Composición microbiana

Application of Macroscopic Principles to Microbial Metabolism

J. A. ROELS, Laboratory for General and Technical Biology, Department of Chemical Engineering, Delft University of

Technology, Julianalaan 67, 2628 BC Delft, The Netherlands

Summary General expressions for mass, elemental, energy, and entropy balances are derived

and applied to microbial growth and product formation. The state of the art of the application of elemental balances to aerobic and heterotrophic growth is reviewed and extended somewhat to include the majority of the cases commonly encountered in biotechnology. The degree of reduction concept is extended to include nitrogen sources other than ammonia. The relationship between a number of accepted measures for the comparison of substrate yields is investigated. The theory is illustrated using a generalized correlation for oxygen yield data. The stoichiometry of anaerobic product formation is briefly treated, a limit to the maximum carbon conservation in product is derived, using the concept of elemental balance. In the treatment of growth energetics the correct statement of the second law of thermodynamics for growing organisms is emphasized. For aerobic heterotrophic growth the concept of thermodynamic efficiency is used to formulate a limit the substrate yield can never surpass. It is combined with a limit due to the fact that the maximum carbon conservation in biomass can obviously never surpass unity. It is shown that growth on substrates of a low degree of reduction is energy limited, for substrates of a high degree of reduction carbon limitation takes over. Based on a literature review concerning yield data some semiempirical notions useful for a preliminary evaluation of aerobic heterotrophic growth are developed. The thermodynamic efficiency definition is completed by two other efficiency measures, which allow derivation of simple equations for oxygen consumption and heat production. The range of validity of the constancy of the rate of heat production to the rate of oxygen consumption is analyzed using these efficiency measures. The energetics of anaerobic growth are treated-it is shown that an approximate analysis in terms of an enthalpy balance is not valid for this case, the evaluation of the efficiency of growth has to be based on Gibbs free energy changes. A preliminary analysis shows the existence of regularities concerning the free energy conservation on anaerobic growth. The treatment is extended to include the effect of growth rate by the introduction of a linear relationship for substrate consumption. Aerobic and anaerobic growth are discussed using this relationship. A correlation useful in judging the potentialities for improvement in anaerobic product

Biotechnology and Bioengineering, Vol. XXII, Pp. 2457-2514 (1980) 0 1980 John Wiley & Sons, Inc. 0006-3592/80/OO22-2457$05.80

2458 ROELS

formation processes is derived. Finally the relevance of macroscopic principles to the modeling of bioengineering systems is discussed.

1. INTRODUCTION

In the methods of analysis used in physics two broad groups of approaches can be distinguished-the corpuscular and the continuum description. The corpuscular description takes into account the fact that particles or objects are the basic units of any physical system. It is based on a more or less careful analysis of the properties of and interactions between the objects composing the system. After performing this analysis the methods of statistical mechanics are invoked to calculate the properties of a system in which a large number of individual objects are present.'-4

The corpuscular description is extremely versatile, but it requires a rather detailed knowledge about the properties of the objects and their interactions. Furthermore, the mathematics involved are quite elaborate.

The continuum approach, which is more usual in engineering studies, is the basis of the so-called macroscopic method for the description of systems in which large numbers of objects are present. It is based on the definition of variables, e.g., concentrations, pressure, and temperature, which are continuous with respect to time and length coordinates and it ignores the corpuscular structure un- derlying the system.

In this report the possibilities of macroscopic methods with respect to the transformation processes in living systems will be analyzed. Microorganisms are, as has been pointed out by several au- t h o r ~ , ' - ~ so-called open systems, they are not in thermodynamic equilibrium and exchange matter and energy with the environment. Hence, a macroscopic description of their behavior should use the formalism of thermodynamics of irreversible processes.

Excellent treatments of this formalism can be found in the literature.'.' The metabolic pattern of microorganisms is characterized by a highly complex network of chemical reactions and, as has been shown by, amongst others, Aris,' the treatment of such patterns is facilitated by the use of matrix operations. Matrix notation will hence be used in this paper. Readers unfamiliar with this formalism are referred to one of the texts on this subject."

To conclude Sec. 1 a short survey of this report will be given. In present day bioengineering a number of elements of macroscopic theory, e.g., mass and elemental and energy balances, are becoming

BIOENGINEERING REPORT: MICROBIAL METABOLISM 2459

increasingly accepted. This report is an attempt to show the general framework from which these principles derive. The results of the applications of these principles to aerobic and anaerobic growth and product formation will be treated and experimental evidence existing in the literature will be analyzed using this formalism.

2. BALANCE EQUATIONS FOR EXTENSIVE QUANTITIES

2.1 . Balance Equation for the Chemical State Vector

An important tool of the macroscopic method is the balance equation. Balance equations can be formulated for each of the extensive properties of the system. Extensive properties are characterized by the fact that they are additive with respect to parts of the system, some examples are mass, energy, volume, and the number of moles of the various components present in the system.

The structure of a balance equation is always the same; in words it can be formulated as:

(1) (accumulation - (in-out (conversion -

in system) transport) + transformation)

Equation (1) states that the total amount of an extensive quantity changes by two kinds of processes-transformation processes inside the system (e.g., chemical reactions) and exchange processes with the systems environment. A convenient starting point for the formulation of balance equations for the amounts of the various chemical species in the system is the formulation of the chemical state vector, C , of the system. In the state vector, the concentrations of each of the n chemical compounds present in the system are identified by one number, representing the number of moles of that compound per unit of system volume. The verbal statement of the balance principle given by eq. ( I ) can, for the chemical state vector, be represented by the following mathematical equation:

C.dV = rA.dV + 1 ja.dS dt r A is the vector of the net production rates of each of the compounds in the system (mol/m3 hr). j, is the vector of the net rates of transport to the system (mol/m2 hr). S and V are the systems surface and volume, respectively.

2460 ROELS

For a system of constant volume, in which no gradients occur, eq. (2) transforms to'

C = r , + @ (3) where CP is the vector of the net rates of transport to the system expressed per unit of system volume (mol/m3 hr).

The reaction pattern inside a system can be specified in terms of a number of independent chemical reactions and their stoichiometric equations.' The net rates of conversion of each of the chemical species can be expressed in terms of the rates of these reactions and their stoichiometry. It can be shown'~'*'' that the following expression holds:

rA = r a (4) If m independent reactions occur, r is an m-dimensional row vector, in which the rates of chemical processes are specified. Each row of the matrix OL expresses the stoichiometry of a reaction in terms of the number of moles of each compound converted per unit reaction rate. It is an m x n matrix of the following structure:

rn

R components

Combination of eqs. (3) and (4) allows the formudion of a balance equation for the chemical state vector of a system of constant volume:

(5 )

Equation (5 ) is the starting point of the derivations which are to follow. The concept of steady state is of fundamental importance in the treatment of open systems.

In a steady state the time derivative of the state vector has become zero, hence it follows from eq. (5):

Q, = - r a (6)

C = r*oL + Q,


A practical implication of eq. (6) is that, for a steady state, the stoichiometry of a reaction pattern can, as far as the net rate of production of each component of the system is concerned, be studied by observations of the exchange flows, @, with the environment. This is a very useful observation, as these flows can in most cases be readily studied experimentally.

2.2. Conserved Quantities: The Elemental Balance Equations

There is a group of extensive quantities, which have the property that net production in the reaction pattern in the system does not take place. These quantities are called conservative quantities. Most biological systems, excluding man, have the property that elements are conserved in all transformation processes possible to the system. This observation results in the formulation of elemental balances, which have become increasingly accepted in bioengineering in the last decade. The earliest paper, known to this author, which contains an explicit treatment of microbial metabolism in terms of stoichiometry, is due to Hoover and Allison; it was published as early as 1940. Another early contribution is due to Battley.13 To this authors knowledge the pioneering articles in biotechnology are most probably due to sol om on^'^ and Harrison.S Important further de- velopments and generalizations are due to Minkevich and Eroshin.I6 Mateles,I7 Harrison et al. , I 8 H o l ~ e , ~ Sukatch and Faust, Takahashi et Kanazawa and Hamer et al.23 studied applications to the single cell protein (SCP) production technology. Wang et al. ,24 Za- briskie, and Zabriskie and Humphrey26 reported studies concerning the bakers yeast production technology. The penicillin fermentation was studied by Heijnen et al.27

Stouthame?8 applied the principles with special reference to the study of microbial energetics. Erickson et al. presented an application to product formation. The state of the art was reviewed by Herbert3 and Cooney et aL3

Roels and Kossen recently presented a rather general analysis of the principle. I Important new generalizations were recently reported by Erickson et al.32

In this report the various proposals for the formulation of the concept of elemental balance will not be treated in detail, rather a new, more general formulation will be derived directly using eq. (5 ) and the conservation principle for elements.

