6. Dynamic Modeling
6.1 Model Development
6.1.1 Stoichiometry
One of the main differences between the steady state and dynamic AD-FTRW model is that in the dynamic model the metabolic intermediates produced are taken into consideration. Every biodegradable fraction – once utilized by its corresponding FOG – produces a different combination of metabolic products (Figure 6.2).
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Figure 6.2, Dynamic Model Metabolic Pathways & Functional Organism Groups
The acetogenesis of >C2 SCFAs produces metabolic products that include Acetic (Ac), Propionic (Pr) and Butyric Acid (Bu) (eg Hx → Bu + Ac + H2), these metabolic products add to the bulk concentration of these SCFAs (Figure 6.2 and Appendix 6.3). Therefore, a unique stoichiometric equation will be derived for each of the biodegradable fractions in FTRW. This will again be done
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by combining the catabolic and anabolic pathways of substrate utilization as described in Chapter 5 (and Appendix 5.1). The dynamic AD-FTRW model is based on the following:
1) A literature survey, which was done to obtain the catabolic pathways of the individual SCFAs, MeOH, EtOH and the slowly biodegradable organics from anaerobic biomass death and hydrolysis (Sbp) (Kalyuznhyi, 1997b; McCarty, 1975; Batstone et al., 2002; Sötemann et al., 2005)
2) Assuming that all the FOGs have a generic composition of C5H7O2N (McCarty 1975), the standard anabolic equation can be described as
( )
2 3 2 5 7 2
5. 8.
. . 2 . .
20 20 20 20
s s s s
x y z
D D D D
C H O x CO NH x z H O C H O N
+ − + → − − +
(6.1) Where
Ds= (4x+y-2z) = Substrate electron donating capacity [e-equivalents/mol] (6.2) 3) All CO2 produced will be in soluble form (H2CO3) thus:
CO2 Produced +H O2 →H CO2 3 (6.3)
4) Analogous to the steady state model, the carbonate system (HCO3-
/H2CO3) acts as the proton source/sink for all of the WABC reactions. Secondly, the contribution of CO32- will be regarded as negligible in the anaerobic digestion pH range.
5) Only the protonated (non-ionic) form of any SCFA can be metabolized, thus un-protonated (ionic) SCFAs need to pick up a proton from the carbonate system before metabolism:
C Hx y−1O2− +H CO2 3 →C H Ox y 2 +HCO3− (6.4)
6) Since the fraction of the SCFA in the un-protonated form is again governed by the influent pH (compare Section 5.1.2), this gets taken into consideration via the F value, as was the case
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in the steady state model. However, in the case of the dynamic model, each of the SCFAs have an individual pKa value, thus Eq 5.9 can be rewritten as:
( )
2 1 2
2 1 2 2 2
1 1
x
x x
C
x x x x
feed ax
C H O
F C H O C H O H
K
−
−
− +
−
= =
+ +
(6.5)
Where
x = 1 to 6 and
Kax = 10-pKax of the SCFA with carbon chain length x.
[H+]feed= Feed proton activity = 10-pHfeed Thus if CxTotal = CxH2x-1O2-
+ CxH2xO2 then:
2 3 2. 2 3
. .
xTotal x x x x
C +F H CO →C H O +F HCO− (6.6)
7) The relationship between the soluble (H2CO3) and gaseous CO2 is described by the following:
H CO2 3←→rh H O2 +CO2( )g (6.7) With rh being the combined association/dissociation rate of carbon dioxide dependant on Henry’s Law (Appendix 6.1, Section 6.1.3)
8) Urea is the sole nitrogen source for AD-FTRW. In the operational pH range (6.5-7.5) >99%
of the ammonia will be in saline (NH4+
) form, thus upon entering the reactor urea decomposes as follows:
2 2 2 3 2 2 4 3
(NH ) CO+2H CO +H OrUrea→CO +2NH ++2HCO− (6.8)
9) Similarly it is assumed that the NaOH dosed for pH control is converted to H2CO3*
Alkalinity upon entering the system:
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NaOH + H2CO3→Na++H2O+HCO3-
(6.9)
Under normal operating conditions, the carbonate system concentration is significantly (10 times) larger than any other weak acid/base concentration; it will be assumed that this will be the primary proton source/sink.
10) The saline ammonia (NH4+) produced from urea decomposition needs to be de-protonated before it can be utilized by the FOGs:
NH4+
+HCO3-
→NH3+H2CO3 (6.10) Both assumptions 8 and 9 were proven experimentally. The assumptions made above will be applied in conjunction with the metabolic pathways to derive the dynamic AD-FTRW model stoichiometry.
