List of Symbols

List of Symbols

ENGLISH a a a a a at b bi A A, [A] Ad [A] eq [A]o B B [B] [B]o c c CA,G CA,I,G

0\,i,L CA,L Co2L C C C CA CAB CAE CAF CAL

[A]o + [P]o for reversible reaction, M primary reactant concentration empirical constant in Equation 4.12 interfacial area fractional approach to steady state, xlxs number of gram-atoms of hydrogen per C-mole of component / empirical constant in Equation 4.12 number of gram-atoms of oxygen per C-mole of component / frequency factor in Arrhenius equation concentration of reactant A, M Arrhenius preexponential factor for cell death, t i m e 1 equilibrium concentration of reactant A, M concentration of reactant A at t = 0, M cell-mass related production factor, t i m e 1 constant in Equation 5.23 concentration of reactant B, M concentration of reactant B at t = 0, M constant in Equation 3.30, K _ 1 s _ 1 empirical constant in Equation 4.12 concentration of A in bulk gas phase, mol/cm3 concentration of A in gas phase at gas-liquid interface, mol/cm3 concentration of A in liquid phase at gas-liquid interface, mol/cm3 concentration of A in bulk liquid phase, mol/cm3 concentration of oxygen in liquid given by Henry's law or related equilibrium equation or data for equilibrium with bulk gas phase, mol/cm3 concentration (or solubility in concentration units), mol/L integration constant constant in Equation 7.33 concentration of A concentration of A in bulk fluid concentration of A in effluent concentration of reactant A in feed concentration of A in liquid phase 121

122

Bioreactor Design Fundamentals

CAs CAsL Q CCRO 2 C* Co 2 CP ACP Cpi C\ C2 d d D D DAB De Di e e0 E EA EaA EaC Ed [E]d [E]„ [E], [E]o [E]o / F Fn Fw FE(%) g g g AgaWe

concentration of A at radius R (solid side of surface) concentration of A at liquid interface with particle inhibitor concentration critical oxygen concentration limiting inhibitor concentration saturation oxygen concentration in liquid for given oxygen partial pressure in gas phase concentration of product heat capacity change for stoichiometric reaction, J / ( m o l K ) heat capacity of stream / 0 2 concentration in water, mL(STP)/L 0 2 concentration in water, mass ppm empirical constant in Equation 4.12 density of a mycelial pellet heat dissipation rate, kW dilution rate, F/VR, t i m e 1 diffusivity (or diffusion coefficient) of A in medium B , cm 2 /s effective diffusivity of A in medium or pores, m 2 /s impeller diameter empirical constant in Equation 4.12 total enzyme concentration reactor effluent rate, volume/time activation energy in Arrhenius equation concentration of enzyme adsorbed on amorphous cellulose concentration of enzyme adsorbed on crystalline cellulose activation energy for cell death rate equation stoichiometric concentration of denatured enzyme active centers, mol/L stoichiometric concentration of normal (undenatured) enzyme active centers, mol/L active enzyme concentration at time / stoichiometric concentration of enzyme active centers total enzyme concentration empirical constant in Equation 4.12 reactor feed rate, volume/time volumetric feed rate to reactor n volumetric rate for cell product stream fermentation efficiency, Equation 5.16 empirical constant in Equation 4.12 maximum value of cell population, X, mass or number per unit volume local acceleration of gravity average Gibbs free energy change p e r mole of electrons transferred in substrate conversion to products, kJ/mol

List of Symbols Age, AgcN AG°' AGÏ Gi G2 G2A G2c Gr h h ΔΛ°, hi Δ//° Δ//° Δ// φ Δ//ί / / / /o It [I] JAz k k k!" k kA &B kAB &A kc kd kd kH kL kx ky