The derivation of an expression for the elemental balances is easily achieved by the introduction of an elemental composition matrix E. If elemental balances for k elements are constructed. this matrix has

2462

the following structure:

C

0

rn P

n e n t

vs

0 E =

ROELS

ai j

7 - J . k chemical elements

aij stands for the number of atoms of atomic speciesj present in one mole of component i.

The change of the amounts (or for a system of constant volume the concentrations) of each of the elements present in the culture is obtained, if the right- and left-hand sides of eq. ( 5 ) are multiplied by matrix E:

(7) The first term at the right-hand side of eq. (7) represents the net production of each of the chemical elements in the transformation processes in the system. By virtue of the definition of a conserved quantity this term should always be zero:

r a - E = 0 (8)

(CaE) = r.a.E + cP.E

Or, as eq. (8) must hold for any vector of reaction rates, r:

cw*E = 0 (9)

Equation (9) is the general form of the principle of the conservation of atomic species; it specifies rn x k relationships between the rn x n stoichiometric constants of the system.

Direct application of eq. (9) to the stoichiometry of bioconversion processes does, however, present some difficulties. In the derivation of eq. (9) it was formally necessary to specify each component present in the system in the systems state vector C. Hence, application of eq. (9) presupposes the elemental composition of each of these components to be specified in the matrix E. In biosystems this presents some difficulty as several hundreds of components take part in metabolism. Up to now, in the application of elemental balances to microbial metabolism, this problem was avoided and only a fairly


limited number of compounds was considered in a so-called gross stoichiometric equation of growth. Although this is intuitively valid, it is interesting to analyze, if such an approach can be theoretically justified.

To perform such an analysis eq. (7) is studied for the systems steady state, hence the left-hand side of eq. (7) becomes zero and it follows:

@E = - r a . E (10) And, by combination of eqs. (10) and (8):

cD*E = 0

Equation (1 1) presents a second statement of the principle of elemental balance, which is restricted to a steady state. It has the advantage of formulating the elemental balance principle in terms of the flows of matter leaving and entering the system. In the application of eq. (1 1) components, which are present in the system, but are not exchanged with the environment in significant amounts, need not be considered.

The reasoning presented above formally validates the treatment of microbial metabolism in terms of a gross stoichiometric equation of growth. It has to be borne in mind that its application is only valid for a system in steady state. Furthermore each component exchanged with the environment in non-negligible amounts needs to be specified in the vector of flows.

2.3. Thermodynamic Treatment of Energy Transformations

The efficiency of the transformation of the energy in the substrate to biomass on growth of microorganisms has received considerable attention in the last decade. Related problems like the heat production in microorganisms have also been studied.

An interesting measure for the efficiency of growth, YATp, has been proposed by Bauchop and E l ~ d e n ~ ~ and it was recently extensively discussed by stout ha me^.^^*^^ YATp expresses the yield of biomass/mol ATP consumed and it is based on knowledge of the amount of ATP produced in the various catabolic pathways in the organism. As such it is not a quantity of a purely macroscopic nature and its merits will not be studied in this paper.

Various proposals for macroscopic efficiency measures have ap- peared in literature and they are all more or less related. The quantity Yavle, as proposed by Payne and his c o ~ o r k e r s ~ ~ - ~ ~ expresses the yield, biomass dry matter (DM)/mol electrons in the substrate, which

2464 ROELS

are available for transfer to oxygen. A second measure proposed by the same authors isYkcal, the yield of biomass DM/kcal heat of combustion of the substrate. These two measures are related as the heat of combustion of an organic compound is known to be proportional to the electrons available for transfer to ~ x y g e n . ' ~ , ~ ~ . ~ ~ The efficiency measure, q, as proposed by Minkevich and EroshinI6 is proportional to Yavle as well as Ykcal. The authors, however, do put limits to q based on thermodynamic considerations. Although the authors in our opinion missed the correct formulation of the second law of thermodynamics for open systems, this approach is very promising and will be used in a somewhat modified form in this present paper. The various efficiency measures have been recently reviewed by NagaL4'

Heat produqtion is the second aspect of the energetics of growth, which has been studied extensively. Cooney et al.41 were among the first to focus attention to the proportionality between heat production and oxygen consumption. Minkevich and Eroshin greatly con- tributed to the understanding of the phenomenon,'6 Imanaka and Aiba,42 Mou and C ~ o n e y , ~ ~ and Wang et al.44 also made contributions to the theoretical and experimental evaluation of the enthalpy balance. Two other important contributions to the thermodynamic analysis certainly merit mentioning. The review due to Wilson and

gives a very detailed thermodynamic analysis of growth and explicitly stipulates the correct formulation of the second law. The treatment of M ~ C a r t y , ~ ~ directed to the analysis of wastewater treatment, gives a detailed analysis of the efficiency of growth in thermodynamic terms. The argument in this report will be very sim- ilar to that treatment.

It will not be attempted to integrate the various contributions in literature into a general theory, rather a direct approach, based on nonequilibrium thermodynamics will be used.

For open systems, such as microorganisms, the first law of thermodynamics' is formulated as:

U = j Q + @.hT (12) Equation (12) states that the system's internal energy per unit volume, u, changes, due to the flow of heat to the system, j , , and the flow of enthalpy, @-hT, associated with the various compounds exchanged with the environment. hT is the transpose of the row vector in which the molar enthalpies of the compounds are specified. For a system in steady state the classic equation for the calculation of heat exchange from an enthalpy balance results:

j , = - @*h' (13)


The second law of thermodynamics allows the formulation of a restriction to the heat production in a process, and it can be derived by the introduction of the Gibbs equation for a system of constant volume:8

T i = - c a p ' (14)

In this equation p r stands for the transpose of the row vector in which the partial molar thermodynamic potentials of the system's components are specified. s is the system's entropy per unit volume.

For a system in steady state, s and u become independent of time and eq. (14) can be written

c.p' = 0 (15) By combination of equations (15) and (3) it follows:

r w p T + @.pT = 0 (16) The left-hand side of eq. (16) contains two contributions, the entropy production, II,, being equal to - rwp'/Tand the entropy exchange with the environment, cP.pT/T. The second law of thermodynamics stipulates H, to be larger or equal to zero, hence it follows:

@ * p T 2 0 (17) Equation (17) is equivalent with the common notion, that for a chemical reaction to proceed at constant temperature and pressure a negative free-enthalpy change (Gibbs free-energy change) should be associated with that reaction.

Minkevich and Eroshin's efficiency measure, q, is based on the assumption that j , can never exceed zero. Hence, endothermic reactions, which are known to exist, are excluded. It is, formally speaking, based on an incorrect statement of the second law, as is clear from comparison of eqs. (17) and (13).

3. CONCEIT OF ELEMENTAL BALANCE AND ITS APPLICATION TO GROWTH AND PRODUCT FORMATION IN

MICROORGANISMS

3.1. General Expressions for Growth with Product Formation

The formalism developed in Sec. 2 is thus general, that in principle very complex cases of growth and product formation can be treated. This results, however, in complicated equations, which offer little advantage over the more formal equations. Therefore only situations of limited complexity will be studied in more detail. An organism

2466 ROELS

is considered, which grows on one sole source of carbon and energy, which may contain nitrogen. One sole source of nitrogen is supplied, it may also contain carbon. One product is excreted, CO,, H20, and 0, are the only other components, which may be exchanged with the environment.

In terms of the formalism developed in Secs. 2.1 and 2.2 the system will be considered to be a given quantity of microorganisms DM. Growth is assumed to be balanced.'' The organism exchanges, macroscopically speaking, an exact replica of itself with the environment; it is characterized by its gross elemental composition formula CHb,O,,Nd,. The concept of the C mol of ~rganisrn,~' the amount containing 1 mol carbon, is adopted.

Figure 1 gives a schematic representation of the system and the possible flows to and from the system. Only the elements C, H, 0, and N are considered and indeed these elements comprise in most cases about 95% of the cellular mass and the various other exchange flows. Equation ( 1 1 ) can be directly applied to this specific case with:

and

E =

- 1 bi C I di a 2 b z c2 dz a 3 b3 c3 d3 a 4 64 c4 d4 0 0 2 0 1 0 2 0

- 0 2 I 0 .

In the present case there are seven flows and eq. ( 1 1) specifies four equations between the flows, hence only three flows are independent

Fig. 1 . System and flows for the macroscopic analysis of microbial growth.


variables. Which flows are to be chosen independently strongly depends on the application one has in mind. First the case will be considered in which the flow of biomass, the flow of substrate, aZ, and the flow of product, a3, are known. The standard routines of linear algebra now allow eq. (1 1) to be solved for the remaining flows, the results are given in Table I.

In the equations several conventions are adopted. The macroscopic yield factors Y h , the number of C mol biomass producedl mol substrate taken up, and YLx, the number of mol biomass produced/mol product produced, are introduced. Yix and YLx are defined by:

YLx = I@I/@3/ (20)

By analogy, macroscopic yield factors for C 0 2 , Y& and oxygen, YAx can be defined.

