Intraparticle mass-transfer resistance and apparent time stability of immobilized yeast alcohol dehydrogenase

The stability and, consequently, the lifetime of immobilized enzymes (IME) are important factors in practical applications of IME, especially so far as design and operation of the enzyme reactors are concerned. In this paper a model is presented which describes the effect of intraparticle diffusion on time stability behaviour of IME, and which has been verified experimentally by the two-substrate enzymic reaction. As a model reaction the ethanol oxidation catalysed by immobilized yeast alcohol dehydrogenase was chosen. The reaction was performed in the batch-recycle reactor at 303 K and pH-value 8.9, under the conditions of high ethanol concentration and low coenzyme (NAD⁺) concentration, so that NAD⁺ was the limiting substrate. The values of the apparent and intrinsic deactivation constant as well as the apparent relative lifetime of the enzyme were calculated.The results show that the diffusional resistance influences the time stability of the IME catalyst and that IME appears to be more stabilized under the larger diffusion resistance.List of Symbols C _A, C_B, C_E mol · m^–3 concentration of coenzyme NAD⁺, ethanol and enzyme, respectively - C _p mol · m³ concentration of reaction product NADH - d _p mm particle diameter - D _eff m² · s^–1 effective volume diffusivity of NAD⁺ within porous matrix - k _d s^–1 intrinsic deactivation constant - K _A, K_A, K_B mol · m^–3 kinetic constant defined by Eq. (1) - K _A ^x mol · m^–3 kinetic constant defined by Eq. (5) - r _A mol · m^–3 · s^–1 intrinsic reaction rate - R m particle radius - R _v mol · m^–3 · s^–1 observed reaction rate per unit volume of immobilized enzyme - t _E s enzyme deactivation time - t _r s reaction time - V mol · m^–3 · s^–1 maximum reaction rate in Eq. (1) - V ^x mol · m^–3 · s^–1 parameter defined by Eq. (4) - V _f m³ total volume of fluid in reactor - w _s kg mass of immobilized enzyme bed - factor defined by Eqs. (19) and (20) - kg · m^–3 density of immobilized enzyme bed - unstableness factor - effectiveness factor - Thiele modulus - relative half-lifetime of immobilized enzyme Index o values obtained with fresh immobilized enzyme     

Two mathematical models for the development of a single microbial pellet

A mathematical model for pellet development of filamentous microorganisms is presented, which simulates in detail location and growth of single hyphal elements. The basic model for growth, septation and branching of discrete hyphae is adopted from Yang et al. [2, 23]. Exact solutions to the intracellular mass-balance equations of a growth-limiting key component is given for two types of either branched or unbranched cellular compartments. Furthermore, the growth model was extended in regard to the external mass-balance equations of limiting substrates (oxygen, glucose) under the assumption that the substrates can enter the denser regions of the pellet only diffusively. Penetration of the substrates into the more porous outer regions of the pellet occurs more easily due to microeddies in the surrounding fluid. Chipping of hyphae from the pellet surface by shear forces was included in the model as well. The application of shear forces leads to a marked smoothing of the simulated pellet surface. The development of pellets from spore germination up to late stages with cell-lysis due to shortage of substrates in the pellet centre can be described. The effects of various model parameters are discussed.List of Symbols A _i algebraic coefficient (i = 1, 2,..., 6) - B _i algebraic coefficient (i = 1, 2,..., 6) - C _i mass-concentration of component i (i = O₂, S) (gl^–1) - C _i,crit concentration of substance i critical for lysis (i=O₂, S) (gl^–1) - C _i,stop concentration of substance i below which cells are inactivated (gl^–1) - C(l _i,t) intracellular concentration of the key component at site l _i and time t (gl^–1) - C _m maximal intracellular concentration of the key component (gl^–1) - C _X Concentration of dry biomass (gl^–1) - D intracellular diffusion coefficient of the key component (m² h^–1) - D _max,i maximal molecular diffusion coefficient of substrate i (i = O₂, S) (m² h^–1) - D _eff,i effective diffusion coefficient of component i (i = O₂) (m² h^–1) - d _h cross-sectional diameter of hyphae (m) - k production coefficient for the key component (h^–1) - K _s Monod coefficient for glucose (gl^–1) - k ₀ Monod coefficient for oxygen (gl^–1) - L _c total length of a compartment (m) - L _i total length of branch i (i=1, 2, 3) (m) - l _i position on branch i (i=1, 2, 3) - L _m maximal length of a segment (m) - m _i maintenance coefficient of substrate i (h^–1) - N _m maximal number of segments in a compartment - n _iR number of tips of type i in layer R, i=1, 2 - p auxiliary variable (see Eq. (7)) - P _Br probability that a hypha is chipped off (%h^–1) - pO ₂ partial pressure of oxygen in the liquid phase (%) - Q auxiliary variable (see Eq. (8)) - Q _i uptake rate of substrate i (i = O₂, S) (gl^–1 h^–1) - q auxiliary variable (see Eq. (7)) - R index of radial layer (R=1, 2, 3,..., R _max) - r radius (m) - r _crit critical radius, Eq. (15) (m) - r _max pellet radius (m) - r _tip distance from the pellet centre to the tip position (m) - r _thr threshold radius (m) - s auxiliary variable (see Eq. (7)) - S index for glucose - t time (h) - v _R volume of layer R (1) - Y _Mi observable yield coefficient of biomass on substrate i (gg^–1) - Y _Xi yield coefficient of biomass on substrate i (gg^–1) Greek Letters _i actual tip expansion rate (m h^–1) - _i,m actual maximal extension rate of tip i (i=1, 2) (m h^–1) - _1y lysis rate (h^–1) - _m maximal tip extension rate (m h^–1) - auxiliary variable in Eq. (2) - auxiliary variable in Eq. (3) - auxiliary variable defined in Eq. (4) (m^–1) - _shear shear force parameter - _R overall specific growth rate in layer R (h^–1) - _m maximal specific hyphal growth rate (h^–1) - cell volume density (l cell volume per 1) - _crit critical cell volume density in Eq. (15) - _S shear force parameter - _X cell mass density (g dry weight per 1 wet cells) - (C _i) growth kinetics on substrate i - proportional factor in Eq. (34) (l g^–1) We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.     