123

standard-state Gibbs free energy of combustion of component I, kJ/C-mole standard-state Gibbs free energy of combustion of nitrogen source to yield N 2 , kJ/mol Gibbs free-energy change for stoichiometric reaction at biological standard state and system temperature, J/mol standard-state Gibbs free-energy change for reaction 1 glucose concentration cellobiose concentration concentration of amorphous cellulose concentration of crystalline cellulose Grashof number, Equation 4.13 Hill coefficient Planck constant, J s standard-state heat of combustion of component /, kJ/C-mole enthalpy of stream /, kJ/mol standard-state enthalpy change, J/mol enthalpy change at biological standard state enthalpy of activation, J/mol enthalpy of activation at zero pressure inhibitor concentration light intensity inhibitor concentration incident light intensity threshold, or compensation, light intensity concentration of substance I, M molar flux of A in the z direction, relative to bulk flow, mol/(cm2-s) general rate constant constant in Equation 5.55 first-order reaction rate constant giving rate in (moles stoichiometric reaction)/(m 3 s) Boltzmann constant, J/K specificity constant for A, L/(mols) specificity constant for B, L/(mols) reciprocal Dalziel coefficient for A and B reciprocal Dalziel coefficient gas concentration mass-transfer coefficient, cm/s specific reaction rate constant for enzyme deactivation specific death rate constant, t i m e 1 Henry's law constant, units of pressure liquid concentration mass-transfer coefficient, cm/s molar liquid-phase mass-transfer coefficient, mol/(cm 2 s) molar gas mass-transfer coefficient, mol/(cm 2 s)

124

Bioreactor Design

ko k0 kopp k\ /c4, k5 K K Ka Ka\ Ka2 Ke KH Ki KiA Kic Kiu KiA KmA Kma ^mB K'mA Ks Ks Ks KSei{ KT Kw Κγ K\ K\ Ki K\ K2 K2 K2 K3, K4 / L m m m

Fundamentals

zero-order reaction-rate constant catalytic constant, s - 1 apparent value of k0, catalytic constant first-order reaction-rate constant rate constants for Equations 3.49 and 3.50 reaction equilibrium constant for simplest stoichiometric equation yielding one mole of product, units variable Monod-type constant, Equation 5.65, units of light intensity acid dissociation constant acid dissociation constant for first ionization acid dissociation constant for second ionization equilibrium constant constant for solubility of C 0 2 in aqueous solutions, mol/(Latm) inhibition constant substrate inhibition constant competitive inhibition constant for / uncompetitive inhibition constant for / cellobiose inhibition constant Michaelis constant for A, mol/L Michaelis constant for conversion of component a, units of concentration apparent Michaelis constant for B specific Michaelis constant for substrate inhibition Monod constant for substrate, concentration units constant in Equation 5.21 constant in Equation 5.22 effective modified form of Monod constant, Equation 5.37 equilibrium constant at temperature T [H + ] [ O H ] equilibrium constant for enzyme denaturation constant in Haldane equation, Equation 3.10, M _ 1 s _ 1 equilibrium constant for simplest integral stoichiometric equation for enzyme heat denaturation first acid dissociation constant constant in Equation 5.66 constant in Equation 5.66 second acid dissociation constant constant in Haldane equation, Equation 3.10 constants in Haldane equation, Equation 3.10, M _ 1 length characteristic length of system maintenance coefficient, t i m e 1 empirical parameter between - 1 and oo7 but not 0, in Equation 5.20 constant in Han-Levenspiel equations

List of Symbols MB Mw n n n N N NAR NAz No Pi p P Pi P0 Pt [P]o Po Pe pKa pKH pKw qP qs Q r vp rp rs rx R R R Re s S S S S Δ5°' S\

125

molecular weight of solvent B Weisz modulus, Equation 6.16 number of streams order of reaction constant in Han-Levenspiel equations number of cells mixing impeller rotation speed, rotations/time flux of A in radial direction at radius = R molar flux of A in the z direction with respect to fixed coordinates, mol/(cm2*s) cell population at time zero partial pressure of component / absolute pressure, kPa or atm, as specified mechanical power input, kW pressure of stream /, kPa absolute, unless otherwise specified reactor outlet pressure, kPa total system pressure initial concentration of product, mol/L mixing power input Péclet number, Equation 4.17 -logioK* - logio KH - logio Kw specific product appearance rate, Equation 7.11 specific substrate consumption rate, Equation 5.13 rate of heat production, kW radius volumetric product formation rate, moles or mass per unit volume per unit time radius of pellet or particle volumetric substrate consumption rate, moles or mass per unit volume per unit time rate of growth of cells, mass or number per unit volume per unit time universal gas constant, 8.314 J/(mol K), unless otherwise specified equivalent radius of particle or slab volumetric recycle flow rate Reynolds number, Equation 4.14 impeller tip speed, length/time substrate concentration surface area of element normal to diffusion of A reactor scaling factor salinity, parts per thousand by mass entropy change at biological standard state salinity, 5/(34.4 mass [%o], parts per thousand)

