The Chemical Engineering Journal, 37 (1988) B31 - B37

Effectiveness Factors for Bioparticles with Monod Kinetics

GRAHAM ANDREWS*

Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260 (U.S.A.)

(Received July 10, 1987)

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ABSTRACT

Bioparticles include floes, pellets, granules, biofilms on solid particles, biomass in porous particles, and gel-immobilized cells. For these particles a distinction is made between the limiting nutrient, the one which is exhausted first inside a large particle, and the rate- controlling component, the nutrient or inhibitory product that most influences the metabolic rate in the particle. A variable transformation for inhibitory products and a new definition of the Thiele modulus lead to a single graph for effectiveness factor valid whenever the rate-controlling compo- nent follows Monod or linear inhibition kinetics.

1. INTRODUCTION

Many biochemical processes use not dispersed cells, but macroscopic particles containing many cells. Some of the particles occur naturally, including floes of bacteria and yeast, the pellets made by some molds, and the granules that form in anaerobic sludge blanket reactors. Others represent artificial cell-immobilization, including biofilms on solid support particles, flocculent biomass or single cells growing inside porous support particles, and cells entrapped in gel beads. What all of these particles have in common is that molecular diffusion is the only way that nutrients can reach most of the cells and that products can get out. Since liquid-phase dif- fusion is a slow process, cells deep inside a large particle may be inactive either because they are deprived of some essential nutrient

*Present address: Biotechnology Group, Idaho National Engineering Laboratory, P.O. Box 1625, Idaho Falls, ID 83415, U.S.A.

0300-9467/88/$3.50

or because a product accumulates to inhibiting concentrations.

The situation can be analyzed using the effectiveness factor concept developed by chemical engineers studying heterogeneous catalysis. Analytical solutions for simple first- order and zero-order approximations to the inherent microbial kinetics are readily avail- able [l, 21. Various approaches have been suggested for the more complex, non-linear Monod kinetic equation and the equivalent Michaelis-Menten equation for immobilized enzymes [3, 41. More difficult situations such as penicillin production [5] and the sequential processes involved in anaerobic wastewater treatment [6] have been treated numerically.

A question that has been overlooked is that of which components (nutrients and inhibitory products) are most critical in determining the rate of metabolic activity inside the particle. Even in a simple case like the production of ethanol from glucose it is unclear whether ethanol production deep inside a particle would be stopped by ethanol inhibition or by glucose exhaustion. The tendency has been either to include all possible components in the kinetic equation and to solve the many resulting equations on a computer, a procedure that provides little physical insight into the problem, or to select a single likely candidate as the limiting component and use the available analytical solutions.

This paper presents a new procedure which is a compromise between mathematical complexity and applicability to many real fermentations. A variable transformation allows inhibitory products to be treated on the same basis as nutrients, and criteria are developed for choosing which two compo- nents are most critical in fixing the metabolic activity in the particle. A single graph is then

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given for the effectiveness factor, based on a new definition for the Thiele modulus. It allows the calculation of optimum particle sizes, biofilm thicknesses etc., whenever one component follows Monod or linear inhibi- tion kinetics and the other component approximates zero-order kinetics.

2. THEORY

2.1. General case: limiting and rate-control- ling components

Consider a spherical particle which may be a biofilm growing on a solid spherical support of radius R, (Fig. 1) or a floe, pellet, gel sphere or porous support in which case R = 0. The mass-transfer resistance in the liquid phase around the particle is assumed to be negligible [l]. For each nutrient diffusing into the particle a mass balance over an element dr gives

C=S at r=R+L

(0) Thin Film

Other components

Fig. 1. Concentration profiles in the biofilm.

(1)

dC -= 0 at r=R dr

The equation for an inhibitory product diffusing out of the biomass is similar but the right-hand side is negative. This can be avoided, and the equations made identical, by the variable transformation C equals B minus actual product concentration, where p is the concentration that stops production. Note that C = 0 now stops metabolism for a product just as it does for a substrate. The liquid-phase concentration S results from the same transformation. In this work, nutrients and inhibitory products are combined under the name “components”.

Solving eqn. (1) in general is difficult because the specific consumption or produc- tion rate of component i qi is a function of many different component concentrations. The first step towards obtaining an analytical solution is to identify the two components that have the greatest influence on the metabolic rate in the particle. The procedure for doing this is familiar from chemical reac- tion engineering, but is given in detail because the significant components may be different from the “limiting nutrient” in dispersed growth cultures.

The biomass specific growth rate Jo = q 1 YI = q2Y2 = . . . where Yi, Y2 etc. are the observed yields for each component. The qi values can therefore be eliminated between the equations for the m different compo- nents. Integrating twice gives (m - 1) equa- tions relating the concentration profiles in the particle (see Fig. 1) to each other:

D,Y,(S, - C,) = D,Y,(S, - C,) . . .

