Effect of Cell Growth on Oxygen Demand — Ethanol to Acetic Acid: 04-BS-13, December 2013, Q2

Question 2: Effect of Cell Growth on Oxygen Demand — Ethanol to Acetic Acid (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.

Quantity	Value
Biomass yield, $Y_{XS}$	0.14 g biomass/g ethanol
Product yield, $Y_{PS}$	0.92 g acetic acid/g ethanol
Degrees of reduction $\gamma$ (per C)	ethanol 6, acetic acid 4, biomass 4.2
MW ethanol / acetic acid / biomass (ash-free)	46 / 60 / 24.6 g/(c)mol
No-growth reference reaction	1 mol O₂ per mol ethanol (given equation)

Find. The O₂ demand per mole of ethanol consumed with growth, compared against the no-growth reference reaction, and whether growth increases or decreases the oxygen requirement.

Approach. The no-growth equation is a complete electron balance on its own (all of ethanol's available electrons end up in acetic acid or O₂). With growth, some of those electrons are diverted into biomass instead, so a degree-of-reduction (available-electron) balance — not a full atom balance, since the given yields do not by themselves close C/H/O/N exactly — is the direct route to the actual O₂ requirement: available electrons from ethanol = electrons fixed in biomass + electrons fixed in product + electrons transferred to O₂ (4 electron-equivalents per mole O₂).

Biomass and product formed per mole of ethanol (46 g). $$m_X=Y_{XS}(46)=0.14(46)=6.44\ \text{g}\ \Rightarrow\ c=\frac{6.44}{24.6}=0.2618\ \text{cmol biomass},$$ $$m_P=Y_{PS}(46)=0.92(46)=42.32\ \text{g}\ \Rightarrow\ n_P=\frac{42.32}{60}=0.7053\ \text{mol acetic acid}.$$
Available electrons supplied by 1 mol ethanol (2 carbons, $\gamma=6$/C). $$\text{AE}_{\text{ethanol}}=6\times2=12\ \text{e}^-\text{-equivalents.}$$
Electrons routed to biomass and to product. $$\text{AE}_X=\gamma_X\cdot c=4.2(0.2618)=1.100,\qquad \text{AE}_P=\gamma_P\cdot(2\,n_P)=4(2)(0.7053)=5.643.$$ (Acetic acid has 2 carbons per mole, so its electron content is $\gamma_P$ times its carbon-mole quantity, $2n_P$.)
Close the balance on O₂. Remaining electrons must transfer to oxygen, at 4 electron-equivalents per mole O₂: $$4\,a_{\text{growth}} = 12 - 1.100 - 5.643 = 5.257\ \Rightarrow\ a_{\text{growth}}=\boxed{1.314\ \text{mol O}_2/\text{mol ethanol}}.$$
Compare with the no-growth reference. The given equation $\text{C}_2\text{H}_6\text{O}+\text{O}_2\rightarrow\text{C}_2\text{H}_4\text{O}_2+\text{H}_2\text{O}$ is itself a closed electron balance ($12=8+4$, i.e. all ethanol converts to acetic acid, $a_{\text{no growth}}=1$): $$\frac{a_{\text{growth}}-a_{\text{no growth}}}{a_{\text{no growth}}}\times100=\frac{1.314-1}{1}\times100=\boxed{+31.4\%}.$$

Quantity	Result
Biomass formed	0.262 cmol/mol ethanol
Acetic acid formed	0.705 mol/mol ethanol
Electrons to biomass / product	1.100 / 5.643 e⁻-eq
O₂ demand, with growth	1.314 mol O₂/mol ethanol
O₂ demand, no growth (given rxn)	1.000 mol O₂/mol ethanol
Effect of growth on O₂ demand	increases it by 31.4%

Concept & Theory

Topic. Degree-of-reduction (available-electron) balances for partial-oxidation product formation with concurrent growth.

Key relations. $\gamma=4c+h-2o-3n$ per mole of carbon (reference states CO₂, H₂O, NH₃ $=0$); electron balance $\text{AE}_S=\text{AE}_X+\text{AE}_P+4\,n_{\text{O}_2}$, since O₂ accepts exactly 4 electron-equivalents per mole (two O atoms, 2 e⁻ each).

Why this works. Growth diverts a fraction of the substrate's carbon (and its electrons) away from the target product into new cell material. Because biomass's degree of reduction (4.2) is lower than ethanol's (6) but the product route it displaces (acetic acid, $\gamma=4$) was itself already the electron sink that balanced the no-growth reaction, less product is formed per mole of substrate when growth occurs — and the electrons that would have gone into that "missing" product must instead be respired to O₂. The net effect is counter-intuitive at first glance (diverting substrate to biomass, itself an electron sink, still increases total oxygen demand) but follows directly once the balance is worked mole-for-mole.

Common pitfall. Assuming that because biomass "absorbs" some of the substrate's electrons, oxygen demand must fall. The correct comparison is against how much product is no longer formed (and hence how many electrons are no longer parked in that product) — not against the biomass sink in isolation. A second pitfall is applying $\gamma$ values per mole of the whole molecule instead of per mole of carbon; multiplying by the number of carbon atoms in each species is a required step.

Source. Shuler & Kargi, Bioprocess Engineering: Basic Concepts, Ch. 6 (Degree of reduction and available-electron balances).

Question 2 of 10: Effect of Cell Growth on Oxygen Demand — Ethanol to Acetic Acid

Question 2: Effect of Cell Growth on Oxygen Demand — Ethanol to Acetic Acid (20 marks)