In the equation for the oxygen flow the generalized degrees of reduction of substrate, ys, of biomass, yx, and of product, yp, are introduced. These are defined by analogy to the definition of degree of reduction used by Eroshin and his coworker^,'^^^^ who defined the value for growth with ammonia as a nitrogen source. In the present case the degree of reduction is modified to apply to any nitrogen source. The generalized degrees of reduction are given by:

(21)

(22)

(23)

yx = 4 + bl - 2 ~ 1 - d1[(4U4 + 64 - 2~4)/d41 ys = (4Uz + bz - ~ C Z - dz[(4U4 + b4 - 2~4)/d4]}/~2 yp = ( 4 ~ 3 + b3 - 2C3 - d3[(4~4 + b4 - 2~4)/d4]}/~3

The rationale for the definition is the concept of available electrons as introduced by Payne and his coworker^^^,^' and Eroshin and his coworker^.'^,^^ As can be seen the degree of reduction of the biomass is defined dependent on the nature of the nitrogen source used, this is a convenient procedure, as it makes specification of the degree of reduction of the nitrogen source and assumptions about the degree of reduction of the nitrogen present in the biomass unnecessary.

The equations given in Table I can be used to calculate the flows of oxygen, carbon dioxide, and nitrogen source, once the flows of biomass, substrate, and product are known. However, the existence of accurate gas analyzers often results in a situation where these flows can be readily obtained. In that case a combination of these flows with the often also measured flow of substrate can be used

TA

BL

E I

Cal

cula

ted

Flow

s fo

r th

e G

ener

al C

ase

Flow

C

ompo

nent

Fl

ow (i

n m

ourn

3 hr)

@I

biom

ass

@*

subs

trat

e

@, pr

oduc

t

inde

pend

ent

inde

pend

ent

inde

pend

ent

- 1 (d

, - f

d + *

) ni

trog

en so

urce

d

q

y J

X

Ypx

@4

@5

oxyg

en

Q6

carb

on d

ioxi

de

-

@7

wat

er


to estimate the flows of biomass and product. By some rearrange- ments of the equations in Table I the following equations are obtained, if the nitrogen source is NH3 and product and substrate do not contain nitrogen:

@I = @2[Ro(4 - yp.RQ) + ady, - ys)l/(yx - yP) (24) Q3 = Q2[- R0(4 - yx.RQ) - a2(yx - ys)l/[a3(yx - yp)l (25)

These equations show that knowledge of the respiratory quotient, RQ, the ratio of the carbon dioxide production to the oxygen consumption, and Ro, the ratio of the oxygen consumption to the substrate consumption, allows direct estimation of the biomass production rate and the production formation rate. The nature of the equations shows, as yx - y p appears in the denominator at the right- hand side of eqs. (24) and (25), that the method becomes less accurate if yx approaches yp. Procedures of this kind have been proposed for the detection of unwanted alcohol formation in aerobic bakers yeast produ~tion.~ There is a variety of other applications of the equations presented in Table I, but their use in particular applications will be left to the reader.

3.2. Aerobic Growth on a Sole Source of Carbon and Energy without Product Formation

In this section the case of growth on a single source of carbon and energy not containing nitrogen will be studied. Three simple nitrogen sources are considered and product formation is excluded. In Table I1 the equations for the various flows are given. The equations were obtained from the general equations of Table I by substitution of composition data for the nitrogen sources and putting YLx equal to infinity. In the equations another version of the substrate yield factor has been introduced, YZx the yield expressed as C mol biomass produced/C mol substrate, it can also be regarded as the fractional conservation of the substrates carbon in the biomass. The equations for oxygen consumption appear to be the same, irrespective of the nitrogen source. The reader should note, however, that the equations are different, as yx depends on the kind of nitrogen source supplied. The equations for yx in the three cases studied are also given in Table 11. Eroshin and his coworker^'^.^^ pointed out that the degree of reduction with respect to NH, as the nitrogen source does vary little, if biomass of various sources is compared. In Table 111 some literature data on biomass elemental composition are summarized and the degrees of reduction calculated with respect to NH,, HNO,, and

2470 ROELS


N, are seen to vary very little. The relative standard error is less than 5% in all cases. A good approximation to the average composition of biomass is the elemental composition formula CH, .*- Oo.sNo.,. In calculations account has to be taken of the biomass fraction consisting of elements other than C, H, 0, and N. This fraction will be termed ash and assumed to be 5% of the biomass DM. If it is assumed that yx indeed is more or less constant, the equations presented in Table I1 allow the formulation of a general equation for the relationship between YLx and the yield on substrate Y%, this relationship can be formulated as

YLx = (4/yx)*[q0/(l - qo)l (26)

In this equation qo is the efficiency factor of Minkevich and Eroshin.I6 It is given by

qo = yxYYys (27)

The correlation is numerically different for the three nitrogen sources as yx is dependent on the nitrogen source used.

In Figure 2 the relationship according to eq. (26) is shown for growth with HNO, and NH, as the nitrogen sources. Some fairly recent experimental data on substrate yield and oxygen yield, compiled in Table IV, are plotted in Figure 2. As can be seen there is good agreement between theory and practice. It is interesting to compare the relationship given in Figure 2 with the concept of yield on available electrons, Yavl, as proposed by Payne and his cowork- e r ~ . ~ ~ . ~ ' YaVl, is easily identified to be given by

Yav/e = (YYys )*Mx (28)

where M , is the molecular weight of the biomass g/C mole. As is apparent from eq. (28) a plot of YLx vs. Yavie should have the same shape as the general correlation given in Figure 2. This was shown to be the case by NagaL40 The property of Yav,, and of qo will be discussed further in Sec. 4.

The general correlation presented in Figure 2 and also by eq. (26) can be used in a variety of ways:

1) For the case of growth on a substrate of known degree of reduction with a known nitrogen source and known absence of product formation the equation can be used to estimate oxygen consumption if the substrate yield factor is known.

2) In the case of measured oxygen yield and measured substrate yield the relationship can be used to check the consistency of the data. In the case of discrepancy and, of course, positive indications

h, 5

TAB

LE 1

11

N

Elem

enta

l C

ompo

sitio

n an

d D

egre

e of

Red

uctio

n fo

r Bio

mas

s of

Var

ious

Sou

rces

Deg

ree

of r

educ

tion

(yx)

El

emen

tal

Org

anis

m

form

ula

NH

3 H

NO

3 N

Z R

ef.

Can

dida

util

is

C. u

tilis

C

. util

is

C. u

tilis

K

lebs

iella

aer

ogen

es

KI. a

erog

enes

K

l. a

erog

enes

K

I. a

erog

enes

Sa

ccha

rom

yces

cer

evis

iae

S. ce

revi

siae

S

. cer

evis

iae

Para

cocc

us d

enitr

ifica

ns

P. d

enitr

ifica

ns

Esch

eric

hia

coli

Pseu

dom

onas

C12

B A

erob

acte

r aer

ogen

es

Ave

rage

4.45

4.

15

4.34

4.

15

4.23

4.

15

4.30

4.

15

4.12

4.

20

4.28

4.

19

3.96

4.

07

4.27

3.

98

4.19

sa

= 0

.13

(3%

)

5.25

5.

75

5.86

5.

75

5.99

6.

07

5.66

6.

07

5.40

5.

56

5.64

5.

79

5.54

5.

99

6.11

5.

98

5.78

s

= 0

.26

(4.5

%)

4.75

4.

75

4.91

4.

75

4.89

4.

87

4.81

4.

87

4.60

4.

71

4.79

4.

79

4.59

4.

79

4.96

4.

73

4.79

s

= 0

.10

(2.1

%)

30

30

30

30

30

30

30

30

48

49

50

51

52

36

36

15

w

a s s

tand

s for

the

stan

dard

err

or o

f the

est

imat

e of

yx

(in p

aren

thes

es th

e st

anda

rd d

evia

tion

expr

esse

d as

per

cent

of

the

aver

age)

.


I

efficiency factor qo

Fig. 2. Relation between biomass yield on oxygen and the efficiency factor, qo. Theory and experimental results for growth with: glucose, NH, (0); acetate, NH3 (8); glycerol, NH3 (0); citrate, NH3 (H); ethanol, NH, ( x ) ; methanol, NH, (0); dodecane, NH, (e); methane, NH, (@); gluconate, HN03 (A); pentane, HNO, (A); methane, HNO, tr).

that no measuring errors are present, the deviations may indicate the presence of undetected other carbon sources or products. If the amount of carbon converted to product is known, for example from a total organic carbon measurement of the broth filtrate, the equations presented in Sec. 3.1 can be used to estimate the degree of reduction of the product and hence can provide indications of its chemical nature.