= DmYA% - Cm) (2)

The limiting component is defined as the component whose concentration reaches zero first. It will be identified by the subscript 1. From eqn. (2) with CI = 0 and all other C > 0 it is clear that the limiting component is the one with the lowest value of the product DYS. Comparing this with the case of a dispersed growth batch culture, where the limiting component is the one with the smallest value of YSO, shows the importance of the mass-transfer parameter D.

The need to identify a separate “rate- controlling” component is well illustrated by the production of ethanol (subscript p) from glucose (subscript s). Typical parameter values

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are p = 100 g l-l, K, = 0.02 g l-l, Y,/Y, = 0.48 (94% of the theoretical yield) and D,/D, = 0.54 (the ratio in water). Assume linear inhibition kinetics for ethanol and consider a point near the reactor inlet (or at the start of a batch culture) where the ethanol con- centration in the liquid is zero and the glucose concentration is 150 g 1-l. The transformed product concentration S, = p so (DYS),/ (DYS), = 0.39 which proves glucose is the limiting component. But let us consider the point inside the particle where the glucose concentration is C, = 1 g I-‘. From eqn. (2) C, = 61 g 1-l. The effect of glucose on the biomass growth rate is (p/ii) = C,/(C, + K,) = 0.98, while the effect of ethanol is (p/,ii) = C&j = 0.61. Clearly product inhibition is having a greater affect on the metabolic rate in the particle despite the fact that glucose is the limiting component. This is the usual situation in product-inhibited fermentations, except when the substrate concentration is low.

In general the rate-controlling component is defined as the one whose concentration first reaches the value at which p is restricted to 80% of its maximum value (80% is an arbitrary choice but seems reasonable). For Monod kinetics this means the rate-control- ling component has the lowest value of DY(S - 4K) (see eqn. (2); this component will hereafter be given no subscript), and the kinetic expression should be written

P _c 4= Y=4K+C ___ I-UC,) (3)

H(C,) is the Heaviside function. It equals unity for CI > 0 and zero for CI < 0. When the limiting and rate-controlling components are the same this term is redundant.

When the rate-controlling process is product inhibition the Monod function can be replaced by Levenspiel’s [7] suggestion for product inhibition kinetics q = Q(C,/fi)“. The 4X in the above expression is then replaced by 0.8 1’n p. The effectiveness factor for these kinetics is given in standard texts on chemical reaction engineering. The linear inhibition (n = 1) case is covered by the low concentration asymptote of eqn. (3) with K replaced by p. It is interesting to note that some product-inhibition data [8] could be fitted by a Monod function of the trans- formed product concentration.

In aerobic processes oxygen is usually both the limiting and rate-controlling component because of its low value of S. This is why the formation of films and floes is usually discouraged in aerobic fermentations except in special cases such as citric acid production where the low oxygen concentration inside pellets of Aspergillus fungi may stimulate citrate formation. Aerobic wastewater treat- ment is a special case because the concentra- tion of organic matter is then only one order of magnitude higher than the solubility of oxygen, not two or three as in most fermenta- tions. For a typical carbohydrate-type waste- water the DY value for oxygen is 8 times that for the organic matter, and the K value is much lower. So the organics may become rate controlling if their concentration falls below 70 mg ll’ and limiting if it is below 50 mg 1-l (approximately).

2.2. Effectiveness factors Problems involving simultaneous diffusion

and reaction are greatly simplified if the dimensionless distances (Thiele moduli) are defined appropriately for the reaction kinetics involved. The correct procedure for the kinetics of eqn. (3) is to multiply the distances by (qX/DKf2)“2. (Note that F will be defined as the dimensionless distance from the particle surface, not its center).

2$l-E)-2ln~~ l/2

(4)

(D Ys), E=l---

DYS

The parameter E relates the concentrations of the limiting and rate-controlling compo- nents. Its extreme values are E = 0 when one component is both limiting and rate controlling, and E = 1 when there is none of the limiting component is the liquid phase.

Rewriting eqn. (1) with the new variables and assuming D and X are constant through- out the particle,

=Kf2(0 + Y-?)~- ’ WC,) K+C

(5)

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C=S at P=O

dC -= dr 0 at r;=[

The second boundary condition has been changed to accommodate the discontin- uous nature of the function H(CI). The quantity .$ is the (dimensionless) active depth, the distance into the biomass to which the limiting component penetrates. For a thin film 4: is the same as the film thickness 0 (see Fig. l(a)). For a thick film (Fig. l(b)) [ is the depth at which CI = 0 or (from eqn. (2)) C = ES. These statements define what is meant by “thin” and “thick” films in this paper.

