calculated by the difference be- tween the total COD and the solubilized COD. In the initial solubilized organic matter, the fraction solid that solubilized quickly in a batch run is also con- sidered.

The Chen and Hashimoto and Monod equations can be applied to homogeneous systems, where the substrate is solubilized. In this case, reaction 1 refers to a particulate substrate, and consequently the equa- tions developed for substrates solubilized cannot be applied. The experimental results obtained in fed- batch fermentors seem to correlate satisfactorily to the model developed previously, where a constant growth rate of microorganisms to the depletion of the substrate is taken into account. The particles or layers of substrate are considered slabs with the same thick-

LOPEZ CABANES

ness. The expression used is the same as eq. (50), but (COD) in this case refers to the particulate substrate, instead of the total organic matter. The optimum values obtained in the correlation of the experimental data by a Flexible Simplex Program are the following: ~ = 0.223 d- ~, R = 0.365, VC = 6.5%.

For reaction 2, the Chen and Hashimoto model is appropriate because the substrate is solubilized. Tak- ing into account that the initial solubilized COD equals 10.10 g/l, the following equation can be writ- ten:

solubilized x COD(g/l)/10.10

1~ = ,u,, K + (I - K)solubilized x COD(g/l)/10.10'

(51)

The values of the specific growth rate/~ of microor- ganisms for the solubilized substrate are calculated using the following expression, assuming that the let- mentor can be considered as a differential reactor:

specific growth rate of microorganisms

formation rate of microorganisms

microorganism concentration

solubilized organic matter

reaction time (1 d)

average concentration of organic" matter that has been solubilized

Table 4 shows the values of solubilized and partic- ulate COD corresponding to the beginning and to the end of the reaction stage (1 d). From this table, the following can be deduced: (a) the degraded amount of solubilized organic matter (= X - Z + Y - U), and (b) the average concentration of organic matter that has been solubilized (= 10.10 - V + 39.15 - I4'). It can also be observed that the differences between Z and X and U and Y (corresponding to initial and final concentrations) are small, in accordance with the assumption concerning the fermentor as differential. Equation (51) can be written as

X - Z + Y - U = - (1 d)(10.10- V + 39.15 - W)" (52)

The deduced values of/~, determined in this way, are also presented in Table 4. By a Flexible Simplex Program, the following optimum values were deter- mined: # = 0.408 d - i, K = 2.257, (refractory coeffic- ient was considered nil), VC = 5.6%.

In accordance with the parameters obtained, the washout of microorganisms for the solubilized COD fermentation should take place at a residence time equal to 1/~ (ca. 4.3 d). This value is similar to those considered by Pavlostathis and Gasset (1986) and Rozzi and Verstraete (1981).

CONCLUSIONS

The performance equations for fed-batch fermen- tors in homogeneous and particulate systems under steady state have been deduced in this paper. When

Fermentation in fed-batch reactors

Table 4. COD values for determination of the kinetic parameters corresponding to the fermentation of the solubilized organic matter in the fed-batch runs

2125

Residence time (d)

5 7 t0 15 20 25

(g COD/I) X (solubilized COD of the sewage sludge

digested) 7.03 5.62 3.80 3.04 2.50 1.92 Y (particulate matter COD of the sewage

sludge digested) 36.90 21.41 19.47 17.05 17.02 18.07 Z (solubilized COD at the beginning of the

stage reaction in the fed-batch digester) Z = (RX + M)/(R + I) 7.65 6.26 4.44 3.51 2.88 2.24

U [particulate matter COD at the begin- ning of the stage reaction in the fed- batch reactor, U = (RY + N)/(R + 1)] 37.35 23.94 21.44 18.52 18.13 18.92

V (mean value of solubilized COD in the digester for the reaction time)

V = (Z + X)/2 7.34 5.94 4.12 3.27 2.69 2.08 W [mean value of particulate matter COD

in the digester for the reaction time, W = (U + Y)/2] 37.13 22.68 20.46 17.78 17.58 18.49

(specific growth rate of microorganisms, d -Z ) 0.222 0.153 0.105 0 . 0 7 1 8 0 .0512 0.0408

M (solubilized COD in the feed) = 10.10 g COD/I. N (particulate matter COD in the feed) = 39.15 g COD/I.

the controlling step of the fermentation of particulate K substrates is the solubilization, as occurs with the sewage sludge, the residence time distribution of the M* particulate substrate must be considered. It has been M~o tested that the experimental results of the anaerobic digestion of sewage sludge, obtained with fed-batch n fermentors, can be correlated satisfactorily with a kin- etic expression corresponding to a shrinking core O F model in fiat-shaped particles, considering the time rA residence distribution of the particulate substrate and r v the microorganism formation as a consequence of the digestion of the substrate, rv*

CA CAf CAin CAO

Cp Cpo

Cx C x f

C xi.

Cxo

(COD)

(COD)o (CODLo k

NOTATION

concentration of substrate A, kg/m 3 rx

final concentration of substrate A, kg/m 3 R initial concentration ofsubstrate A, kg/m a R* concentration of substrate A, correspond- S ing to the feed, kg/m 3 Sb product concentration, kg/m 3 t* product concentration, corresponding to t, the feed, kg/m 3 U microorganisms concentration, kg/m 3 final microorganisms concentration, V kg/m 3

initial microorganisms concentration, VM kg/m 3

microorganisms concentration corres- Vo ponding to the feed, kg/m 3 VR chemical oxygen demand, kg 0 2 / m 3 or VC g O2/I W initial value of (COD) final value of (COD) at time infinity X kinetic constant, (kg sust. m3/(s kg micro.)

dimensional constant of the Chen and Hashimoto model particle mass per unit of surface, kg/m 2 initial particle mass per unit of surface, kg/m 2

total concentration of particles, number/m 3

objective function reaction rate of substrate, kg/(s m 3) reaction rate of product formation, kg/(s m 3)

reaction rate of a substrate in a particle per unit of time and per unit of surface, kg subs/(part s m 3 ) reaction rate of microorganisms, kg/(s m 3) recycle ratio of fed-batch reactor refractory coefficient total surface of all the particles, m 2 substrate concentration, kg/m 3 residence time of a particle, s fermentation time, d initial value of solid substrate, k g O 2 C O D / m s

methane volume evolved per unit of reac- tion volume average value of solubilized substrate, kg 0 , COD/I initial volume, m 3 reactor volume, m 3 variation coefficient average value of solid substrate, kg 02 C O D / m 3 final value of solubilized substrate, kg 02 C O D / m 3

2126 R. FONT and J. M. LOPEZ CABANES

XA fractional conversion of substrate )(A mean conversion of substrate A (XA)p fraction conversion in a particle Y final value of solid substrate, kgO2

C O D / m 3 Yp yield coefficient, kg product/kg substrate Yx yield coefficient, kg microorganisms/kg

substrate Z initial value of solubilized substrate, kg 02

C O D / m 3

Greek letters # specific growth rate of microorganism, d - /~ maximum value of/~, d - i

z mean residence time, d

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