Again in complete analogy to the cases treated in Sec. 3.1, in those cases where the oxygen flow and the carbon dioxide flow are known, the theory presented here can be used to calculate biomass growth curves. As in this case the number of independent variables is two, the number of flows which has to be measured is only two. Any

TA

BL

E IV

Ex

perim

enta

l V

alue

s of

Yix

, Yb,

andY

.,/.

Org

anis

m

S. ce

revi

siae

E

. col

i Pe

nici

llium

chr

ysog

enum

Pe

. chr

ysog

enum

A

zoto

bact

er v

inel

andi

c"

C. u

tilis

Pseu

dom

onas

jluo

resc

ens

Rho

dops

eudo

mon

as s

pher

oide

s S

Aspe

rgill

us a

wam

ori

Aspe

rgill

us n

idul

ans

P. d

enitr

$can

s"'

C. u

tilis

Ps.

jluor

esce

ns

A. a

erog

enes

A

. aer

ogen

es

Can

dida

boi

dini

i C

. util

is Ps

. jlu

ores

cens

C

. boi

dini

i K

lebs

iella

sp.

M

ethy

lom

onas

met

hano

lica

Can

dida

N-1

7 H

anse

nula

pol

ymor

pha

"Bac

teria

" (I

)

Can

dida

lipo

lytic

a M

etha

ne b

acte

ria"'

Met

hylc

occu

s ca

psul

atus

M

etha

ne b

acte

ria

Subs

trat

e

yix

(mol

DM

/C m

ol

subs

trat

e)

Yb*

(mol

DM

/mol

02

) YW

/, (g

DM

lavl

e)

gluc

ose

gluc

onat

e ac

etat

e

glyc

erol

ci

trat

e et

hano

l

met

hano

l

pent

ane

dode

cane

m

etha

ne

0.59

0.

62

0.52

0.

56

0.30

0.

59

0.44

0.

52

0.62

0.

70

0.45

0.

42

0.32

0.

66

0.34

0.

61

0.61

0.

43

0.52

0.

47

0.60

0.

46

0.45

0.

47

0.41

0.

38

0.63

0.

56

I .34

1.

61

1.93

1.

55

0.50

I .6

4 1.

16

1.81

1.

55

2.64

1.

76

0.87

0.

57

1.17

1.

25

0.82

0.

82

0.51

0.

47

0.70

0.

66

0.49

0.

46

0.54

0.

54

0.27

0.

36

0.32

3.63

3.

81

3.20

3.

44

1.85

3.

63

2.71

3.

20

3.81

4.

31

3.02

2.

58

1.97

3.

48

2.78

2.

50

2.50

1.

76

2.13

1.

93

2.46

1.

89

1.85

1.

81

1.64

1.

17

1.94

1.

72

53

54

55

56

57

58

58

59

;a

56

56

50

b~

58

58

60

34

61

58

58

61

62

63

64

65

21

66

67

68

69

a N

itrog

en s

ourc

e is

NH

,, ex

cept

for

cas

es in

dica

ted

by (

I),

HN

O,

and

(2),

N,.


combination of two flows suffices to specify all the remaining flows. As was recently pointed out, also the flow of the nitrogen source can in some cases be advantageously

There is an obvious limit7' to the value of Y:x in the cases treated in the present sections. The substrate is the only source of the element C, hence Y&, the fractional conservation of the element C in the biomass, cannot exceed unity, Y:: 5 1. This restriction also defines a maximum to the value of Y,, on substitution of Y'& 5 1 in eq. (26) it follows:

this equation only holds if ys > yx. For ys 5 yx Y:, can have any value.

using a somewhat different approach.

Equation (29) was derived earlier by the present

3.3. Anaerobic Growth with Product Formation

As much as has been published concerning the application of elemental balances to aerobic growth as little has recently been published concerning anaerobic growth. This is understandable as much of the work done recently has been concerned with the study of SCP production. Owing to the rising prices of crude oil a renewed interest is growing with respect to the production of bulk chemical products by the anaerobic metabolic routes in microorganisms. An example is the production of ethanol. The anaerobic formation of one product is easily analyzed using the concept of elemental balances. All flows to the system are completely specified if two flows are known. An equation for the rate of product formation can be obtained by putting the oxygen flow, a5, of Table I equal to zero:

This equation can be used to estimate the rate of product formation if the biomass yield factor on substrate and the rate of substrate consumption are known. Equation (30) can be reformulated to an equation for the fractional conversion of the substrates carbon to product, ep, as a function of Y%:

ql = yslyp - Y X Y X / Y P ) (31)

Equation (3 1) leads to the conclusion that the maximum fractional conversion of the substrates carbon to one single product can never exceed the ratio of the substrate degree of reduction to that of the

2476 ROELS

product. For the anaerobic formation of ethanol from glucose this results in a maximum conservation of 2/3, for the formation of methane 1/2. This maximum is reached if YYx approaches zero. The reasoning can also be made to apply to mixtures of products, if a C mol averaged degree of reduction is used.

A common value for the substrate yield factor YYx in the anaerobic formation of ethanol is 0.14 (C mol biomass/C mol produ~t).~ The fractional conversion of substrate to product is calculated to be 0.57 for that case, being 85% of the potential maximum. In view of the already mentioned renewed interest in the anaerobic formation of products a further study of the potentialities of macroscopic methods in anaerobic product formation studies may prove rewarding. The subject will be pursued somewhat further in the following sections.

3.4. Conclusion

From the foregoing it will have become clear that elemental balances are an invaluable tool in the description of the systems commonly encountered in bioengineering. It is as fundamental as stoichiometry in chemical reaction engineering systems. The theory seems to be well developed and the field is open to the development of specific applications. The strategy for the application of the tool can be summarized as follows:

1) Verify that the system is defined such that a (pseudo) steady- state assumption is justified.

2) Construct a list of all n components and their elemental composition, which the system exchanges with the environment in non- negligible amounts.

3) Determine the number of elements (k ) for which elemental balances can be constructed.

4) Choose the most convenient set of n - k flows to be obtained experimentally.

5 ) Apply the balance principle given by eq. ( 1 1) and obtain the unknown flows by the solution of the matrix equation.

There is a significant further problem associated with the application of elemental balances. It applies to instances in which more flows are measured than are minimally needed to calculate the remaining ones. In this case a statistical procedure can be applied to a more optimal estimate of all measured and unknown flows. The procedure can also be used to detect systematic errors in the meas- urements of one or more of the flows and may also be used for identification of the existence of unremarked exchange flows with the environment. It is beyond the scope of this report to treat this


powerful method in further A field, where further devel- opments may prove especially rewarding, is the realm of the anaerobic product formation.

4. ENERGETIC ANALYSIS OF MICROBIAL GROWTH AND PRODUCT FORMATION

4.1. Introduction

In this section some energetic aspects of the growth of microorganisms will be dealt with. Two simple cases will be treated, aerobic heterotrophic growth on a single source of carbon and energy with simple nitrogen sources and the anaerobic formation of products. In both cases only two variables, for example, the yield factor for biomass on substrate, Y k , and the biomass flow, a,, remain unknown after the application of the elemental balance principle.

The application of the first law of thermodynamics, the energy balance introduces one new unknown flow, the heat flow j , , but also introduces one new equation, the enthalpy balance according to eq. (13). Hence, the introduction of the first law of thermodynamics does not diminish or increase the number of unknown flows. For a system in steady state the enthalpy balance will not provide any further information if two flows are already known. Equation (13) allows one to calculate the heat flow if Yyx and a, are known. On the other hand the heat flow in combination with any other of the flows allows calculation of all remaining flows. The second law of thermodynamics, e.g., eq. (17), specifies an inequality, which must always be satisfied. This allows the calculation of a maximum value Yix can never exceed. It thus also allows definition of the thermodynamic efficiency of the growth process. It is in principle possible to define other, less fundamental efficiency measures, for example the one proposed by Minkevich and Eroshin,I6 where the maximum yield is defined with respect to zero heat flow (or zero oxygen flow) to the environment. These efficiency measures are very useful for some types of calculations. In order to obtain values for the maximum yield consistent with the second law and with a positive heat flow to the environment thermodynamic data are needed. In Table V some data are shown. The enthalpy values refer to the pure components in the standard state and are expressed per C mol of the various substances. The free enthalpy (also often termed Gibbs free energy) values refer to 1M aqueous solutions of the respective compounds. The procedure used for the calculation of the data for biomass is summarized in Appendix I. Especially in

2478 ROELS

TABLE V Thermodynamic Properties of Some Compounds in the Standard State

Component ho IJ?