Equation (5) cannot be solved analytically due to the non-linear form of the Monod kinetics. The procedure followed here is to derive various asymptotic solutions and then to interpolate between them. The results are expressed in terms of a particle effectiveness factor defined by

actual consumption (or production) of component in particle

77= consumption (or production) if the particle were all biomass exposed

to liquid-phase conditions

= 4n(R + L)2D(dCldr)lR +L

(4a/3)(R + L)3Xq(Si)

= _ 3(K + S) dC - KS(B + r)f2 dF ,,

(6)

Although the derivations are done here for spherical particles, the solutions for biofilms growing on flat plates can be found directly as a special case. A flat plate effectiveness factor is defined by

consumption (or production) rate of component per unit area

771 = consumption (or production) rate if whole film were exposed to liquid-

phase conditions

2.3. Thick films on large particles: 0 > t and ((3 + y) 9 <

We are now interested only in the region F< g < (8 + y) where CI > 0, so eqn. (5) can be re-written

(8)

C=S at F=O

C=ES and g=OatY=[

Integrating once and substituting the value of g at T; = 0 into eqn. (6) (g is dC/dr)

3

rl= e+y (9)

It is the simplicity of this result, combined with the work of Bischoff [9] that dictates the form of the function f given by eqn. (4).

2.4. Zero-order kinetics:

S % K; q = qH(C,); f2 = Z(l -E);

e=L (71x II2 i 1 WI&

“Zero-order” kinetics means that any bio- mass exposed to the limiting component will be metabolizing it at its maximum possible rate. There is, in effect, no separate rate- controlling component and we need only be concerned with the limiting component (note that the Thiele moduli, 8, y and i;, reduce to a form which contains parameter values only of the limiting component). The effectiveness factor is equal to the fraction of the particle volume that consists of active biomass.

(10)

For “thin” biofilms all the biomass is active because it is all in contact with the limiting component (Fig. l(a)). So 8 replaces 4; in the numerator of eqn. (10). To evaluate t for thick films (Fig. l(b)) eqn. (5) must be solved (with K 4 C) to obtain the concentration profile.

(e + r-t)3 T; -- (e + r)(e +7-q 2 I

v e+Y = Lim B - Y+” 3 (11)

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Since we know 0 at F = t this gives

by definition that CL =

t2 l- i

2t 3(8 + y) = 1 1 (12)

If this equation has a solution in the range 0 < t < 19 then the film is thick. If it does not then the film is thin.

2.5. First-order kinetics: S < K;

f= (1 -E2)1’2; q = ZH(C,)

Equation (5) can be solved in this case in the range 0 < F < t where H(CI) = 1, using the variable transformation y = C(y + 8 - F). The concentration profile for the rate-controlling component is found to be

c(e + y - ?)

S(O + Y)

= f(e + Y - 8 cash f( l- F) + sinh f( ,$ - F)

f(0 + 7 - t) cash f[ + sinh fl

(13) The effectiveness factor from eqn. (6) is:

3

i

f(0 + y - $j tanh f[ + 1

’ = f(e + 7) f(e + 7 - t) + tanh ft

1 -

fv + 7) t (14)

Substituting the requirement that C = ES at ? = r (eqn. (2) with C1 = 0) into (13) gives:

sinh fE e+r (e + y - [) cash f.$ + ~ = - f E

(15)

The general solution is found by solving this equation for ,$ and substituting in eqn. (14) (computationally it is easier to solve for the quantity ft). If there is no solution in the range 0 < g < 0 then the biofilm is thin and c = 8.

The limiting cases for extreme values of E are as follows. When E = 0 (f = 1) the film is always thin in the sense used here, whatever ,ts actual thickness. The limiting component is now the same as the rate-controlling com- ponent, so CI approaches zero asymptotically but never actually reaches it (mathematically

it is clear that eqn. (15) has no finite solu- tion).

When E -+ 1 (f + 0) the hyperbolic func- tions in eqns. (14) and (15) can be expanded as power series and terms higher than (f,$)2 dropped. This gives

n=E/l-il- &]‘I

12(1- 3(::y)

2

= E(l+E)

(16)

It is not a coincidence that these become identical to the zero-order solution (eqns. (10) and (12)) when E = 1. The condition E -+ 1 means that the concentration of the limiting component approaches zero, and so does the active depth (the dimensionless active depth [ does not approach zero owing to the definition of the function f). The active region is so thin that the concentration of the rate-controlling component does not vary, but remains close to S throughout. Consequently 4 remains close to s(S) which creates a pseudo-zero-order kinetic situation. This must be true whatever the form of the kinetic equation for q.