126

Bioreactor Design Fundamentals

Δ5* S, [S] S* SE Sp Sp SR So So Si S2 Sc Sh t td tm T Ti T0 u umz v V/ vm v'" V, V m AV* V VA Vp Vp VR Vs VPi w W W x x xA xA,i Xi xs

entropy of activation, J/(mol K) substrate concentration limiting substrate concentration concentration of substrate in effluent surface area of pellet or particle surface area of slab outer surface area of particle concentration of substrate in feed concentration of substrate at time zero concentration of substrate 1 concentration of substrate 2 Schmidt number, Equation 4.15 Sherwood number, Equation 4.11 time doubling time mixing time temperature, K temperature of stream /, K reactor outlet temperature, K fluid velocity, cm/s net molar velocity of mixture in z direction, cm/s reaction rate, (moles stoichiometric reaction)/(m 3 s) molar volume of stream / at temperature Γ; and pressure P, m3/mol reaction rate at optimal pH reaction rate based on volume of reactor, (moles stoichiometric reaction)/(m 3 s) limiting reaction rate, (concentration)/time molar volume change of activation, m3/mol constant in Equation 3.17 solute A molar volume at normal boiling point, cm3/mol volume of pellet or particle volume of slab volume of reaction mixture in reactor volume of solid vapor pressure of pure component / at system temperature thickness of peripheral growth zone in a mycelial pellet yield constant, Equation 5.25 thickness of culture (depth) fractional conversion mole fraction in liquid phase mole fraction of A in bulk liquid phase mole fraction of A in liquid phase at gas-liquid interface mole fraction of component / in the liquid phase fractional conversion of reactant at steady state

List of Symbols X X X* XE Xmax Xp XR Xt Xo Xo Xn Xn YA yA,i y av / e Y PIS Ypix Y six Yxis z z Z\ zt [Z]

127

cell concentration, mass or number per unit volume mass of a mycelial pellet limiting cell concentration concentration of cells in effluent maximum cell concentration under substrate supply limitation mass of active peripheral zone in mycelial pellet concentration of cells in recycle terminal cell concentration concentration of cells in feed concentration of cells at time zero concentration of cells in outlet of reactor n concentration of cells in reactor n mole fraction of A in bulk gas phase, mol/cm3 mole fraction of A in gas phase at gas-liquid interface cell yield, grams dry mass per mole of electrons transferred in substrate reaction to yield products ratio of amount of product produced to amount of substrate consumed, mass/mass or mole/mole ratio of mass of product produced per mass of cells appearing ratio of mass of substrate converted per mass of cells produced ratio of mass of cells produced per mass of substrate converted distance in z direction approach to equilibrium, [A] - [A] eq , for reversible reaction, M initial value of z, M value of z at time t, M concentration of product Z, M

GREEK

ai

coefficient of [A]' in n u m e r a t o r of rational rate expression,

ß ßi yt

fraction of cells in product recycled coefficient of [A]1' in denominator of rational rate expression, M _ 1 degree of reduction of c o m p o n e n t / or activity coefficient as specified in text extinction coefficient for cells, Equation 7.64 effectiveness factor, E q u a t i o n 6.11 viscosity of solvent B, cp film effectiveness factor, E q u a t i o n 6.23 overall efficiency, E q u a t i o n 6.18 effectiveness factor for particle, Equation 6.17 thermodynamic efficiency constant in Equation 5.22 residence time in reactor

€ TJA 17B t]f Ύ]Ο Ύ]Ρ r\th λ r

128 μ μ μ μτη ^meff μ0 v v νι p Δρ T φ φ φΗ φί φΝ φο φ5 φχ

Bioreactor Design Fundamentals Arrhenius activation energy, J/mol fluid viscosity, g/(cms) specific growth rate, t i m e 1 maximum specific growth rate constant, t i m e 1 effective modified form of maximum specific growth rate constant, /x,m, Equation 5.38 uninhibited value of /xm, t i m e 1 specific product formation rate, Equation 5.14 kinematic viscosity, Equation 4.16, cm2/s stoichiometric coefficient of reactant or product, positive for products and negative for reactants liquid density liquid density - gas density residence time association factor of solvent B, dimensionless Thiele modulus, Equation 6.12 heat production rate, kJ/s vector rate for stream / into a reactor, positive for input and negative for output, kJ/mol net rate of flow of nitrogen source into system, mol/s net uptake rate of 0 2 , mol/s net rate of substrate flow into system, C-mole/s net rate of cell flow out of system, C-mole/s