Heat Gen and Carbon Conversion & Heat Gen Problems

Prediction of heat evolution

Energy balance for aerobic growth: Q0 kγs − aQ0/4 = cQ0φγb + dQ0wγp

Fraction of available electrons incorporated into biomass

ξb = cφγb/(kγs)

Fraction of available electrons incorporated into product

ξp = dγp/(kγs)

Fraction of available electrons transferred to oxygen

ε = a4/(kγs)

Energy fraction meanings

ξb is the fraction of substrate energy stored in biomass; ξp is the fraction stored in product; ε is the fraction released as heat

Normalized energy balance

Divide every term in the energy balance by the term for the substrate’s energy

Energy fractions relationship

The fractions of substrate energy stored in biomass

Metabolic heat

Metabolic heat sets cooling requirements for bioreactors that contain cells

YH

YH is metabolic heat evolved per cell mass produced

Heat of combustion of oxygen

104 kcal/mol O2 = 435.24 kJ/mol O2

Heat-generation relationship

1/YH = ΔHC/YX/S − ΔHS

Rate of heat evolution from growth

QGr = Hbiomass rGr

Heat of combustion of substrate

ΔHS is proportional to the sum of metabolic heat

Respiration yielding cells

Respiration yielding cells have 1/YH kJ/g cell

Substrate combustion

Substrate (1 mole) + k O2 → m CO2 + n H2O

Substrate combustion coefficient k

k is the moles of O2 needed to combust 1 mole of substrate

Cell combustion

Cells (1 mole) + t O2 → u CO2 + w H2O + yN2

Cell combustion coefficient t

t is the moles of O2 needed to combust 1 mole of cells

Heat of combustion from oxygen

Use 435.24 kJ/mol O2 together with the moles of O2 needed for combustion and molecular weight to calculate ΔHS or ΔHC

Yeast heat-generation example

A yeast is grown on glucose with YX/S = 0.192 g cells/g S

Yeast growth equation

C6H12O6 + aO2 + bNH3 → cC6H10NO3 + dC2H5OH + eH2O + fCO2

Yeast molecular weight

144 g/mol

Ethanol molecular weight

46 g/mol

Glucose molecular weight

180 g/mol

Nitrogen balance in yeast example

b = c

Carbon balance in yeast example

6 = 6c + 2d + f

Oxygen balance in yeast example

6 + 2a = 3c + d + e + 2f

Hydrogen balance in yeast example

12 + 3b = 10c + 6d + 2e

Converting yeast YX/S

0.192 g X/g S converts to 0.24 mol X/mol S

Stoichiometric coefficient c in yeast example

c = 0.24

Stoichiometric coefficient b in yeast example

b = c = 0.24

Growth/product rate ratio

The ratio of rates of cell growth to ethanol production is proportional to g cells/g ethanol

Molar rate conversion in yeast example

Convert g cells and g ethanol to moles to obtain the ratio of their stoichiometric coefficients

Stoichiometric coefficient d in yeast example

d = 1

Carbon calculation in yeast example

6 = 6(0.24) + 2(1) + f

Hydrogen calculation in yeast example

12 + 3(0.24) = 10(0.24) + 6(1) + 2e

Oxygen calculation in yeast example

6 + 2a = 3(0.24) + 1 + 2.16 + 2(2.56)

Final yeast stoichiometric equation

C6H12O6 + 1.5O2 + 0.24NH3 → 0.24C6H10NO3 + C2H5OH + 2.16H2O + 2.56CO2

Yeast example substrate heat of combustion

ΔHS = 14.508 kJ/g S

Yeast example cell heat of combustion

ΔHC = 21.1574 kJ/g S

Yeast example metabolic heat yield

YH = 0.01838 kJ/g cells

Yeast example cell growth rate

rcell = 0.306 g cells/L·hr

Yeast example heat evolution rate

QGr = 16.65 kJ/L·h

Corynbacterium example

Corynbacterium (C8H17O7N) is grown in a batch reactor for production of threonine (C4H9O3N) with glucose as the carbon source

