Problem Statement: Growth of yeast on ethanol can be described by the following reaction:
CH3CH2OH + a O2 + b NH3 ---------> c C6H10NO3 + d H2O + e CO2 + f CH3COOH (ethanol) (yeast) (acetic acid)The nitrogen source comes from ammonia, which in practice is derived from ammonium sulfate (NH4)2SO4. The fermentor is continuously monitored with an oxygen gas sensor and a carbon dioxide gas sensor. From these measurements, one can construct oxygen uptake rate (OUR=moles of O2 consumed per unit time) and carbon dioxide evolution rate (CER=moles of CO2 produced per unit time), which in turn yields the respiratory quotient (RQ=CER/OUR). Furthermore, pH is controlled at a constant level (pH=7) with the addition of either NH4OH (which is alkaline) or HCl (which is acid) as needed.
Given the RQ and the rate of addition of acid/base Q, find the biomass yield Yx (i.e., the number of grams of biomass produced per gram of ethanol consumed) in terms of RQ and Q. Do you expect acid or base to be added during fermentation?
Estimate the rate of heat evolution in terms of RQ and Q, taking the heat of reaction to be 27 kcal per mole of oxygen-released electron. In practice, this gives you the rate of heat removal. If you design a fermentor, make sure you include adequate cooling equipment. Set up the equations that will lead to a solution.
The dissociation constant for acetic acid is [H+]*[Ac-]/[AcH]=1.85*10-5M.
Species Formula MW
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Ethanol C2H6O 46
Yeast C6H10NO3 144
Ammonium Sulfate (NH4)2SO4 132
Acetic Acid C2H4O2 60
Solution:
Elemental balance for the stoichiometry:
Solve the above six equations for six unknowns (a, b, c, d, e, and f).
These stoichiometric coefficients are the various yields from the substrate
expressed on a molar basis.
Finally,
CH3CH2OH + a O2 + b NH3 ----> c C6H10NO3 + d H2O + e CO2 + f CH3COOH
C: 2 = 6*c + 1*e + 2*f
H: 6 + 3*b = 10*c + 2*d + 4*f
N: 1*b = 1*c
O: 1 + 2*a = 3*c + 1*d + 2*e + 2*f
In addition, RQ provides the fifth relationship.
RQ=CER/OUR=e/a
The sixth relationship comes from the pH control.
Base must be added to neutralize the acid that is formed via two routes:
1) when ammonia is taken up and leaves a H+ behind for each NH3
(NH4)2SO4 ----> 2NH3 + 2H+ + SO42-
and 2) when acetic acid is produced, taking into account that acetic is a
weak acid whose extent of dissociation depends on pH.
Thus, the sixth equation is the sum of these two acid contributing factors.
Q/OUR=(b+f)/a
where Q is the number of moles of base added per unit time.
Yx=c*MWyeast/MWethanol=c*144/46=3.13*c
Once we set up the governing equations, let Mathcad rip.
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