Baker's Yeast Fermenter — Rate of Heat Removal: 04-BS-13, December 2013, Q3

Question 3: Baker's Yeast Fermenter — Rate of Heat Removal (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.

Check: assumes the standard bioprocess-engineering rule of thumb of ≈460 kJ of metabolic heat released per mole of O₂ consumed during aerobic growth (Cooney's correlation) — the exam does not supply a heat-of-reaction value directly, so this textbook constant (Shuler & Kargi, Ch. 6) is taken as given.

Quantity	Value
Fermenter volume, $V$	50 000 L
Biomass yield, $Y_{XS}$	0.5 g/g (dry, incl. 5% ash)
Specific growth rate, $\mu$	0.45 h⁻¹
Yeast concentration, $X$	10 g/L
MW sucrose / biomass (ash-free)	342 / 25.01 g/(c)mol
Heat evolved per mole O₂ consumed	≈460 kJ/mol (assumption, see callout)

Find. The rate of heat removal (kW, or kJ/h) needed to hold the fermenter at constant temperature.

Approach. First close the stoichiometric equation (C, H, N, O atom balances, with $Y_{XS}$ and the ash correction fixing $c$) to get the O₂ demand per mole sucrose consumed. Then convert the given growth rate and cell concentration to a volumetric substrate-consumption rate via $Y_{XS}$, scale by the O₂ stoichiometric coefficient and the fermenter volume to get a total O₂ uptake rate, and finally apply the heat-per-mole-O₂ rule (since at steady temperature, all metabolic heat generated must be removed).

Biomass coefficient $c$ (ash-corrected, per mole sucrose). Dry biomass per mole sucrose $=Y_{XS}(342)=171$ g; ash-free $=171(0.95)=162.45$ g: $$c=\frac{162.45}{25.01}=\boxed{6.495\ \text{cmol biomass/mol sucrose}}.$$
N, C, H balances → $a$, $d$, $e$. $$a=0.17c=1.104,\qquad d=12-c=5.505,$$ $$e=\frac{22+3a-1.83c}{2}=\frac{22+3.312-11.887}{2}=6.713.$$
Oxygen balance → $b$. $$2b = 0.55c+2d+e-11 = 3.572+11.009+6.713-11=10.295\ \Rightarrow\ b=\boxed{5.147\ \text{mol O}_2/\text{mol sucrose}}.$$
Volumetric growth and substrate-consumption rates. $$r_X=\mu X = 0.45(10)=4.5\ \text{g biomass/(L}\cdot\text{h)},\qquad r_S=\frac{r_X}{Y_{XS}}=\frac{4.5}{0.5}=9.0\ \text{g sucrose/(L}\cdot\text{h)}=\frac{9.0}{342}=0.02632\ \text{mol/(L}\cdot\text{h)}.$$
Volumetric and total O₂ consumption rate. $$r_{\text{O}_2}=b\cdot r_S=5.147(0.02632)=0.1355\ \text{mol O}_2/\text{(L}\cdot\text{h)},$$ $$\dot n_{\text{O}_2}=r_{\text{O}_2}\cdot V = 0.1355(50\,000)=\boxed{6773\ \text{mol O}_2/\text{h}}.$$
Heat generation (= heat removal) rate. $$\dot Q = \dot n_{\text{O}_2}\times 460\ \frac{\text{kJ}}{\text{mol O}_2} = 6773(460)=3.116\times10^6\ \text{kJ/h} = \boxed{865\ \text{kW}}.$$

Quantity	Result
Biomass coefficient, $c$	6.495 cmol/mol sucrose
O₂ coefficient, $b$	5.147 mol/mol sucrose
Substrate consumption rate	9.0 g/(L·h)
Total O₂ consumption rate	6773 mol O₂/h
Rate of heat removal required	3.12×10⁶ kJ/h ≈ 865 kW

Concept & Theory

Topic. Combined stoichiometric + energy balance for an aerobic batch/continuous fermenter operating at fixed temperature.

Key relations. $r_X=\mu X$ (specific growth rate); $r_S=r_X/Y_{XS}$; O₂ stoichiometric coefficient from atom balance; Cooney's rule $\dot Q_{\text{metabolic}}\approx460\ \text{kJ}$ per mole O₂ consumed, essentially independent of substrate or organism because it reflects a roughly constant heat release per available-electron transferred to oxygen.

Why this works. At constant fermenter temperature the accumulation term in the energy balance is zero, so all metabolic heat generated by respiration must be removed by the cooling system at the same rate it is produced — converting a mole-based O₂ uptake rate directly into a heat-removal duty via the empirical kJ/mol-O₂ constant sidesteps the need to know the full heat of combustion of every species individually.

Common pitfall. Forgetting to scale the per-litre O₂ uptake rate by the full 50 000 L fermenter volume before applying the heat-per-mole constant — the rate $r_{\text{O}_2}$ is intensive (per litre), but the cooling duty is an extensive quantity for the whole vessel.

Source. Shuler & Kargi, Bioprocess Engineering: Basic Concepts, Ch. 6 (Fermentation energy balances, heat of reaction estimation via O₂ consumption).

Question 3 of 10: Baker's Yeast Fermenter — Rate of Heat Removal

Question 3: Baker's Yeast Fermenter — Rate of Heat Removal (20 marks)