Corynbacterium growth rate

0.3 g cells/L-h

Corynbacterium threonine production rate

0.4 g threonine/L-h

Corynbacterium energy condition

Half of the energy in the carbon source is released as heat

Corynbacterium growth equation

aC6H12O6 + bNH3 + cO2 → C8H17O7N + dC4H9O3N + eH2O + fCO2

Corynbacterium carbon balance

6a = 8 + 4d + f

Corynbacterium nitrogen balance

b = 1 + d

Corynbacterium hydrogen balance

12a + 3b = 17 + 9d + 2e

Corynbacterium oxygen balance

6a + 2c = 7 + 3d + e + 2f

Corynbacterium rate-to-stoichiometry relationship

The ratio of molar growth and production rates equals the ratio of the corresponding stoichiometric coefficients

Corynbacterium product coefficient

d = 2.68

Corynbacterium ammonia coefficient

b = 3.68

Corynbacterium hydrogen relationship

e = 6a − 15.04

Degree of reduction of glucose in Corynbacterium example

γS = 4

Degree of reduction of biomass

γb = 4.214

Degree of reduction of threonine

γP = 4.286

Degree of reduction of oxygen

γO2 = -4

Degree of reduction of NH3

γNH3 = 0

Degree of reduction of H2O

γH2O = 0

Degree of reduction of CO2

γCO2 = 0

Reductance balance

6aγS + bγNH3 + cγO2 = 8γb + 4dγP + eγH2O + fγCO2

Reductance balance note

Each term must also be multiplied by the number of carbons in the molecule

Oxygen coefficient from reductance balance

c = 3a

Corynbacterium oxygen coefficient

c = 18.72

Corynbacterium glucose coefficient

a = 6.24

Corynbacterium carbon dioxide coefficient

f = 18.72

Corynbacterium water coefficient

e = 22.4

Final Corynbacterium stoichiometric equation

6.24C6H12O6 + 3.68NH3 + 18.72O2 → C8H17O7N + 2.68C4H9O3N + 22.4H2O + 18.72CO2

Fraction of glucose energy converted to biomass

0.214

Fraction of glucose energy converted to product

0.286

Fraction of glucose energy converted to heat

0.5

Energy fraction relationship in Corynbacterium example

Biomass fraction + product fraction + heat fraction = 1

Cell yield based on glucose

YX/S = biomass molecular weight/(a × substrate molecular weight)

Corynbacterium YX/S

YX/S = 0.214 g X/g S

Cell yield based on oxygen

YX/O2 = c × biomass molecular weight/(oxygen molecular weight)

Corynbacterium YX/O2

YX/O2 = 0.399 g X/g O2

Glucose combustion equation

C6H12O6 + O2 → CO2 + H2O

Glucose CO2 coefficient during combustion

6 mol CO2 are produced per mole of glucose

Glucose H2O coefficient during combustion

6 mol H2O are produced per mole of glucose

Glucose oxygen requirement for combustion

6 mol O2 are required to combust 1 mol C6H12O6

Glucose heat of combustion

ΔHS = 14.508 kJ/g substrate

Corynbacterium cell combustion equation

C8H17O7N + O2 → 8CO2 + 8.5H2O + 0.5N2

Corynbacterium cell CO2 coefficient

8 mol CO2 are produced per mole of cells

Corynbacterium cell H2O coefficient

8.5 mol H2O are produced per mole of cells

Corynbacterium cell oxygen requirement

8.75 mol O2 are required to combust 1 mol of cells

Corynbacterium cell heat of combustion

ΔHC = 15.935 kJ/g cells

Corynbacterium substrate heat of combustion

ΔHS = 14.508 kJ/g substrate

Corynbacterium metabolic heat yield

YH = 0.0193 kJ/g cells

Corynbacterium cell growth rate

0.3 g cells/L-h