Cooling-Duty Estimate for an Alginate Fermentation: 04-BS-13, Undated paper, Q6

Question 6: Cooling-Duty Estimate for an Alginate Fermentation (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. Continuous (chemostat) fermenter, 4 L working volume, 28°C, NH₃ N-source; alginate production rate 5 kg/hr; agitator power input 1.5 kW; 4 mol H₂O consumed (per the assumed reaction basis, not separately quantified in the source).

Find. An estimate of the cooling duty (heat that must be removed to hold the fermenter at 28°C).

Approach. A fermenter's steady-state heat balance is $Q_{\text{cooling}} = Q_{\text{metabolic}} + Q_{\text{agitation}} - Q_{\text{sensible/evaporative}}$. All of the agitator's shaft work, however, is dissipated as heat in a well-mixed, mechanically-agitated vessel (essentially 100% conversion of shaft power to heat via viscous dissipation), so this term is fully quantifiable and is reported as the computable lower bound on the cooling duty.

Mechanical (agitation) heat load. Shaft work delivered by the impeller against a highly viscous broth is dissipated entirely as heat: $$Q_{\text{agitation}} = P_{\text{mix}} = \boxed{1.5\text{ kW}}$$
Metabolic heat load (not computable from the given data). The standard method would be $Q_{\text{metabolic}} = 460\ \text{kJ/mol O}_2 \times \text{OUR}$ (mol O₂/h), exactly as used in Q3(d); this requires either the O₂ uptake rate or a fully balanced sucrose→alginate+CO₂+H₂O+cells reaction with numeric O₂ and CO₂ coefficients. Only the alginate production rate (5 kg/hr) and "4 moles of water consumed" are given, with no accompanying O₂, CO₂, or sucrose coefficients — the source is missing the numeric data needed to close this term.
Total estimated cooling requirement. $$Q_{\text{cooling}} \geq Q_{\text{agitation}} = \boxed{1.5\text{ kW}}$$ (a lower bound; the true duty is this plus whatever metabolic heat the aerobic oxidation of sucrose generates, which the question's emphasis on "viscosity…considerable…energy input due to mixing…cannot be neglected" suggests is the intended teaching point — in highly viscous, mixing-intensive bioprocesses like alginate fermentation, agitation heat can rival or dominate metabolic heat, unlike in low-viscosity fermentations where metabolic heat is usually the larger term.)

Quantity	Value
Agitation heat load (computable)	1.5 kW
Metabolic heat load	not computable from given data (needs OUR or full balanced reaction)
Cooling requirement (estimate)	≥ 1.5 kW (agitation-dominated lower bound)

Check

The source gives no oxygen-uptake rate, air-flow rate, or fully balanced sucrose-to-alginate reaction (only "4 moles of water are consumed" with no matching O₂/CO₂/sucrose coefficients), so the metabolic heat term of the standard fermenter heat balance cannot be computed rigorously from the data as extracted. Per the "solve by method, don't fabricate a number" rule, only the mechanical (agitation) term is reported as a firm, computable figure; the question's own framing (repeated emphasis that mixing energy "cannot be neglected") is read as the intended teaching point — that for viscous polysaccharide fermentations, agitation heat is a first-order, not secondary, contributor to the cooling duty.

Topic: Fermenter heat balance — metabolic vs. mechanical (agitation) heat sources.

Key relations: $Q_{\text{cooling}}=Q_{\text{metabolic}}+Q_{\text{agitation}}-Q_{\text{sensible/evap}}$; $Q_{\text{metabolic}}\approx460$ kJ/mol O₂ consumed; $Q_{\text{agitation}}=P_{\text{shaft}}$ (viscous dissipation, effectively 100% conversion to heat).

Why this works: In a closed, liquid-full agitated vessel, shaft work has nowhere to go except into heating the fluid (no net PdV work, no significant kinetic energy accumulation), so it appears as heat exactly like metabolic heat and must be removed by the same cooling system.

Common pitfall: Assuming agitation heat is always negligible next to metabolic heat — true for low-viscosity fermentations (e.g. baker's yeast) but false for viscous polysaccharide broths (xanthan, alginate, dextran) where mixing power per unit volume is far higher.

Source: Shuler & Kargi, Bioprocess Engineering: Basic Concepts, Ch. 6, 9 (fermenter energy balance, non-Newtonian broth rheology and mixing power).

Question 6 of 9: Cooling-Duty Estimate for an Alginate Fermentation

Question 6: Cooling-Duty Estimate for an Alginate Fermentation (20 marks)