(kJ/C mol) (W/C mol)

Acetic acid C2H402 (aq)" Ammonia NH3

( a 4 Ammonium ion NH,' (as) Biomass CHl.800.sNo.z (aq) Carbon dioxide c02 (g) Bicarbonate ion HCOj- ( a d Citric acid C6H807 (as) Dodecane CIZH26 (1) Ethane C2H6 (g) Ethanol C2H60 (as) Formic acid ( 3 3 2 0 2 ( a d Fumaric acid C4H.404 (aq) Glucose C6H1206 (as) Glycerol C3HsO3 (1) Lactic acid C3H603 (as) Malic acid C&Os ( a d Methane CH4 (g) Methanol CH4O (1) Nitric acid HNO3 ( a d Nitrogen N2 (9) Oxalic acid CZH204 (aq) Oxygen 0 2 (g) Pentane C5H12 (as) Propane C3Hs (g) Succinic acid C4H604 ( a 4 Water HzO (1)

a aq = aqueous; I = liquid; g = gaseous.

H' (PH = 7)

By extrapolation of data for lower hydrocarbons.

- 244.8 - 46.3 - 80.9 - 133 - 91.4 - 394 - 692

- 29.3 - 42.4 - 139 -410.6

-211 - 222.3

- 75.0 - 239 - 173

0 -414

0 - 34.6 - 34.7

- 286

- 186.4 - 16.7 - 26.7 - 79.6 - 67.1 - 395 - 588 - 195

2Sb - 16.4 - 90.9 - 335.2 - 151.3 - 153.1 - 163.0 - 173.0 - 141.0 - 50.9 - 176.5 - 110.7

0 - 338

0 - 3.9 - 7.8 - 172.8 - 238 - 40.5

the case of the free enthalpy balance it should be emphasized that the free enthalpy values at the concentrations actually encountered in the process should be used so the values obtained can only be considered an approximation.

4.2. Aerobic Heterotrophic Growth without Product Formation

In the case of aerobic heterotrophic growth without product formation eq. (17) allows the calculation of a maximum to Y:x, which can never be surpassed due to the restrictions put by the second law. The maximum value of YYx consistent with the second law will be termed wr.


Two other restrictive values will be used in the treatment to follow, we, the maximum value of Yyx consistent with heat transport to the environment, and wo, the maximum value of Yyx consistent with oxygen uptake from the environment. It has, however, to be emphasized that there exist no a priori laws of nature forbidding reactions with the uptake of heat or oxygen generation. In Table VI the numerical values of of, we, and wo are given for a variety of substrates and growth with the nitrogen sources NH,, HN03, and nitrogen. In the calculations the elementary formula of the biomass was assumed to be given by CH,.,Oo.sNo.,. All values refer to ash- free dry weight. An example of such a calculation is given in Ap- pendix 11. In Table VII the general expressions for we and wo are recorded as these will be used later on. The numerical values given in Table VI allow the following conclusions:

0 The w, values do not deviate more than 10% from the we values and hence the maximum yield according to the second law does not deviate too much from that according to zero heat exchange with the environment. This validates the approach of Minkevich and Eroshin.'6 In terms of thermodynamics the observation can be formulated as follows: For aerobic growth

Substrate Y.

Oxalic acid 1 Formic acid 2 Malic acid 3 Citric acid 3 Succinic acid 3.5 Gluconic acid 3.67 Lactic acid 4 Acetic acid 4 Glucose 4 G 1 y c e r o 1 Methanol 6 Ethanol 6 Dodecane 6.17 Pentane 6.4 Propane 6.67 Ethane 7 Methane 8

4.67

TABLE VI Calculated Values of of, we, and wo

Nitrogen source

NH3

Of we 00

0.28 0.25 0.24 0.53 0.56 0.48 0.71 0.72 0.70 0.72 0.79 0.84

0.88 0.93 0.96 0.89 0.90 0.96 1.00 0.97 0.96 1.15 1.14 1.12 1.46 1.50 1.44 1.38 1.42 1.44 1.37 1.40 1.48 1.42 1.45 1.53 1.47 1.53 1.60 1.54 1.62 1.67 1.72 1.84 1.91

HNO3

0, 0, 00

0.24 0.21 0.17 0.47 0.48 0.35 0.62 0.52 0.61 0.52 0.69 0.61

0.63 0.81 0.69 0.78 0.77 0.69 0.87 0.83 0.69 1.00 0.97 0.81 1.27 1.28 1.04 1.21 1.21 1.04 1.20 1.20 1.07 1.24 1.24 1.11 1.29 1.31 1.15 1.34 1.38 1.21 1.50 1.57 1.38

0.24 0.22 0.21 0.47 0.48 0.42 0.62 0.63 0.62 0.63 0.70 0.73

0.77 0.82 0.84 0.78 0.78 0.84 0.88 0.84 0.84 1.01 0.98 0.98 1.28 1.30 1.26 1.22 1.23 1.26 1.20 1.21 1.29 1.25 1.25 1.34 1.30 1.32 1.40 1.35 1.40 1.46 1.52 1.59 1.67

2480 ROELS

TABLE VII Expressions for the Maximum Values of YYx Consistent with Various Restrictions

Restriction Nitrogen

h,' + 394 + 143(bz/az) 302.5 + 1436, - 383dl h,' + 394 + 143(b2/uz) 302.5 + 143bj + 30dl h,' + 394 + 143(b2/a2)

302.5 + 143bl

the entropy flow associated with matter is small as compared to that associated with heat. o, is approximately proportional to the degree of reduction of the substrate. This is shown in Table VIII where the values of Table VI are analyzed statistically. The error of an estimate assuming proportionality between o, and ys is not significantly different from that of the least-squares estimates in each of the three cases considered. For the nitrogen sources N2 and NH3 the equation oo = we is approximately valid, the residual error of that estimate not differing significantly from that of the least-squares estimate of we from the relationship with ys. For HN03 the relationship o, = oo is not valid, the most reliable proportionality relationship being we = 1.18 coo.

The consequences of this observation will be discussed later on. The various expressions formulated in this section can now be used to define a number of efficiency measures.

4.2.1. Thermodynamic efficiency of growth

The values of of calculated in Sec. 4.2 are the maximum values of the yield allowed by the second law of thermodynamics. The thermodynamic efficiency of the growth process can now be defined as:

Equation (32) was used in the analysis of some recent yield data which have been published in the literature. In the calculations it


was assumed that the composition of the biomass could in all cases be adequately represented by the average formula CH,.,Oo.,No.,. An ash content of 5% was assumed.

In Table IX the data and the calculated efficiencies are summarized. The data are also shown graphically in Figure 3 where the thermodynamic efficiency is plotted against the degree of reduction of the substrate for growth with NH, as a nitrogen source. Although Figure 3 shows the existence of appreciable variation the trend seems to be adequately represented by the dotted line. The thermodynamic efficiency seems to be low for the highly reduced as well as highly oxidized substrates, although the latter conclusion rests on only very little data. The tendency for the thermodynamic efficiency to be low for highly reduced substrates is, however, clear. A reason for this drop in thermodynamic efficiency can be easily obtained, if it is recognized that, due to carbon limitation, YYx can never exceed 1, hence the thermodynamic efficiency is subject to

TABLE VIII CoIrelations of of, o,, and oo with ys and 0, with wo

Nitrogen source Correlation

NH3 Of = 0.09 + 0.21y, we = 0.07 + 0 . 2 2 ~ ~ 0, = 0 . 2 4 ~ ~ WO = 0 . 2 4 ~ ~ 0, = 00

Residual standard error of estimate Remarks

0.045 least-squares

0.047 least-squares

0.058

0.058

estimate

estimate

-

HNO3 ~f = 0.08 + 0 . 1 8 ~ ~ 0.042 least-squares 0, = 0.06 + O . l 9 y , 0.039 least-squares

estimate

0, = 0.2oy,

0, = 1 . 1 8 ~ ~ 00 = 0 . 1 7 ~ ~

estimate 0.041

0.041 -

N1 W, = 0.09 + 0.18y, 0.041 least-squares 0, = 0.06 + O.l9y, 0.040 least-squares 0, = 0.21y, 0.058 00 = 0.21y, - 0, = wo 0.058

estimate

estimate

TABL

E IX

Ex

peri

men

tal V

alue

s of

Y:,

and

the

Ther

mod

ynam

ic E

ffic

ienc

y, q

th,

as w

ell a

s Ya

v/r a

nd ye

Org

anis

m

Subs

trat

e

Pseu

dom

onas

oxa

laiic

us

Pseu

dom

onas

sp.

Ps

eudo

mon

as d

eniir

ifica

ns

A. a

erog

enes

P

s. d

eniir

ifica

ns

Ps.

jluo

resc

ens

Ps.

den

itrifi

cans

P

s. d

enitr

ifica

ns

Pseu

dom

onas

sp.

Ps

. den

trifi

cans

A

. aer

ogen

es

Ps.

jluo

resc

ens

Pseu

dom

onas

sp.

C

. uiil

is P

s. jl

uore

scen

s C

andi

da tr

opic

alis

S.

cer

evis

iae

S. c

erev

isia

e E

. col

i Pe

nici

llium

chr

ysog

enum

Azo

ioba

cter

vin

elan

dii

C. u

iilis

Ps.

fluo

resc

ens

oxal

ate

form

ate

citr

ate

mal

ate

fum

arat

e su

ccin

ate

gluc

onat

e" '

lact

ate

acet

ate

gluc

ose

gluc

ose"

'

7s

yyx

qlh

1.00

0.