Immobilization of lactase for the continuous hydrolysis of whey permaete

The production of lactose-based sweeteners is considered very promising. Fungal lactase has been immobilized on crosslinked chitin to develop a process for the continuous hydrolysis of demineralized whey permaete. The optimization of lactase immobilization on chitin and chitosan was performed, activities of 4 · 10⁵ and 2.2 · 10⁵ u/kg at yields of 33 and 23% were obtained for both supports, respectively. The chitin based catalyst was selected for further studies and a procedure was developed for in-situ enzyme immobilization. The kinetic behaviour of the catalyst was determined to propose a kinetic model for the initial rate of lactose hydrolysis. Pseudo steady-state and long term operation of packed bed reactors with chitin-immobilized lactase ranging from small laboratory to pre-pilot unit was carried out. The results are discussed and compared with commercial immobilized lactases. Preliminary economic evaluation for the production of ultrafiltered whey protein and hydrolyzed lactose syrup, within a dairy industry in Chile, was satisfactory in terms of profitability, both for the chitin immobilized lactase developed and for a commercial immobilized lactase.List of Symbols a moles/m³ glucose concentration in Eq. (1) - C _i US$ total annual cost (without considering plant depreciation) - D US$ annual depreciation - F m³/h flowrate - h m³/h volumetric mass transfer coefficient - i moles/m³ galactose concentration in Eqs. (1) and (2) - K _A moles/m³ dissociation constant for glucose in Eq. (1) - K _A moles/m³ dissociation constant for glucose in Eq. (1) - K _I moles/m³ inhibition constant for galactose in Eqs. (1) and (2) - K _m moles/m³ Michaelis constant for substrate in Eqs. (1) and (2) - k _D h^–1 first-order thermal deactivation constant - P kg dry weight of catalyst - PV US$ net present value - R % discounted cash-flow rate of return - s moles/m³ substrate concentration - s₀ moles/m³ feed substrate concentration - S _n US$ annual sales income - TC US$ total capital income - t _1/2 h catalyst half-life - v moles/h · kg initial rate of reaction - V _MAX moles/h · kg maximum reaction rate in Eqs. (1) and (2) - V _MAX moles/h · kg maximum reaction rate in Eq. (1) - ¯V _max moles/h initial rate of reaction - V _R m³ reaction volume free of catalyst particles - X substrate degree of conversion = s₀–s/s₀ - Damkoehler number = ¯V _MAX /h k _m - moles/(m³ · h) reactor productivity in Eq. (3)     

Biotransformation of cephalosporin C to 7-aminocephalosporanic acid with coimmobilized biocatalyst in a batch bioreactor

Biotransformation of cephalosporin C (CPS-C) to 7-aminocephalosporanic acid (7-ACA) was carried out with coimmobilized permeabilized cells of Trigonopsis variabilis and Pseudomonas species entrapped in Ca-pectate gel beads. Good aeration and stirring during the process was assured. The analysis of this complicated biochemical process in a heterogeneous system was based on the identification of individual effects (internal diffusion, reaction) running simultaneously. A spectrophotometric method was proposed for the determination of 7-(-ketoadipyl amido) cephalosporanic acid (CO-GL-7-ACA) and 7-ACA. The reaction-diffusion model containing dimensionless partial differential equations was solved by using the orthogonal collocation method. A good agreement between experimental values and values predicted by the mathematical model was obtained. Numerical simulations were performed on the basis of following the two assumptions:- several times higher activity of both cells,- hydrogen peroxide was continuously supplied in the bioreactor.List of Symbols A m² surface of the bead - c _i mol/dm³ concentration of component in the bead and/or in the solution - c _i0 mol/dm³ initial concentration of component in the solution - c _l0 mol/dm³ initial concentration of CPS-C in the solution - C _jl orthogonal collocation weights of the first derivation - D _ei m²/s effective diffusion coefficient of the components - D _jl orthogonal collocation weights of the second derivation - k ₅ dm³/(mol · s) kinetic parameter of non-enzyme reaction - K _inh mol/dm³ inhibition parameter for the first enzyme reaction - K _i dimensionless Michaelis constant for the first and second enzyme reaction, defined in Eq. (7) - K _l dimensionless inhibition parameter for the first enzyme reaction, defined in Eq. (7) - K _mi mol/dm³ Michaelis constant for the first and second enzyme reaction - n number of beads - P( _i ) symbol of dimensionless reaction rate, defined in Eq. (13) - r m radial coordinate inside the bead - R m radius of the bead - R(c _i ) mol/(dm³ · s) symbol for reaction rate, defined in Eq. (6) - t s time - V _max mol/(dm³ · s) max. reaction rate for the first and second enzyme reaction - V _L dm³ volume of solution excluding the space occupied by beads - voidage in batch bioreactor - _P porosity of the bead - i dimensionless effective diffusion coefficient of the components, defined in Eq. (7) - dimensionless time, defined in Eq. (7) - _mi Thiele modulus, defined in Eq. (7) - _i dimensionless concentration, defined in Eq. (7) - dimensionless radial position inside the bead, defined in Eq. (7) - _l0 initial dimension concentration of CPS-C, defined in Eq. (9), (10) - _i0 initial dimension concentration of component, defined in Eq. (9), (10) The authors wish to thank Dr. P. Gemeiner of Slovak Academy of Sciences for rendering of pectate gel. This work is supported by Ministry of Education (Grant No. 1/990 935/93).     