Metabolic Pathways
In the steady state AD-FTRW model, the anabolic and catabolic pathways are linked through the E value since the yield and death rates are modeled in a single stoichiometric equation. Because the growth and death processes are modeled separately in the dynamic AD-FTRW model, the catabolic and anabolic rate equations are linked through the true organism yield, Y [mgCODBiomass/mgCODUtilized]. However, in the case of the dynamic model, an individual yield is assigned to each of the functional organism groups. From Figure 6.2 it can be noted that acetogenesis is subdivided into a further 5 functional organism groups to model the utilization of the wide range of SCFAs. Since the weak acid/base chemistry reactions happen significantly faster than the biological processes it was decided to incorporate these into the biological process steps, rather than using separate algebraic equations to solve them. This is one of the primary distinctions of the
|dynamic AD-FTRW model if compared to its predecessors. The derivation of the acetogenic biological processes in the dynamic AD-FTRW model is represented in Appendix 6.3; the remainder (acetoclastic methanogenesis, acidogenesis and hydrogenotrophic methanogenesis) is derived in much the same manner as presented in Appendix 5.3. The biological growth processes in the dynamic AD-FTRW model can be summarized as follows;
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• Acetogenesis of Hexanoic Acid (AcHx)
( ) ( )
( ) ( )
( ) ( ) ( )
6 12 2 6 11 2 2
2 3 4 4 8 2
2 4 2 2 3 5 7
5 1 5 10 34.
8. 8. 8.
5. 2. 5 5.
8. 8.
5 5. 10 10. 5. 8.
8. 8. 8.
acHx
Hx Hx AcHx
AcHx AcHx AcHx
Hx AcHx r AcHx
AcHx AcHx
AcHx AcHx Hx AcHx
AcHx AcHx AcHx
F F Y
C H O C H O H O
Y Y Y
F Y Y
H CO NH C H O
Y Y
Y Y F Y
C H O H HCO C H O
Y Y Y
−
+
−
− −
+ +
+ −
+ + →
− − −
+ + + + 2N
(6.11)
• Acetogenesis of Valeric Acid (AcVa)
( ) ( )
( ) ( )
( ) ( ) ( )
5 10 2 5 9 2 2
2 3 4 3 6 2
2 4 2 2
10 1 10 20 59.
13. 13. 13.
10. 2. 10. 1
13. 13.
10 1 20 1 10. 13.
13. 13. 13.
acVa
Va Va AcVa
AcVa AcVa AcVa
Va AcVa r AcVa
AcVa AcVa
AcVa AcVa Va AcVa
AcVa AcVa AcVa
F F Y
C H O C H O H O
Y Y Y
F Y Y
H CO NH C H O
Y Y
Y Y F Y
C H O H
Y Y Y
−
+
− −
+ +
+ −
+ + →
− − −
+ + + HCO3−+C H O N5 7 2
(6.12)
• Acetogenesis of Butyric Acid (AcBu)
( ) ( )
( ) ( ) ( )
4 8 2 4 7 2 2 2 3 4
2 4 2 2 3 5 7 2
1 2 5.
2 1 2 1
AcBu
Bu Bu AcBu Bu
AcBu AcBu AcBu AcBu
AcBu AcBu Bu AcBu
r
AcBu AcBu AcBu
F F Y F
C H O C H O H O H CO NH
Y Y Y Y
Y Y F Y
C H O H HCO C H O N
Y Y Y
− +
−
− −
+ + + +
− − −
→ + + +
(6.13)
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• Acetogenesis of Propionic Acid (AcPr)
( ) ( )
( ) ( ) ( )
( )
Pr Pr Pr Pr
3 6 2 3 5 2 2 4
Pr Pr Pr
Pr Pr Pr Pr
2 4 2 2 2 3
Pr Pr Pr
Pr Pr
3 5 7 2
Pr
10 1 10 30 51.
7. 7. 7.
10 1 30 1 10 8. 10.
7. 7 7.
10. 7.
7.
Ac rAc
Ac Ac Ac
Ac Ac Ac
Ac Ac Ac
Ac Ac
F F Y
C H O C H O H O NH
Y Y Y
Y Y Y F
C H O H H CO
Y Y Y
F Y
HCO C H O N Y
− +
−
− −
+ + + →
− − − −
+ +
+ − +
(6.14)
• Acetogenesis of Ethanol (AcEtOH)
( )
( ) ( )
3 2 2 4 3 2 3
2 4 2 2 5 7 2
15 19.
