Heat Gen & Carbon Conversion

Degree of reduction

γ

Inorganic molecule degree of reduction

γ

Degree of reduction for carbon-containing molecules

γ = number of equivalents of electrons available for transfer to oxygen / gram-atom of carbon

Degree of reduction elements

Carbon = 4; Sulfur = 6; Hydrogen = 1; Oxygen = -2; Phosphorous = 5; Nitrogen = -3

Gram-atom of carbon

1 gram-atom of carbon-12 is 12 grams of carbon-12

General cell-growth mass balance

CHmOn + a O2 + b NH3 → c C HαOβNδ + d C HxOyNz + e H2O + f CO2

Calculating degree of reduction

Calculate γ for every compound in the cell balance

Small molecules in degree of reduction calculation

Start with the small molecules and use the number of valence electrons for inorganics

Degree of reduction of O2

γ = 2(-2) = -4

Degree of reduction of NH3

γ = -3 + (3×1) = 0

Degree of reduction of H2O

γ = 2(1) + (-2) = 0

Degree of reduction of CO2

γ = 4 + (2×-2) = 0

Degree of reduction of substrate

γs = 4 + m − 2n

Degree of reduction of biomass

γb = 4 + α − 2β − 3δ

Degree of reduction of product

γp = 4 + x − 2y − 3z

Degree of reduction of glucose

Glucose is C6H12O6

Glucose degree of reduction calculation

γ = (6×4 + 12×1 + 6×-2) / 6

Glucose degree of reduction

γ = 4

Meaning of glucose degree of reduction

γ = 4 moles of electrons available for transfer to oxygen per gram-atom of carbon

Decane degree of reduction

Decane is C10H22

Decane degree of reduction calculation

γ = (10×4 + 22×1) / 10

Decane degree of reduction

γ = 6.2

Decane clicker question answer

The reductance of decane (C10H22) is 6.2

Energy balance for aerobic growth

Q0 kγs − aQ0 4 = cQ0φγb + dQ0wγp

Heat generation during cell growth

Heat evolves during cell growth due to transfer of electrons to oxygen

Q0

Q0 is the heat of reaction per mole equivalent of electrons going to O2

Q0 approximate value

Q0 ≈ 27 kcal/mol e−

Energy per gram-atom carbon

γ = number of equivalents of electrons available for transfer to oxygen / number of carbon atoms

Energy balance use of degree of reduction

γi is a linear combination of elemental balances and may be convenient to use in place of an elemental balance

Water and reductance balance

Moles of water produced (e) are not easy to measure

Energy per gram-atom carbon in species i

γiQ0 is the energy (kcal) per gram-atom carbon in species i

Mass and energy balance equation 1

Carbon: 1(k) = cφ + dw + f

Mass and energy balance equation 2

Nitrogen: b = cδ + dz

Mass and energy balance equation 3

Reductance balance: kγs − a/4 = cφγb + dwγp

Mass and energy balance equation 4

Respiratory quotient: f/a

Prediction of heat evolution

Energy balance for aerobic growth: Q0kγs − aQ0/4 = cQ0φγb + dQ0wγp

Fraction of available electrons incorporated into product

ξP = dγp / kγs

Fraction of available electrons incorporated into biomass

ξb = cφγb / kγs

Fraction of available electrons transferred to O2

ε = a4 / kγs

Energy stored in biomass

Fraction of substrate’s energy stored in the biomass

Energy stored in product

Fraction of substrate’s energy stored in the product

Energy released as heat

Fraction of substrate’s energy released as heat

Energy fraction relationship

The fractions of substrate energy stored in biomass

Normalized energy balance

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

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

Heat of combustion = 104 kcal/mol O2 = 435.24 kJ/mol O2

Relationship for heat of combustion

1/YH = ΔHC / YX/S − ΔHS

Growth heat rate

QGr = Hbiomass × rGr

Heat of combustion relationship

ΔHS is proportional to the sum of metabolic heat (1/YH) and heat of combustion of cell material (ΔHC)

Respiration yielding cells

Respiration yielding cells has an oxygen requirement of 1/YH in the combustion-based calculation

Substrate combustion

Substrate + oxygen → CO2 + H2O (+ N2)

Cell combustion

Cells + oxygen → CO2 + H2O + N2

Combustion of substrate

ΔHS is the heat of combustion of 1 mole of substrate

Combustion of cells

ΔHC is the heat of combustion of 1 mole of cells

Rate of heat evolution from growth

QGr = Hbiomass rGr

Heat generation calculation

YH can be calculated using the heat of combustion of substrate

Heat of combustion calculation using oxygen

Use 435.24 kJ per mole O2 and the number of moles of O2 needed to combust the substrate or cells

Substrate oxygen requirement for combustion

The number of moles of O2 needed to combust 1 mole of substrate can be calculated from the combustion equation

Cell oxygen requirement for combustion

The number of moles of O2 needed to combust 1 mole of cells can be calculated from the combustion equation

Heat-generation example

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

Heat-generation example cell equation

C6H12O6 + a O2 + b NH3 → c C6H10NO3 + d C2H5OH + e H2O + f CO2

Molecular weight of yeast in example

144 g/mol

Molecular weight of ethanol in example

46 g/mol

Molecular weight of glucose in example

180 g/mol

Heat-generation example nitrogen balance

b = c

Heat-generation example carbon balance

6 = 6c + 2d + f

Heat-generation example oxygen balance

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

Heat-generation example hydrogen balance

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

Converting YX/S to mol X/mol S

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

Stoichiometric coefficient c in heat-generation example

c = 0.24

Stoichiometric coefficient b in heat-generation example

b = c = 0.24

Relationship between growth/product rates and stoichiometric coefficients

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

Converting cell and ethanol rates to molar rates

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

Stoichiometric coefficient d

d = 1

Heat-generation example carbon calculation

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

Heat-generation example hydrogen calculation

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

Heat-generation example oxygen calculation

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

Final heat-generation stoichiometric equation

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

Heat-generation example heat of combustion values

Heat of combustion of oxygen = 435.24 kJ/mol O2; glucose = 14.508 kJ/g glucose; cells = 21.1574 kJ/g cells

Cell yield in heat calculation

YX/S = 0.192 g cells/g S

Substrate heat of combustion

ΔHS = 14.508 kJ/g S

Cell heat of combustion

ΔHC = 21.1574 kJ/g cells

Metabolic heat yield calculation

HGr = 0.01838 kJ/g cells

Cell growth rate

rcell = 0.306 g cells/L·h

Rate of heat evolution

QGr = 16.65 kJ/L·h