Packed-bed immobilized enzyme reactors for complex processes

Utilization of enzymic reactors for biotechnological-biomedical applications is currently developing at a sustained pace.Our present study concentrates on development of procedures for describing the performance of devices where enzyme-catalyzed reactions between two substrates take place, and for the rational design and optimization of the reactors considered. Within this context, an analytical model was developed for immobilized enzyme packed-bed reactors; it takes into account internal diffusion limitations for the cosubstrates, and hydrodynamic backmixing effects. In order to overcome the complex mathematical problems involved, the compartmental analysis approach was employed.Using this model, performance was simulated for various configurations of the enzymic unit, i.e. from a continuously operated stirred tank reactor (CSTR) to an essentially plug flow type. In addition, an experimental method is described for quantitatively assessing the backmixing effects prevailing in the reactor.The procedures established also provide the ground for further developments, particularly for systems where, in parallel to the enzymic reaction, additional processes (e. g. complexation) take place.List of Symbols C _j,i mM Concentration of substrate j in the pores of stage - iD _j cm²/s Internal (pore) diffusion coefficient of substrate j; defined in Eq. (7) - D _e cm²/s Axial dispersion diffusion coefficient - D _j, cm²/s cm²/s Bulk diffusion coefficient for substrate j - E mM Enzyme concentration inside the catalytic pores - J _j,immol/s/cm² Net flux of substrate j taking place from the bulk of stage i into the corresponding pores; defined in Eq. (6) - K _m,1, K _m,2 mM Michaelis-Menten constants for cosubstrates 1 and 2, respectively - k s ^–1 Catalytic constant - k _s cm/s Catalytic constant - n Total number of elementary stages in the reactor - Q cm³/s Volumetric flow rate throught the reactor - r cm Radius of the pore - R _j,i mM/s Reaction rate of substrate j in stage i, in terms of volumetric units - S cm² Internal surface of a pore - S _j,0 mM Concentration of substrate j in the reactor feed - S _j,i–1, S _j,i mM Concentration of substrate j in the bulk phase leaving stages i — 1 and i, respectivley - V _i cm³ Total volume of stage i (bulk phase + pore phase + inert solid carrier) - V cm³ Total volume of the reactor - V _m ^* mmol/s/cm² Maximal reaction rate in terms of surface units; defined in Eq. (8) - V _m mM/s Maximal reaction rate in terms of volumetric units; defined in Eq. (8) - V _p cm³ Volume of one pore - y cm Axial coordinate of the pores - y ₀ cm Depth of the pores - Z cm Axial coordinate of the reactor - Z ₀ cm Length of the reactor - ₁ Dimensionless parameter; defined in Eq. (27) - ₂ Dimensionless parameter; defined in Eq. (27) - ₁ Dimensionless parameter; defined in Eq. (27) - ₂ Dimensionless parameter; defined in Eq. (27) - Ratio between the radius of the enzyme molecule and the radius of the pore (dimensionless) - V₁ Dimensionless parameter; defined in Eq. (21) - v₂ Dimensionless parameter; defined in Eq. (21) - Q Volumetric packing density of catalytic particles (dimensionless) - Ø Porosity of the catalytic particles (dimensionless) - Ø Dimensionless concentration of substrate j in pores of stage i; defined in Eq. (16) - _j,i-1,_j,i Dimensionless concentration of substrate j in the bulk phase of stage i; defined in Eq. (18) - Dimensionless position; defined in Eq. (16) - ² s² Variance; defined in Eq. (33) - Mean residence time in the reactor; defined in Eq. (33)     

Biotransformation of cephalosporin C to 7-aminocephalosporanic acid with coimmobilized biocatalyst in a batch bioreactor

The permeabilized cells of Trigonopsis variabilis CCY 15-1-3 having D-amino acid oxidase (DAAO) activity were used to convert cephalosporin C (CPS-C) into 7-(-ketoadipyl amido) cephalosporanic acid (CO-GL-7-ACA) in a batch bioreactor with good aeration and stirring during the process. The deacylation of 7--(4-carboxybutanamido)-cephalosporanic acid (GL-7-ACA) to 7-cephalosporanic acid (7-ACA) by permeabilized cells of Pseudomonas species 3635 having 4--(4-carboxybutamido)-cephalosporanic acid acylase (GL-7-ACA acylase) activity was performed in a batch bioreactor. A spectrophotometric method for the determination of CO-GL-7-ACA and 7-ACA was proposed. Experimental data were fitted by non-linear regression with parameters optimization. The sorption method (without reaction) was applied for the determination of cephalosporin effective diffusion coefficients in Ca-pectate gel beads. These beads were prepared by dropping a potassium pectate gel suspension of inactive permeabilized cells of Trigonopsis variabilis and Pseudomonas species, crosslinked with glutaraldehyde, into a stirred 0.2 M calcium chloride solution. Concentrations of appropriate cephem components were measured by the refractive method. Values of effective diffusion coefficients were calculated by the Fibbonacci optimization method.List of Symbols c _L mol/dm³ concentration on the surface of a bead - c _L0 mol/dm³ initial cephalosporin concentration - c _L mol/dm³ equilibrium cephalosporin concentration in the solution - c _s1 mol/dm³ concentration of CPS-C - c _s2 mol/dm³ concentration of GL-7-ACA - D _ei m²/s effective diffusion coefficient of the components - K _i mol/dm³ inhibition parameter in Eq. (2) - K _{m i} mol/dm³ Michaelis constant in Eq. (1) - K _{m 2} mol/dm³ Michaelis constant in Eq. (2) - n number of beads - q _n nonzero positive roots in Eq. (7) - r ₁ mol/(dm³·s) rate of the conversion of CPS-S to CO-GL-7-ACA - r ₂ mol/(dm³·s) rate of the conversion of GL-7-ACA to 7-ACA - R m radius of the bead - S( ) symbol for total residual sum of squares in Eq. (1) - t s time - V _{m 1} mol/(dm³·s) max. reaction rate in Eq. (1) - V _{m 2} mol/(dm³·s) max. reaction rate in Eq. (2) - V _L dm³ volume of the solution excluding the space occupied by beads - V _s dm³ volume of beads - y _i mol/(dm³ · s) symbol for experimental data in Eq. (1) - _i mol/(dm³· s) symbol for calculated data in Eq. (1) - _P porosity, defined by Eq. (5) - dimensionless parameter, defined by Eq. (6) The authors wish to thank Dr. P. Gemeiner of Slovak Academy of Sciences for rendering of pectate gel. This work is supported by Ministry of Education (Grant No. 1/990 935/93)     