07

0.26

2.

00

0.18

0.

34

0.18

0.

34

3.00

0.

38

0.54

0.

34

0.49

3.

00

0.37

0.

52

0.33

0.

46

3.00

0.

37

0.53

3.

50

0.39

0.

49

0.41

0.

52

3.67

0.

45

0.67

4.

00

0.32

0.

34

0.37

0.

40

4.00

0.

44

0.49

0.

42

0.47

0.

32

0.36

0.

36

0.40

4.

00

0.59

0.

59

0.57

0.

57

0.62

0.

62

0.52

0.

52

0.56

0.

56

0.30

0.

33

0.59

0.

59

0.44

0.

44

YW/.

1.72

2.

21

2.21

3.

12

2.78

3.

03

2.71

3.

03

2.74

2.

88

3.01

1.

97

2.27

2.

71

2.58

1.

97

2.21

3.

63

3.51

3.

81

3.20

3.

44

1.85

3.

63

2.71

tl. Re

f.

0.28

74

0.

32

74

0.32

74

0.

52

74

0.47

34

0.

51

74

0.46

58

0.

51

74

0.46

74

0.

50

74

0.59

28

0.

33

34

0.38

58

0.

49

74

0.46

58

0.

35

58

0.40

75

0.

61

53

0.59

48

0.

64

54

0.54

55

0.

58

56

0.36

57

0.

61

58

0.45

58

R. s

pher

oide

s A

. aw

amor

i A

. nid

ulan

s Ps

eudo

mon

as s

p.

A. a

erog

enes

C

. tro

pica

lis

C. b

oidi

nii

C. u

tilis

Ps.

jluo

resc

ens

C. u

tilis

C. b

rajs

icae

C

. boi

dini

i K

lebs

iella

sp.

M

. met

hano

lica

Can

dida

N-1

7 H

. pol

ymor

pha

Pseu

dom

onas

C

Pseu

dom

onas

EN

To

rulo

psis

M

ethy

lom

onas

sp.

M

. met

hano

lica

C. t

ropi

calis

C

. lip

olyt

ica

"Bac

teria

" Jo

b 5

Met

hane

bac

teria

M

. cap

sula

tus

Met

hane

bac

teria

M. m

etha

noox

idan

s

benz

oate

gl

ycer

ol

etha

nol

met

hano

l

hexa

deca

ne

dode

cane

pe

ntan

e"'

prop

ane

etha

ne

met

hane

"' m

etha

ne

met

hane

"'

4.29

4.

67

6.00

6.00

6.13

6.

17

6.40

6.

67

7.00

8.

00

0.52

0.

62

0.70

0.

48

0.66

0.

61

0.61

0.

61

0.43

0.

55

0.64

0.

52

0.47

0.

60

0.46

0.

45

0.67

0.

67

0.70

0.

50

0.64

0.

56

0.41

0.

47

0.71

0.

71

0.38

0.

63

0.62

0.

56

0.50

0.

68

0.52

0.

62

0.70

0.

48

0.57

0.

44

0.44

0.

44

0.23

0.

39

0.46

0.

36

0.32

0.

41

0.31

0.

31

0.46

0.

46

0.48

0.

34

0.44

0.

41

0.30

0.

38

0.48

0.

46

0.25

0.

37

0.41

0.

33

0.29

0.

40

3.20

3.

81

4.31

2.

75

3.48

2.

50

2.50

2.

50

1.76

2.

26

2.62

2.

13

1.93

2.

46

1.89

1.

85

2.75

2.

75

2.87

2.

05

2.62

2.

24

1.64

1.

81

2.62

2.

50

1.17

1.

94

1.91

1.

72

1.54

2.

09

0.54

0.

64

0.72

0.

47

0.57

0.

43

0.43

0.

43

0.30

0.

39

0.45

0.

35

0.31

0.

40

0.27

0.

30

0.45

0.

45

0.47

0.

33

0.43

0.

40

0.29

0.

38

0.46

0.

44

0.24

0.

34

0.39

0.

30

0.27

0.

37

59

56

56

74

60

75

61

58

58

76

77

61

62

63

64

65

74

74

78

79

80

75

81

21

74

74

67

68

82

69

83

84

2484

0.9 r c- 6 0.8 P

9 B E

2 U

0.7-

u

z 0.6- t 0

K w 0.5

0.4

0.3

a2-

0.1

ROELS

-

-

-

-

-

-

l.o+, SECOND LAW

\ C -LIMITATION

0 0

0

1 I I L 0 1 2 3 4 5 6 7 8

DEGREE OF REDUCTION v, Fig. 3. Thermodynamic efficiency of aerobic growth with one carbon source as

a function of the substrates degree of reduction and theoretical limits to the efficiency.

the restrictions given by the right-hand part of the solid line in Figure 3. The left-hand part of the solid line is the restriction put by the second law, the efficiency can never exceed unity. Figure 4 shows the results presented in Figure 3 in a somewhat other way: Yyx, the carbon conservation in biomass, is plotted against substrate degree of reduction. The solid line again gives the restrictions due to the second law and total carbon conservation. Both in Figures 3 and 4 the results are seen to be well within the limits posed by both the second law and that due to carbon limitation.


The tendencies observed in Figures 3 and 4 can be summarized as follows. For substrates up to a degree of reduction of about 4.2, the degree of reduction of biomass, the energy content of the substrate is insufficient to allow all substrate carbon to be converted to biomass, even if the thermodynamic efficiency were unity. For substrates with a degree of reduction higher than 4.2 all carbon could be converted to biomass as far as energy requirements are concerned and even CO, could be fixed; the extent to which the energy can be stored in biomass becomes limiting. Hence, the thermodynamic

CARBON LIMITATION

0 0 0 0 0

0' 0 I lAl IUN I

1 0 1 2 3 4

8 I I

5 6 7 8

DEGREE OF REDUCTION ( y s )

Fig. 4. Relationship between substrate yield factor Y:, and substrate degree of reduction and the theoretical limits to YE.

2486 ROELS

efficiency has to decrease on growth on substrates of a high degree of reduction. This has been discussed more extensively by the present author.72 For reasons which are not understood by the present author, the carbon conservation efficiency, Yyx, never exceeds 0.7 and has a maximum of about 0.60 on the average. Furthermore, the thermodynamic efficiency never exceeds 0.7 and has a maximum value of 0.55 on the average. Taking these empirical notions into account the following estimate of biomass yield for substrates of various degrees of reduction on growth on NH, seems reasonable:

Yyx = 0.05 + 0.12 ys ys 5 4.58 (334 Yyx = 0.60 ys > 4.58 (33b)

The values predicted with the present semiempirical correlation can now be compared with data from other sources, notably the review of (see Table X). In general there is fair agreement between the empirical equation and the theory.

Bells5 explains the low values for lactic acid and acetic acid from the growth-inhibiting nature of these compounds. The low values for formic and oxalic acids may be due to the fairly large amount of energy needed to transport these components across the cellular membrane.86

Concluding, it can be stated that eq. (33) can be of value in a preliminary estimation of growth yield for growth with NH3 as a nitrogen source.

It is also possible to speculate about the effect of the nitrogen source on the yield on substrate. If it is assumed that the thermodynamic efficiency does not depend on the nitrogen source used, then, by virtue of the fact that of values for HN03 and N, are lower by about 15% as compared to NH,, the yields are expected to be lower by maximally 15%, if these nitrogen sources are used.

The efficiency of the conversion of a substrate to biomass has also received considerable attention in the literature on wastewater treatment.& A well-known correlation, which is frequently used, is that due to Servizi and Bogan:

g biomass produced = 0.39 g COD consumed

In this equation COD stands for the number of g oxygen consumed on complete combustion of the substrate.