3. DISCUSSION

All the above solutions are plotted against the dimensionless particle radius in Fig. 2. The most important feature of this graph is that the curves for the extreme cases are close together. So, with definitions of 8 and y given by eqn. (4), it can be used to

I I I I I I I I

0.1 0.2 0.5 I 2 5 IO 20 (B+y)

Fig. 2. Particle effectiveness factors.

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find the best particle size or biofilm thickness in any situation that can be approximated by eqn. (3). This includes a number of important cases.

Shieh [2] and others have defined effec- tiveness factors based on biomass volume. The “total particle volume” basis used in eqn. (6) is preferable because it gives the local reactor productivity directly as e,nXq(S). So the maximum overall volumetric produc- tivity will be obtained from a reactor with a high cell concentration in the particles (usually highest with natural films and floes) and high average values of q(S), solids holdup and effectiveness factor.

Figure 2 shows that to maintain a high effectiveness factor (6’ + y) should be less than 2. The corresponding particle radius depends on the kinetic parameters, the cell concentra- tion in the particle, and the liquid-phase con- centrations, but for natural floes it falls between approximately 0.1 mm for aerobic bacterial processes and 2 mm for anaerobic ‘fermentations [l]. It is interesting to note that both activated sludge floes and those formed by Zymomonas mobilis [lo] are approximately the correct size. In continuous reactors the highest average value of q(S) is achieved with plug flow of the liquid (substrate inhibition is an exception). All S values decrease through a plug-flow type of reactor, so (0 + y) = 2 corresponds to a parti- cle size that decreases through the reactor. The solids in fluidized beds tend to stratify naturally in this fashion, which is one reason for the high productivity of this type of reactor [ 111.

Figure 2 also shows that particles consisting of a biofilm on a solid support require a support radius y < 1 as well as control on the film thickness to keep (0 + y) =G 2. The 2 in rocks traditionally used in trickling filters correspond to a y value of several hundred, and therefore produce very low reactor productivity. Solid supports containing a monolayer of cells (0 of order 10P2) also give very low effectiveness factors. Note that a reasonable choice of values, 0 = y = 1, gives a ratio of biofilm volume to support particle volume equal to 7. A packed bed could never accommodate this much biomass without clogging. A fluidized bed would have to expand to seven times the height of the bed of clean support particles [12].

REFERENCES

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6

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8

9 10

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G. F. Andrews and J. Przezdziecki, Biotechnol. Bioeng., 28 (1986) 802.

W. K. Shieh, L. T. Mulcahy and E. J. LaMotta, Trans. Inst. Chem. Eng., 59 (1981) 129. B. Atkinson, Biochemical Reactors, Methuen, New York, 1974. J. Bailey and D. Ollis, Biochemical Engineering Fundamentals, McGraw-Hill, New York, 1986, 2nd edn. Y. Park, M. E. Davis and D. A. Wallis, Biotechnob Bioeng., 26 (1984) 457. R. L. Droste and K. J. Kennedy, Biotechnol. Bioeng., 28 (1986) 1577. 0. Levenspiel, Biotechnol. Bioeng., 22 (1980) 1671. K. J. Lee and P. L. Rogers, Chem. Eng. J. (Lausanne), 27 (1983) B31. K. Bischoff, AZChE J., 11 (1965) 351. C. D. Scott, Fluidized-bed bioreactors using a flocculating strain of 2. mobilis for ethanol production, in K. Venkatsubramanian, A. Constantinides and W. R. Vieth (eds.), Bio- chemical Engineering ZZZ, New York Academy of Science, 1983. S. T. Jones, R. A. Korus, W. Admassu and R. C. Heimsch, Biotechnol. Bioeng., 26 (1984) 742. G. F. Andrews and C. Tien, AZChE J., 25 (1979) 720.

APPENDIX A: NOMENCLATURE

c

D

9 g H K L

P q

s r T; R S X Y

component concentration in the particle effective diffusivity of a component in the particle l- (DYS)J(DYS) function defined by eqn. (4) dC/dr the Heaviside function Monod half-velocity constant biofilm thickness or bioparticle radius product concentration at which q = 0 specific consumption (or production) rate of a component maximum value of q radial position (R + L - r)(tjX/DKf2)1’2 radius of solid support particle component concentration in liquid phase cell concentration cell yield from a component

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Greek symbols

Y R(ljX/DKf2)1’2

ES solids holdup in reactor

I: effectiveness factor L(@X/DKf 2)1’2

t dimensionless active depth

Subscripts i any component 1 limiting component 0 initial value

P product (ethanol) S substrate (glucose)