Production studies on protozoa

Summary In the river Saale and in the terrestrial moss Mnium cuspidatum Leyss. in 1974/75 the annual production of Testacea and loricate ciliated protozoa were investigated.The production was estimated in the Saale-Aufwuchs on a -meso ... oligosaprobic (Kaulsdorf, Thuringia, GDR) and on a -mesosaprobic (Rothenstein, Thuringia, GDR) area of the river. The mosses were investigated in a forest near Jena.The production was estimated on slides and in special productionchambers; the time of exposure was 2 weeks. Investigations concerned annual production of individuals and biomass, the ratio of annual production/standing crop (P/B), numbers of generations per year (G) and mortality (M%/d). In the mosses, the rainfall modified the production and dislocation of the protozoa.The values for production are: Aufwuchs Saale (-meso... oligosaprobic): 24·10⁶ i/m²·a (=1,0 g/m²·a=79·10³ i/m²·d); P/B: 12.6. Aufwuchs Saale (-mesosaprobic): 3.2·10⁶ i/m²·a (=0.35 g/m²·a=81·10³ i/m²·d); P/B: 34.9; G: 22; M: 5%/d. Moss: 145·10⁶ i/m²·a (=0.11 g/m²·a=40.6·10³ i/m²·d); P/B: 8.1; G: 16.5; M: 3.0%/d.     

Inhibition of the acidogenic dissimilation of glucose in anaerobic continuous cultures by free butyric acid

Summary In anaerobic wastewater treatment the separation of fermentative and methanogenic bacteria is aimed at an increased performance of the total digestion process. It is known that the attainable growth rate of the acidogenic population in continuous culture decreases at increasing influent concentrations of glucose. To account for this phenomenon, a new kinetic model was developed that combines substrate and product inhibition. In the present research product inhibition was investigated quantitatively in a continuous culture fermenting 50 mmol/l glucose. Extra acetate and butyrate were added up to 200 mmol/l at different pH values, and it turned out that only free butyric acid inhibited growth. The lower attainable growth rates of cultures producing comparable amounts of butyrate when fed with concentrated influents, strongly indicated substrate inhibition. Evidence is presented that transitions to low-conversion steady states predicted by the kinetic model, play a role and decrease the stability of the culture.Nomenclature D dilution rate, h^-1 - D_att highest D using certain experimental procedure h^-1 - K_i substrate inhibition constant, mol·m^-3 - K_p product inhibition constant mol·m^-3 - K_s substrate saturation constant, mol·m^-3 - P concentration inhibitory product mol·m^-3 - S substrate concentration, mol·m^-3 - S_o influent substrate concentration, mol·m^-3 - S _max ^c substrate concentration at _max ^c , mol·m^-3 - S _max ^h substrate concentration at _max ^h , mol·m^-3 - specific growth rate, h^-1 - experimental realization of at D_att, h^-1 - _max maximum specific growth rate, h^-1 - _max ^c maximum attainable specific growth rate according to combined substrate/product inhibition model, h^-1 - _h ⁰ specific growth rate at S₀ according to Haldane kinetics, h^-1 - _max ^c maximum attainable specific growth rate according to Haldane kinetics, h^-1 - Y_p yield inhibitory product, mol·mol^-1 - Y_x yield biomass, kg dry weight·kg^-1 - bio biomass - EtOH ethanol - gluc glucose - HAc acetate - HBt butyrate - HCap caproate - HFo formate - HPr propionate - HVal valerate - prod produced - lact lactate     