The COD of the amount of substrate used in the production of 1 C mol biomass can be expressed as:

g COD = 8ys/Yyx (34)


TABLE X Compilation of Average Measured and Calculated Data for Yix and YaVIe for

Aerobic Growth on Various Carbon Sources

yi: YWk

av av av av Number Substrate meas. calc. meas. calc. of data Source

Glucose Glucose Other sugars Glycerol Acetate Acetate Lactate Citrate Malate Fumarate Succinate Pyruvate Pentadecane Hexadecane Heptadecane Octadecane Higher n-alkanes Ethanol Methanol Propane Ethane Methane

______ ~

0.54 0.53 3.33 3.26 26 0.57 0.53 3.51 3.26 10 0.52 0.53 3.20 3.26 15 0.61 0.60 3.21 3.16 7 0.41 0.53 2.52 3.26 7 0.38 0.53 2.34 3.26 4 0.43 0.53 2.64 3.26 3 0.40 0.41 3.28 3.36 4 0.40 0.41 3.28 3.36 3 0.41 0.41 3.36 3.36 3 0.42 0.47 2.95 3.30 3 0.42 0.45 3.10 3.32 2 0.56 0.60 2.27 2.41 6 0.50 0.60 2.00 2.41 5 0.47 0.60 1.90 2.41 3 0.51 0.60 2.06 2.41 5 0.53 0.60 2.16 2.41 5 0.57 0.60 2.34 2.46 5 0.56 0.60 2.30 2.46 10 0.71 0.60 2.62 2.21 1 0.71 0.60 2.50 2.11 1 0.59 0.60 1.81 1.85 4

85 Table IX

85 85 85

Table IX 85 85 85 85 85 85 85 85 85 85 85

Table IX Table IX Table IX Table IX Table IX

If the semiempirical equation [eq. (33)] for YYx is adopted and C mol biomass are converted to g biomass it follows:

g biomass/g COD = O.16/ys + 0.37 ys 5 4.58 (35a) g biomass/g COD = 1.94/yS ys > 4.58 (35b)

For substrates with a degree of reduction exceeding 3, eq. (35) is close to Servizi and Bogans correlation. For highly oxidized or highly reduced substrates the correlation will be less reliable, for ethanol (ys = 6) a coefficient of only 0.32 is expected, for methane only 0.24. In wastewater treatment, however, degrees of reduction extremely differing from 4 are hardly expected.

4.2.2. Heat production on aerobic growth

A quantity that is of great practical importance is the heat production on microbial growth as this quantity may determine the pro-

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ductivity of a fermentor in cases where the cooling capacity of the fermentor system is limited.

A convenient method to calculate the heat flow can, in cases without product formation and for the nitrogen sources NH,, HNO,, and N,, be based on the efficiency q,. A stoichiometric equation is formulated in which mol biomass are produced and @,/we mol substrate are consumed. As we is the value of YYx at which the heat generation equals zero, no heat will be generated in the process according to that reaction. This stoichiometric equation is now sub- tracted from the gross stoichiometric equation of growth and stoichiometric equation representing total combustion of ( l h , - l/YYx)@,. C mol substrate is obtained. The heat production in this component equation, being the heat production on growth with a substrate yield, Yyx is now easily shown to be given by

j~ = (l/Yy,r - l/we).@I[hso + 395 + 119(bJa2)] (36)

qe = Yb/w, (37)

(38)

If the efficiency factor q, is defined by

it follows :

j~ = @l'Ah*(l - q e ) / q e

where

Ah = A,dl + 143bl + 302.5 (39) In eq. (391, A, = -383 for NH,; he = 30 for HNO,; A, = 0 for Nz.

In the derivation of these expressions the equations for we as given in Table VII were used.

For an organism of composition formula CH,.,Oo.,No., it follows:

NH3: Ah = +480

N*: Ah = +560

The values of the constants for HNO, and N, are much higher in- dicating that at a given value of q, heat production will be higher on growth with HN03 and N, as compared to NH,.

As we is for most applications very close to w, the trends observed on an analysis of experimental material with respect to qth will also be valid with respect to qe. The following approximation to the ex-


pected values of qc thus results for growth with ammonia as a nitrogen source:

qe = 0.55 if ys 5 4.55 (40a) qe = 2.5ly, if yo > 4.55 (40b)

These equations allow a calculation of the heat production to be expected on substrates of varying degree of reduction. Figure 5 summarizes the results. The figure shows the expected heat production with methane as a carbon source to be higher by a factor of 2.5 to 3 than that on growth with glucose.

As has been explained the high heat production with methane as the carbon source is not due to peculiarities of the organisms growing

HEAT I (kJ/C-moleD I 1.5

DEGREE OF REDUCTION y, Fig. 5. Expected values of the heat production in relation to the substrates degree

of reduction and theoretical limits, NH, as the nitrogen source.

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on methane but is essentially due to the high energy content of the substrate as compared to the carbon content. This excess energy has to be dissipated if no substrates of a low degree of reduction are available. In Figure 5 the lower limits to the heat production due to the restrictions of the second law and carbon conservation have also been drawn. (It was assumed that zero heat production gives a suf- ficiently close approximation to the maximum yield obtainable in view of the second law.)

It is apparent from Figure 5 that even in the limit of complete carbon conservation the minimum heat production on growth on methane is higher than that which is practically achieved on the average for substrates of a degree of reduction of 4. Indeed the maximum qe for growth with glucose actually observed, 0.7, would result in a heat production lower by a factor 2 than that which is minimally achievable on growth with methane as the carbon source.

4.2.3 Relationship between heat production and oxygen consumption on aerobic heterotrophic growth

equation for the oxygen flow can be derived: Completely analogous to the reasoning in Sec. 4.2.2 the following

(41)

where qo is an efficiency factor with respect to oxygen consumption, it is equal to the thermodynamic efficiency proposed by Eroshin16 and Erickson et al.,32 and it is given by

@o = @i*Ao*(I - qo)/qo

qo = y;,/oo (42)

A0 = -(4 + b1 - 2 ~ 1 + hod,)/4 (43) The constant A . is given by

where ho = - 3 for NH,; ho = 5 for HNO,; X o = 0 for N,.

By combination of eqs. (28) and (42), it follows: Equation (41) can also be used to obtain an expression for Yavle.

Yavle = (qouoM,)/ys (44)

Yavl, = (qoM,)/y, (45)

with the aid of the equation for oo presented in Table VII, it follows:

This equation shows that the concept of YaVle is equivalent to the efficiency factor qo, if yx and M , are considered constant.

For growth with N, or NH, as the nitrogen source qo is by a good


approximation equal to q, (Table VIII) and eq. (45) can be approx- imated by

Yavle = ( q e M x ) / y x (46) If the second law and carbon conservation limits to q, are introduced the limits to Y,,,, shown in Figure 6 result. An expected average dependence of Y,,,, on the substrates degree of reduction is obtained by substitution of the empirical q, correlations, according to eqs. (40a) and (40b). The resulting relationship is shown in Figure 6 and compared with data compiled in Table X. The observed drop

0 1 2 3 4 5 6 7 8

DEGREE OF REWCTION v, Fig. 6. Expected values of and theoretical limits to Yavle. Experimental results

for growth with: glucose (0); other sugars (A); glycerol (0); acetate (v); lactate (0); citrate (0); malate (A); fumarate (B); succinate (TI; pyruvate (e); higher n-alkanes ( x ) ; ethanol (0); methanol (a); propane (0); ethane (a); methane (@).

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in Yavle on growth on substrates with a high degree of reduction is seen to follow directly from the reasoning presented above.

We will now derive a relationship between heat production and oxygen consumption. If qe equals qo, comparison of eqs. (41) and (38) results in the following relationship:

[ ~ Q / @ o 1 = ( A dA o I (47) As the approximation q, = qo is valid for the nitrogen sources NH, and N,, eq. (47) will hold for these nitrogen sources. The proportionality constant can be calculated from eqs. (43) and (39) and is seen to be uniquely dependent on the composition of the biomass. For the average composition of biomass the proportionality constant is about 460 for N, as well as NH,. These values agree rather well with the empirical value of 520 of Cooney et and with earlier theoretical estimates of 451,16 and 455.,* This derivation can be generalized quite easily in cases of growth even in complex media and when appreciable amounts of product are formed. It can be shown that heat production and oxygen consumption are proportional as long as all components present obey the relationships that their heat of combustion is proportional to the oxygen consumption on combustion. This is known to hold for most compounds commonly encountered in fermentation systems. One notable exception is HNO,, whose heat of combustion does not follow the general rule; hence the equality of qo and qe does not hold for that nitrogen source. For growth with HNO, the following relationship between oxygen consumption and heat production can be shown to apply:

(48)

As is clear from eq. (48) for growth with HNO, the proportionality constant between heat production and oxygen consumption depends on qe. For qe = 0.55 (a common value) it is equal to 590 or 28% higher than the value for the nitrogen sources N, and NH,.

IjQ/@ol = l.l8IAh/AOl(l - qJ(l - 1.18 qe)

4.3. Anaerobic Product Formation

Interesting applications of the concept of the energy and entropy balance are possible in the treatment of anaerobic growth and product formation processes, although, probably for the reasons discussed in Sec. 3.3, little seems to have been published on this subject. A noteworthy exception is the review of McCarty concerning wastewater treatment.*

It is important to note that the convenient approximations, which


can, to a large degree of accuracy, be used in the case of aerobic growth, being the use of an enthalpy approach instead of a free enthalpy approach and the concept of energy content of a compound being proportional to the available electrons, are no longer valid. Anaerobic growth has to be treated using a free enthalpy balance, using tabulated free enthalpy values.

It is beyond the scope of this treatment to present a general dis- cussion of anaerobic processes, rather attention will be focused on some special processes.

