Klebsiella aerogenes Oxygen Requirement from Glycerol: 04-BS-13, December 2015, Q1

Question 1: Klebsiella aerogenes Oxygen Requirement from Glycerol (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Biomass yield from glycerol, $Y_{XS}$ (dry wt., incl. ash)	0.40 g/g
Ash content of biomass	8%
Biomass formula (ash-free)	CH_1.75O_0.43N_0.22, MW = 23.71 g/cmol
MW glycerol	92 g/mol
$\gamma_S$ (glycerol) / $\gamma_B$ (biomass)	4.67 / 4.23

Find. The oxygen requirement of the culture, in mass terms (g O₂ per g glycerol consumed).

Approach. The yield and ash correction fix the biomass coefficient $c$; the nitrogen balance then gives $b$, and the C/H/O atom balances solve the remaining unknowns $d$, $e$, $a$. The available-electron (degree-of-reduction) balance provides an independent cross-check on $a$ that does not depend on the C/H/O bookkeeping at all.

Biomass coefficient $c$ from the yield and ash correction. Per mole glycerol (92 g), dry biomass formed $=Y_{XS}(92)=0.40(92)=36.8$ g. Only 92% of this is the ash-free CH_1.75O_0.43N_0.22 material (the ash carries no C/H/N/O balance obligation): ash-free mass $=36.8(0.92)=33.856$ g, so $$c=\frac{33.856}{23.71}=\boxed{1.428\ \text{cmol biomass/mol glycerol}}.$$
Nitrogen balance → $b$. NH₃ is the only N source and biomass the only N sink: $$b=0.22c=0.22(1.428)=0.314\ \text{mol NH}_3\text{/mol glycerol}.$$
Carbon balance → $d$. Glycerol supplies 3 C per mole, split between biomass and CO₂: $$3=c+d \;\Rightarrow\; d=3-1.428=1.572\ \text{mol CO}_2\text{/mol glycerol}.$$
Hydrogen balance → $e$. $$8+3b=1.75c+2e \;\Rightarrow\; e=\frac{8+3(0.314)-1.75(1.428)}{2}=\boxed{3.222\ \text{mol H}_2\text{O/mol glycerol}}.$$
Oxygen balance → $a$ (atom-balance route). $$3+2a=0.43c+2d+e \;\Rightarrow\; a=\frac{0.43(1.428)+2(1.572)+3.222-3}{2}=\boxed{1.990\ \text{mol O}_2\text{/mol glycerol}}.$$
Cross-check via the available-electron balance. With NH₃ at zero reference degree of reduction, the substrate's available electrons split between biomass and the O₂ sink (no other product): $$\gamma_S\,(3\ \text{C atoms})=\gamma_B\,c+4a \;\Rightarrow\; a=\frac{4.67(3)-4.23(1.428)}{4}=1.992\ \text{mol O}_2\text{/mol glycerol}.$$ This agrees with the atom-balance value of 1.990 to within rounding of the given $\gamma$ values — a strong, independent confirmation.
Convert to mass terms. Using $a=1.990$ mol O₂/mol glycerol (MW O₂ = 32): $$\text{O}_2\ \text{requirement}=\frac{1.990(32)}{92}=\boxed{0.692\ \text{g O}_2\text{/g glycerol consumed}}.$$

Quantity	Result
Biomass coefficient, $c$	1.428 cmol/mol glycerol
NH₃ coefficient, $b$	0.314 mol/mol glycerol
CO₂ / H₂O coefficients, $d$, $e$	1.572 / 3.222
O₂ coefficient, $a$ (atom / electron balance)	1.990 / 1.992 mol/mol glycerol
Oxygen requirement (mass)	0.692 g O₂/g glycerol

Topic: Elemental and electron (degree-of-reduction) balances for aerobic fermentation stoichiometry.

Key relations: atom balances on C, H, N, O; available-electron balance $\gamma_S n_S=\gamma_B c+4a$ (no product) or $=\gamma_Bc+\gamma_Pf+4a$ (with product); degree of reduction $\gamma=4n_C+n_H-2n_O-3n_N$ per C-mole, referenced to CO₂, H₂O, NH₃.

Why this works: with one unknown empirical biomass formula and no metabolic product, the four atom balances exactly determine the four stoichiometric unknowns $b,c$ (given), $d,e,a$; the electron balance is a linear combination of those same balances, so it functions purely as an arithmetic audit, not a fifth independent equation.

Common pitfall: forgetting the ash correction — ash contributes to the measured dry-weight yield $Y_{XS}$ but carries no C/H/N/O, so only the ash-free fraction may be converted to cmol via the biomass formula's molecular weight.

Source: Shuler & Kargi, Bioprocess Engineering: Basic Concepts, Ch. 5 (elemental balances, degree of reduction).

Question 1 of 10: Klebsiella aerogenes Oxygen Requirement from Glycerol

Question 1: Klebsiella aerogenes Oxygen Requirement from Glycerol (20 marks)