Intensity of microcarrier collisions in turbulent flow

A model is developed, allowing estimation of the share of inelastic interparticle collisions in total energy dissipation for stirred suspensions. The model is restricted to equal-sized, rigid, spherical particles of the same density as the surrounding Newtonian fluid. A number of simplifying assumptions had to be made in developing the model. According to the developed model, the share of collisions in energy dissipation is small.List of Symbols b parameter in velocity distribution function (Eq. (28)) - c _K factor in Kolmogoroff spectrum law (Eq. (20)) - D _t(r _p ) m²/s characteristic dispersivity at particle radius scale (Eq. (13)) - E(k, t) m³/s² energy spectrum as function of k and t (Eq. (16)) - E _K (k) m³/s² energy spectrum as function of k in Kolmogoroff-region (Eq. (20)) - E _p dimensionless mean kinetic energy of a colliding particle (Eq. (36)) - E _cp dimensionless kinetic energy exchange in a collision (Eq. (37)) - G(x, s) dimensionless energy spectrum as function of x and s (Eq. (16)) - G _B(x) dimensionless energy spectrum as function of x for boundary region (Eq. (29)) - G _K(x) dimensionless energy spectrum as function of x for Kolmogoroff-region (Eq. (21)) - g m/s² gravitational acceleration - I _cp dimensionless collision intensity per particle (Eq. (38)) - I _cv dimensionless volumetric collision intensity (Eq. (39)) - k l/m reciprocal of length scale of velocity fluctuations (Eq. (17)) - K dimensionless viscosity (Eq. (13)) - n(2) dimensionless particle collision rate (Eq. (12)) - n(r) l/s particle exchange rate as function of distance from observatory particle center (Eq. (7)) - r m vector describing position relative to observatory particle center (Eq. (2)) - r m scalar distance to observatory particle center (Eq. (3)) - r _pm particle radius (Eq. (1)) - s dimensionless time (Eq. (10)) - SC kg/ms³ Severity of collision (Eq. (1)) - t s time (Eq. (2)) - u(r, t) m/s velocity vector as function of position vector and time (Eq. (2)) - u(r, t) m/s magnitude of velocity vector as function of position vector and time (Eq. (3)) - u _r(r, t) m/s radial component of velocity vector as function of position vector and time (Eq. (3)) - u _r (r, t) m/s magnitude of radial component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u _gsm/s superficial gas velocity - u(r) m/s root mean square velocity as function of distance from observatory particle center (Eq. (3)) - u_r(r) m/s root mean square radial velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s root mean square latitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s Root mean square longitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - w(x) dimensionless root mean square velocity as function of dimensionless distance from observatory particle center (Eq. (11)) - V _pm³ particle volume (Eq. (36)) - w(2) dimensionless root mean square collision velocity (Eq. (34)) - w ^* parameter in boundary layer velocity equation (Eq. (24)) - x dimensionless distance to particle center (Eq. (9)) - x ^* value of x where G _Band G _K-curves touch (Eq. (32)) - x _K dimensionless micro-scale (Kolmogoroff-scale) of turbulence (Eq. (15)) - volumetric particle hold-up - m²/s³ energy dissipation per unit of mass - m²/s kinematic viscosity - kg/m³ density - (r) m³/s fluid-exchange rate as function of distance to observatory particle center - Latitudinal co-ordinate (Eq. (5)) - Longitudinal co-ordinate (Eq. (5))     

Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

A necessary condition is found for the intermediate temperatures and substrate concentrations in a series of CSTR's performing an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The common case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows that intermediate stream temperatures play as important a role as intermediate substrate concentrations when optimizing in the presence of nonisothermal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. For negligible rate of deactivation, the qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing function of the previous ratio of consecutive concentrations.List of Symbols C _E,0 mol.m^–3 Initial concentration of active enzyme - C _E,i mol.m^–3 Concentration of active enzyme at the outlet of the i-th reactor - C _S,0 mol.m^–3 Initial concentration of substrate - C _S,i mol.m^–3 Concentration of substrate at the outlet of the i-th reactor - Da _i Damköhler number associated with the i-th reactor ((V _i.k_v,0.C_E,0)/(Q.C_S,0)) - Da _min Minimum value of the overall Damköhler number - Da _tot Overall Damköhler number - E _d J.mol^–1 Activation energy of the step of deactivation of the enzyme - E _m J.mol^–1 Standard enthalpy change of the step of binding of substrate to the enzyme - E _v J.mol^–1 Activation energy of the step of enzymatic transformation of substrate - i Integer variable - j Dummy integer variable - k Dummy integer variable - k _d,i s^–1 Kinetic constant associated with the deactivation of enzyme in the i-th reactor (k _d,o·exp{–E _d/(R.T _i}) - k _d,0 s^–1 Preexponential factor of the kinetic constant associated with the deactivation of the enzyme - K _m,i mol.m^–3 Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (k _m,o·exp{–E _m}(R.T _i}) - K _m,0 mol.m^–3 Preexponential factor of the Michaelis-Menten constant associated with the binding of substrate to the enzyme - k _v,i s^–1 Kinetic constant associated with the transformation of the substrate by the enzyme in the i-th reactor (k _v,o·exp{–E _v/(R.T _i})) - k _v,0 s^–1 Preexponential factor of the kinetic constant associated with the transformation of the substrate by the enzyme - N Number of reactors in the series - Q m³.s^–1 Volumetric flow rate of reacting liquid through the reactor network - R J.K^–1.mol^–1 Ideal gas constant - T _i K Absolute temperature at the outlet of the i-th reactor - T ₀ K Initial absolute temperature - V _i m³ Volume of the i-th reactor - v _max mol.m^–3.s^–1 Maximum rate of reaction under saturation conditions of substrate - x _i Normalized concentration of substrate (C_S,i/C_{S, 0}) - x _i,opt Optimum value of the normalized concentration of substrate - y _i Dimensionless temperature (exp{–T ₀/T _i}) - y _i,opt Optimum value of the dimensionless temperature Greek Symbols Dimensionless preexponential factor associated with the Michaelis-Menten constant (K _m,0/C_s,0) - Dimensionless activation energy of the step of enzymatic transformation of substrate (E _v/R.T₀)) - Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E _m/(R.T₀)) - Dimensionless activation energy of the step of deactivation of the enzyme (E _d/(R.T₀)) - Dimensionless deactivation preexponential factor ((k _d,0.C_S,0)/(k_v,0.C_E,0)     

Effects of dissolved oxygen concentration on lipase production by Rhizopus delemar