5 3
3. 3. 2
5 1 10 1
3. 3.
AcEtOH
AcEt r
AcEt AcEt
AcEt acEt
AcEt AcEt
CH CH OH Y H O NH HCO H CO
Y Y
Y Y
C H O H C H O N
Y Y
+ −
+ − + + + →
− −
+ +
(6.15)
Unlike longer chain SCFAs, acetic acid is metabolized directly to CH4 and CO2. Acetoclastic methanogenesis can be described as follows:
• Acetoclastic Methanogenesis (Am)
( ) ( )
( ) ( ) ( )
2 4 2 2 3 2 4 2
4 2 3 5 7 2
5 1 5 5 11
2. 2. 2.
5 5 5 3. 5 5 2
2 2. 2
Ac Ac Am rAcEt
Am Am Am
Am Am Ac Am
Am Am Am
F F Y
C H O C H O NH H O
Y Y Y
Y Y F Y
CH H CO C H O N
Y Y Y
− +
− −
+ + + →
− − − −
+ + +
(6.16)
Another biodegradable fraction in the feed that directly undergoes methanogenesis is Methanol. This substrate is directly converted to methane as follows:
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• Methanogenesis of Methanol (Mm)
( ) ( ) ( )
3 4 3
4 2 3 2 5 7 2
10 3.
30 1 5 9. 5 33.
12 6. 6.
rAcEt
Mm
Am Mm Mm
Am Mm Mm
CH OH NH HCO Y
Y Y Y
CH H CO H O C H O N
Y Y Y
+ −
+ + →
− − +
+ + +
(6.17)
Hydrogen gas produced by acetogenesis and acidogenesis is the single most inhibitory metabolic product produced in the anaerobic process. Apart from the negligible fraction of the dissolved hydrogen gas that escapes to the gas phase, the only means of reducing the dissolved hydrogen concentration is via the following:
• Hydrogenotrophic Methanogenesis (Hm)
( )
( ) ( )
2 4 3 2 3
4 2
10 5 3.
10
7. 20.
10 1 10 15 11.
4 20.
Hm rAcEt
Hm Hm
Am Am
Hm
Am Am
H NH HCO Y H CO
Y Y
Y Y
CH H O Z
Y Y
+ − −
+ + + →
− +
+ +
(6.18)
Parallel to microbial growth, endogenous respiration (organism death) occurs. The biodegradable fraction of the dead organism mass is known as biodegradable particulate COD (Sbp). Sbp is first hydrolyzed and then forms the substrate for the last of the FOGs, namely acidogenesis. Traditionally hydrolysis and acidogenesis are modeled as two separate processes in anaerobic digestion (Batstone et al., 2002 & Sötemann et al., 2005). However in AD-FTRW, the only biodegradable particulates (Sbp) that enter the system are those produced from dead organism mass. Thus it will be assumed that Sbp has the same composition as that of active biomass i.e. C5H7O2N. The contribution of this organism group is so small that it was decided to model both hydrolysis and acidification in a single step:
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• Hydrolysis and Acidogenesis (AD)
( )
( ) ( ) ( ) ( )
*
5 2 2
**
2 4 2 2 3 4 5 2
1 5 1
2 1 2 1 1 1
ad rAd
z
ad ad
ad ad ad ad
z
ad ad ad ad
C H O N Y H O
Y Y
Y Y Y Y
C H O H HCO NH C H O N
Y Y Y Y
− +
+ − →
− + − −
+ + + +
(6.19)
Where
C5H7O2N* = Biodegradable particulates (Sbp) of dead organism mass C5H7O2N** = Active acidogenic biomass (ZAd) produced from Sbp utilization Organism Death
Organism death is described by the theory of endogenous respiration (Dold et al., 1980). This theory states that parallel to organism growth, a fraction of the biomass dies at a continuous rate. A fraction (f) of this ‘dead biomass’ is regarded as unbiodegradable and is known as endogenous residue (Ze).
The remainder (1-f) of the dead biomass is biodegradable particulate (Sbp) and is hydrolyzed to form substrate for acidogenesis (Eq 6.20). The death process can be described as follows:
(
1)
. .rdj
Aj bp e
Z → −f S +f Z (6.20)
Where
ZAj = The active mass of the jth FOG [mol/L]
rdj = Rate of organism death of the jth FOG. [mol/L.d]