First the aerobic formation of various products from glucose will be discussed. The nitrogen source for growth will be assumed to be ammonia.

It is convenient to split the overall stoichiometric equation of anaerobic growth into two parts. In the first reaction the biomass is formed with consumption of all of the nitrogen source. No product is formed in the reaction.

In the second reaction the remaining substrate is converted with production of product, carbon dioxide, and water.

The free enthalpy change of the reaction resulting in the formation of biomass precursors can be calculated in a straighforward manner. For glucose as substrate it comes out to be -25 kJ/mol biomass produced. The free enthalpy change of the biomass synthesis reaction is negative, although small, hence, as far as the second law is concerned, this reaction could proceed without a second energy- generating reaction. But apparently this free enthalpy change is too small for efficient growth, hence the energy-generating reaction in which the product is formed is needed to provide the additional dissipation. In Table XI the free enthalpy change of various anaerobic product formation reactions starting with glucose are summarized. As can be seen the free enthalpy changes on formation of methane, ethanol, and acetic acid are largest, hence these products are efficient as far as the anaerobic generation of free energy for the growth reaction is concerned. The formation of for example glycerol as the sole product is less beneficial.

The formation of products with a degree of reduction lower than that of the biomass would require CO, fixation, hence the anaerobic formation of these compounds as sole products is not excepted. As is also clear from Table XI the enthalpy change of the product formation process can be largely different from the free enthalpy change and hence becomes an incorrect quantity for judging dissipation in terms of the second law.

In the treatment of aerobic growth the thermodynamic efficiency

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TABLE XI Free Enthalpy and Enthalpy Change for Various Anaerobic Product Formation

Reactions Starting with Glucose

Free enthalpy change Enthalpy change Product (kJ/C mol glucose) (W/C mol glucose)

Methane - 67 - 24 Methanol - 16 + 16 Ethanol - 38 - 13 Glycerol - 9 + 5 Acetic acid - 53.6 - 34

of the growth process was shown to be an interesting variable that followed some regularities, the thermodynamic efficiency being in most instances about 0.55 except for substrates of a high degree of reduction for which carbon conservation is the governing factor. It is interesting to see if such general tendencies also exist in anaerobic processes. The thermodynamic efficiency of anaerobic growth will be defined analogous to that of aerobic growth, the free energy of combustion conserved in the biomass relative to the sum of the free energy of combustion conserved in the biomass and the dissipation in the process. Using the stoichiometry considerations treated in Sec. 4.2, this quantity can be calculated from a free enthalpy balance if the substrate yield factor Yjx is known.

Table XI1 summarizes the results of such calculations for a fairly limited number of literature data on anaerobic growth. The procedure of calculation is essentially the same as for aerobic growth (Appendix 11). As can be seen, the thermodynamic efficiency does not vary much if one exception, growth on formate, is left out. The efficiency is in all cases about 0.70 although the substrate yield factor varies from 0.057 to 0.26. Again the concept of the thermodynamic efficiency can most probably be advantageously used for a preliminary estimation of growth yields in anaerobic product formation processes. The thermodynamic efficiency on anaerobic growth seems to be 10 or 15% higher than the aerobic growth efficiencies.

An enthalpy balance can serve to calculate heat production on anaerobic growth. As an example the alcohol fermentation will be treated. On aerobic growth of the yeast Succharornyces cerevisiue on glucose with ammonia as a nitrogen source a typical growth yield Yyx = 0.57 is obtained, this corresponds to a qc value of 0.59 and a heat production per unit biomass produced of 334 kJ/C mol biomass produced. On anaerobic growth a typical yield factor of 0.14 is obtained, this corresponds to a heat production of 90 kJ/C mol pro-

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duced. These values make clear that anaerobic growth results in a smaller heat production than aerobic growth, even if the heat production is expressed per unit biomass produced. This conclusion is expected to be of a fairly general validity for most cases of anaerobic growth. This phenomenon is of particular importance in the bakers yeast production process, where, due to the Crabtree or the Pasteur effects, metabolism may become partially anaerobic. In that case heat production per unit biomass produced has to be expected to become lower with increasing fermentative action. This contrasts the conclusions of a recent publication of Wang et a1.* These authors report increasing heat production with increasing fermentation ac- tivity.

4.4. Conclusion

In this section the relation between various efficiency measures proposed in literature, the concept of Yavlc and Minkevich and Eroshins6 q were analyzed in terms of thermodynamics of open systems. It was shown that the quantities q and YaVl, are proportional and contain to a good degree of approximation the same information as the thermodynamic efficiency, q t h , which is derived from the formally more correct free enthalpy balance. It is shown that the formulation of efficiency measures different from qth, notably qe, the efficiency with respect to enthalpy transformation, and lo, an efficiency with respect to oxygen exchange, allows the formulation of simple expressions from which oxygen consumption and heat production can be easily calculated. These concepts also allow an analysis of the validity of the principle of constancy of the ratio of the rate of heat production to the rate of oxygen consumption, which is shown to be not valid for growth with the nitrogen source HNO,.

An analysis of literature data shows some regularities to exist. The thermodynamic efficiency of aerobic growth is shown to be about 0.55 except for highly reduced substrates, where carbon limitation takes over. The maximal carbon conservation seems to be 0.6 on the average. As far as carbon conservation, heat production, and oxygen consumption are concerned substrates with a degree of reduction between 4 and 6 are, for growth with ammonia as a nitrogen source, to be considered optimal for biomass production (SCP).

Straightforward application of these observations to the case of wastewater treatment results in a relationship between sludge production and chemical oxygen demand as generally accepted in literature on the subject. The concept of the thermodynamic efficiency


of growth is also of value in the treatment of anaerobic processes. A limited literature survey indicates that the efficiency of anaer-

obic growth is in most cases around 0.7. It is also shown that an enthalpy balance gives an incorrect expression of dissipation in anaerobic processes. Heat production, even when expressed per unit of biomass dry matter produced, is, compared with aerobic growth, lower in anaerobic processes.

In future, further development of the application of energy and entropy balances mainly has to be directed to the treatment of anaerobic growth, the treatment of aerobic growth being well ad- vanced. Important questions to be solved in the treatment of aerobic growth are the why of a maximum carbon conservation of about 0.6 and a thermodynamic efficiency of 0.55. It is the belief of this author that macroscopic methods can provide no clues to those matters; it is a subject where biochemical considerations come into the pic- ture.

The thermodynamic treatment of anaerobic growth has to be extended to cover a larger amount of data. Furthermore, uncertainties exist as far as the free enthalpy change at the concentrations oc- curring in actual practice are concerned. This may affect the calculations, the present treatment refers to free enthalpy changes at unit molality of the reactants.

5. LINEAR RELATIONSHIP FOR SUBSTRATE CONSUMPTION

Since the advent of continuous culture various workers have emphasized that the yield constant, the amount of biomass produced per unit of substrate consumed, is dependent on the specific growth rate. This led to the introduction of the concept of maintenance energy by Herbert@ and later on by Pirt." The introduction of the maintenance concept resulted in the following linear relationship for the dependence of the substrate consumption rate on the culture's growth rate:

-rs = rx lY , + m,Cx (49) The equation defines the true yield coefficient Y,, being the maximum amount of biomass produced per unit substrate consumed and the maintenance coefficient, m *, the amount of substrate consumed for maintenance purposes per unit time and per unit biomass present in the culture.

The physiological significance of the constants Y , and m, has been the subject of much d i s c ~ s s i o n ~ ~ ~ ~ ~ ~ ~ between workers on the

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theoretical aspects of microbial energetics and the constancy of Y , and m , has been doubted. However, for technological applications eq. (49) only needs to be considered a fairly good approximation to the relationship between r s and r x , the physiological significance is much less important. In a number of cases significant amounts of a product are formed from the substrate and eq. (49) is no longer adequate in all cases. For these cases eq. (49) can be generalized fo1,91.

- r s = rx /YSx + rp /Ysp + msCx (50) Equation (50) defines a yield constant for product, it is the maximum amount of product that is formed per unit substrate converted. For a system in steady state, eq. (50) can be translated into a relationship, which applies to the various flows of compounds to the system:

as = -@xlYsx - + m,C, (51) Equation (51) was .obtained from eq. (50) by application of the steady-state condition given by eq. (6).

If eq. (51) is combined with the equations for the relationships between the various flows in a system with product formation (Sec. 3.1, Table I), expressions for the flows of oxygen, carbon dioxide, nitrogen source, and water can be derived. The two most important equations, those for oxygen consumption and carbon dioxide production, take the form:

Qo = -@xlYox - + moCx (52) QC = @JYCx + - mcCx (53)

The constants Y,, Y,, mo, Y,, Y,, and m , can be expressed in terms of Y,, Y,, and m,. The equations are given in Table XIII.

The derivation presented above shows that once the linear law for substrate consumption is adopted, linear relations result for the flows

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