Summary The influence of the concentration of oxygen on lipase production by the fungus Rhizopus delemar was studied in different fermenters. The effect of oxygen limitation ( 47 mol/l) on lipase production by R. delemar is large as could be demonstrated in pellet and filamentous cultures. A model is proposed to describe the extent of oxygen limitation in pellet cultures. Model estimates indicate that oxygen is the limiting substrate in shake flask cultures and that an optimal inoculum size for oxygen-dependent processes can occur.Low oxygen concentrations greatly negatively affect the metabolism of R. delemar, which could be shown by cultivation in continuous cultures in filamentous growth form (D_optimal=0.086 h^-1). Continuous cultivations of R. delemar at constant, low-oxygen concentrations are a useful tool to scale down fermentation processes in cases where a transient or local oxygen limitation occurs.Symbols and Abbreviations CO Oxygen concentration in the gas phase at time = 0 (kg·m^-3) - C_{O 2i} Oxygen concentration at the pellet liquid interface (kg·m^-3) - C_{O 2i} Oxygen concentration in the bulk (kg·m^-3) - D Dilution rate (h^-1) - ID_{O 2} Diffusion coefficient for oxygen (m²·s^-1) - dw Dry weight of biomass (kg) - f Conversion factor (rs _{O 2} to oxygen consumption rate per m³) (-) - k Radial growth rate (m·s^-1) - K Constant - kla Volumetric mass transfer coefficient (s^-1) - klA Oxygen transfer rate (m^-3·s^-1) - kl Mass transfer coefficient (m·s^-1) - K _{O 2} Affinity constant for oxygen (mol·m^-3) - K _w Cotton plug resistance (m^-3·s^-1) - M Henry coefficient (-) - NV Number of pellets per volume (m^-3) - R Radius (m) - RO Radius of oxygen-deficient core (m) - RQ Respiration quotient (mol CO₂/mol O₂) - rs _{O 2} Specific oxygen consumption rate per dry weight biomass (kg O₂·s^-1[kg dw]^-1) - rX Biomass production rate (kg·m^-3·s^-1) - SG Soytone glucose medium (for shake flask experiments) - SG ₄ Soytone glucose medium (for tower fermenter and continuous culture experiments) - V Volume of medium (m^-3) - X Biomass (dry weight) concentration (kg·m^-3) - XR _o Biomass concentration within RO for a given X (kg·m^-3) - Y _{O 2} Biomass yield calculated on oxygen (kg dw/kg O₂) - Thiele modulus - Efficiency factor =1-(RO/R)³ (-) - Growth rate (m^-1·s^-1·kg^1/3) - Dry weight per volume of pellet (kg·m^-3)     

Optimizing control of a continuous stirred tank fermenter using a neural network

Feedforward neural networks are a general class of nonlinear models that can be used advantageously to model dynamic processes. In this investigation, a neural network was used to model the dynamic behaviour of a continuous stirred tank fermenter in view of using this model for predictive control. In this system, the control setpoint is not known explicitly but it is calculated in such a way to optimize an objective criterion. The results presented show that neural networks can model very accurately the dynamics of a continuous stirred tank fermenter and, the neural model, when used recursively, can predict the state variables over a long prediction horizon with sufficient accuracy. In addition, neural networks can adapt rapidly to changes in fermentation dynamics.List of Symbols F Dimensionless flow rate (F/ V₀) - F m³/h Flow rate - F ₀ m³/h Inlet flow rate - J Objective cost function - K _i Dimensionless constant in Eq. (3) (k _i /s₀) - k _i kg/m³ Substrate inhibition constant in Haldane model - k _m Dimensionless constant in Eq. (3) (k _s /s₀) - k _m kg/m³ Substrate inhibition constant in Haldane model - n prediction horizon - S Dimensionless substrate concentration (s/s₀) - s kg/m³ Substrate concentration - t h Time - v Dimensionless volume (V/V₀) - V m³ Liquid volume in fermenter - W _ij , W _jk Weight matrices in neural network - X Dimensionless biomass concentration - x kg/m³ Biomass concentration - Y Biomass/substrate yield coefficient - Weighting factor in Eq. (4) - Dimensionless specific growth rate (/ ) - 1/h Maximum specific growth rate - 1/h Specific growth rate - Dimensionless time ( t)     

Development of a new method for testing strength of cells against fluid shear stress

A new method for testing the strength of cells against fluid shear stress by using a long capillary column was proposed. The trajectories of cells in the column were simulated by introducing the Brownian motion model. The Brownian motion was performed by the generation of random numbers. The mean exposure time to shear stress and the mean shear stress acting on the surface of cells were discussed by the result of computer simulation. The mean shear stress acting on the surface of cells flowing in the capillary column was estimated as 4/3-fold of the shear stress at the column wall provided that the ratio of the cell radius to the column radius does not exceed 0.08. The effectiveness of this new method for testing the strength of cells against fluid shear stress was shown.List of Symbols a m radius of cell - c constant - E distribution function - L m length of capillary column - M number of division - N number of division - p probability - Q m³/s flow rate - R m radius of capillary column - r m radial position - t s time - T s exposure time - T _m s mean exposure time - T ₀ s mean residence time - m/s axial velocity - u _m m/s cross-sectional flow velocity - z m axial position - s^–1 shear rate - _w s^–1 shear rate at wall - Pa s viscosity - spherical coordinate - spherical coordinate - Pa shear stress - _m Pa mean shear stress - _w Pa shear stress at wall     

A new technique for continuous measurement of the heat of fermentation

Summary A special temperature control system has been developed and applied to continuous measuring of the heat evolved during a fermentation process. In this system, the fermentation broth was overcooled by a given constant cooling water flow. The excess heat removed from the fermentor was then made up by an immersion electrical heater. The action of the temperature controller was precisely monitored as it varied in response to the amount of heat produced by the microbial activities.The technique was used for determining the heat evolution byEscherichia coli grown on glucose. The ratio between quantities of total heat release and total oxygen consumption has been determined to be 0.556 MJ/mol O₂.The newly developed technique can be employed as an online sensor to monitor the microbial activities of either aerobic or anaerobic fermentation systems.Symbols C_c Heat capacity of cooling water (MJ/kg · °C) - C_p Heat capacity (MJ/kg · °C) - I Current of immersion heater (A) - K Constant in Equation (2) (h) - K Constant in Equation (13) (m³ · h · °C/MJ) - Q_c Flow rate of cooling water (m³/h) - Heat of agitation (MJ/m³ · h) - Heat dissipated by the bubbling gas (MJ/m³ · h) - Heat removal by the action of controller (MJ/m³ · h) - Heat of fermentation (MJ/m³ · h) - Heat loss to the surroundings (MJ/m³ · h) - Q_pass Constant average power dissipated by the immersion heater (MJ/m³ · h) - Fluctuating power dissipated by the immersion heater (MJ/m³ · h) - Power dissipated by the immersion heater (MJ/m³ · h) - T Temperature of fermentation broth (°C) - Constant average temperature of fermentation broth (°C) - Fluctuating temperature of fermentation broth (°C) - T_a Temperature of the ambient air (°C) - T_c Inlet temperature of cooling water (°C) - U₁A₁ Specific heat transfer coefficient for determination of heat loss to the surroundings (MJ/m³ · h · °C) - U₂A₂ Specific heat transfer coefficient for cooling surfaces (MJ/m³ · h · °C) - U₃A₃ Constant in Equation (16) (MJ/m³ · h · °C) - V Voltage of immersion heater (V) - V_L Liquid volume (m³) - OUR Oxygen uptake rate (mol O₂/m³ · h) Greek Letters H_fo The ratio between the total heat release and the total oxygen uptake (MJ/mol O₂) - _c Density of cooling water (kg/m³) - Time constant defined in Equation (6) (h) - _iM_iC_pi Heat capacity of system components (fermentation broth + fermentor jar + stainless steel) (MJ/m³ · °C)     

Control of wheat root growth. The effects of excision on growth,wall rheology and root anatomy

Excision and subsequent incubation of the apices (1 cm) of wheat (Triticum aestivum L.) seedling roots in simple media severely reduced elongation from 28 mm·(24 h)^-1 in intact roots to a maximum of 2 mm·(24 h)^-1 in excised roots. The reduction in growth was accompanied by a loss of cell turgor in the growing zone but was correlated with a hardening of the cell walls in this region. Rheological properties were measured as percent extensibility (both plastic and elastic) using a tensiometer, and as instantaneous volumetric elastic modulus ( _i) using the pressure probe. Excision decreased plastic and elastic properties with a half-time of some 60 min. Plastic extension was reduced from 2.5% to 0.9% and elastic from 4.8% to 2.6% for an 8-g load. By contrast, _i was increased by excision. The observed reduction in root elongation rate was accompained by a reduction in mature cell length from 240 m to 40 m and a shortening of the zone of cell expansion.Symbol _i instantaneous volumetric elastic modulus     

Hydrodynamic characteristics of two-phase inverse fluidized bed

Hydrodynamic characteristics of a new mode of liquid-solid fluidization, termed as inverse fluidization in which low density floating particles are fluidized with downward flow of liquid, are experimentally investigated. The experiments are carried out with low density particles (<534 kg/m³) which allow high liquid throughputs in the system. During the operation, three regimes, namely, packed, semi-fluidization and fully fluidization are encountered. Empirical correlations are proposed to predict the pressure drop in each regime. A computational procedure is developed to simulate the variation of pressure drop with liquid velocity.List of Symbols Ar modified Archimedes number, d _p ³ (– _s)g/² - d _p particle diameter, mm - f friction factor (eq. 2) - g acceleration due to gravity, m/s² - H total bed height, m - H _c height of the column, m - H_f height of fluidized bed, m - H₀ height of initial bed, m - H_p height of the packed bed, m - (p) pressure drop across the bed, N/m² - (p) _f pressure drop across fluidized bed section, N/m² - (p) _p pressure drop across the packed bed section, N/m² - (p) _sf total pressure drop in semifluidization regime, N/m² - Re Reynolds number, d _pU ₁/ - Re_m modified Reynolds number, d _pU ₁/(1– _p) - U ₁ superficial liquid velocity, m/s - U_mf minimum fluidization velocity, m/s - U_osf onset fluidization velocity, m/s Greek Letters _f voidage of fluidized bed - _p voidage of packed bed - liquid viscosity, kg/ms - liquid density, kg/m³ - _s particle density, kg/m³     

Process engineering aspects of the bioleaching of copper ores

The bioleaching of minerals is a complex process that is affected by a number of biological, mineralogical, electrochemical and engineering factors. This work presents and discusses the most significant process engineering aspects involved in the bacterial leaching of copper ores, i.e. bacterial population, type of mineral and particle size, nutrients and inhibitors, oxygen and carbon dioxide, temperature and pH, leaching kinetics and operation mode.It is concluded that more work is needed in this area in order to gain a deeper insight in the many factors that govern this process. This would allow to significantly improve its overall productivity.List of Symbols C _L kg/m³ dissolved oxygen concentration - C ^* kg/m³ equilibrium oxygen concentration - d, e, f, g % percentage of C, H, O and N in the cell - D m impeller diameter - K consistency index - K _S, K₁, K_c constants - k _La h^–1 volumetric oxygen transfer coefficient - M _b mol/kg biomass apparent molecular weight - N s^–1 rotation frequency - n behavior index - P kg/m³ ungassed agitation power, product concentration - P _g kW/m³ gassed agitation power - p % pulp density - Q m³/h air flow rate - S kg/m³ limiting substrate concentration - W kg/(m³ · h) mass transfer rate per unit volume - X cells/cm³ biomass concentration - Y _o g cells/g Fe oxygen cell yield - Y _x g cells/g Fe substrate cell yield - h^–1 specific growth rate - _m h^–1 maximum specific growth rate     

Enhancement of product selectivity via enzyme immobilization in sequential degradation reactions of polymeric substrates

The balance equations pertaining to the modelling of a slap-shaped bead containing immobilized enzyme uniformly distributed which catalyzes the sequential reactions of degradation of a polymeric substrate were written and analytically solved in dimensionless form. The effect of the Thiele modulus on the selectivity of consumption of each multimeric product was studied for a simple case. Whereas plain diffusional regime leads to lower selectivities than plain kinetic regime, improvements in selectivity of species A _i relative to species A_i+1 may be obtained at the expense of higher Thiele moduli within a limited range when the diffusivity of A _i is larger than that of A _i +1, or when the pseudo first order kinetic constant describing the rate of consumption of A _i is lower than that of A_i+1.List of Symbols A _i polymeric substrate containing i monomeric subunits - C _i mol·m^–3 normalized counterpart of C _i - C _i mol·m^–3 concentration of substrate A _i - C _i,0 mol·m^–3 initial concentration of substrate A _i - C _i,0 normalized counterpart of C _i,0 - D _ap,i m²·s^–1 apparent diffusivity of substrate A _i - k _i s^–1 pseudo-first order rate constant - K _m,i mol·m^–3 Michaelis-Menten constant associated with substrate A _i - L m half-thickness of the catalyst slab - N number of monomeric subunits of the largest substrate molecule - Th Thiele modulus - V _i mol·m^–3·s^–1 rate of rection of substrate A _i - V_max,i mol·m^–3·s^–1 maximum rate of reaction under saturating conditions of substrate A _i - x m longitudinal coordinate - S _i,i+1 selectivity of enzyme with respect to substrates with consecutive numbers of monomeric subunits Greek Symbols _i ratio of maximum rates of reaction - _i ratio of apparent diffusivities     

Analysis of gas exchange in seedlings of Acer saccharum: integration of field and laboratory studies

In the field, photosynthesis of Acer saccharum seedlings was rarely light saturated, even though light saturation occurs at about 100 mol quanta m^-2 s^-1 photosynthetic photon flux density (PPFD). PPFD during more than 75% of the daylight period was 50 mol m^-2 s^-1 or less. At these low PPFD's there is a marked interaction of PPFD with the initial slope (CE) of the CO₂ response. At PPFD-saturation CE was 0.018 mol m^-2 s^-1/(l/l). The apparent quantum efficiency (incident PPFD) at saturating CO₂ was 0.05–0.08 mol/mol. and PPFD-saturated CO₂ exchange was 6–8 mol m^-2 s^-1. The ratio of internal CO₂ concentration to external (C _i /C _a ) was 0.7 to 0.8 except during sunflecks when it decreased to 0.5. The decrease in C _i /C _a during sunflecks was the result of the slow response of stomates to increased PPFD compared to the response of net photosynthesis. An empirical model, which included the above parameters was used to simulate the measured CO₂ exchange rate for portions of two days. Parameter values for the model were determined in experiments separate from the daily time courses being sumulated. Analysis of the field data, partly through the use of simulations, indicate that the elimination of sunflecks would reduce net carbon gain by 5–10%.List of symbols A measured photosynthetic rate under any set of conditions (mol m^-2 s^-1) - A _m (atm) measured photosynthetic rate at saturating PPFD, 350 l/l CO₂ and 21% (v/v) O₂ (mol m^-2 s^-1) - C constant in equation of Smith (1937, 1938) - C _a CO₂ concentration in the air (l/l) - C _i CO₂ concentration in the intercellular air space (l/l) - C _i ^/* C _i corrected for CO₂ compensation point, i.e., C _i -I ^*, (l/l) - CE initial slope of the CO₂ response of photosynthesis (mol m^-2 s^-1/(l/l)) - CEM CE at PPFD saturation - E transpiration rate (mmol m^-2 s^-1) - F predicted photosynthetic rate (mol m^-2 s^-1) - G leaf conductance to H₂O (mol m^-2 s^-1) - I photosynthetic photon flux density (mol m^-2 s^-1) - N number of data points - P _m predicted photosynthetic rate at saturating CO₂ and given PPFD (mol m^-2 s^-1) - P _ml predicted photosynthetic rate at saturating CO₂ and PPFD (mol m^-2 s^-1) - R _d residual respiratory rate (mol m^-2 s^-1) - T _a air temperature (°C) - T _l leaf temperature (°C) - V reaction velocity in equation of Smith (1937, 1938) - V _max saturated reaction velocity in equation of Smith (1937, 1938) - VPA vapor pressure of water in the air (mbar/bar) - VPD vapor pressure difference between leaf and air (mbar/bar) - X substrate concentration in equation of Smith (1937, 1938) - initial slope of the PPFD response of photosynthesis at saturating CO₂ (mol CO₂/mol quanta) - (atm) initial slope of the PPFD response of photosynthesis at 340 l/l CO₂ and 21% (v/v) O₂ (mol CO₂/mol quanta) - I ^* CO₂ compensation point after correction for residual respiration (l/l) - PPFD compensation point (mol m^-2 s^-1)     

Slow conductance changes due to potassium depletion in the transverse tubules of frog muscle fibers during hyperpolarizing pulses

Summary When hyperpolarizing currents are applied between the inside and outside of a muscle fiber it is known that there is a slow transient decrease (300- to 600-msec time constant) in the measured fiber conductance sometimes referred to as creep which is maximal in K₂SO₄ Ringer's solutions and which disappears on disruption of the transverse tubular system. An approximate mathematical analysis of the situation indicates that these large, slow conductance changes are to be expected from changes in the K⁺ concentration in the tubular system and are due to differences in transport numbers between the walls and lumen of the tubules. Experiments using small constant-voltage and constant-current pulses (membrane p. d. changes 20 to 30 mV) on the same fibers followed by an approximate mathematical and more exact computed numerical analysis using the measured fiber parameters and published values of tubular system geometry factors showed close agreement between the conductance creep predicted and that observed, thus dispensing with the need for postulated changes in individual membrane conductances at least during small voltage pulses. It is further suggested that an examination of creep with constant-voltage and constant-current pulses may provide a useful tool for monitoring changes in tubular system parameters, such as those occurring during its disruption by presoaking the fibers in glycerol.Table of main symbols used R, T, F Gas constant, Temperature in °K and the Faraday - a Fiber radius - r Radial distance from the center of the fiber (cf. Fig. 2A) - t Time in sec - V ₁,V ₂ Voltages measured by electrodes 1 and 2 (cf. p. 248) - Longitudinal fiber space constant ( ²=R _m a/2R _i ) - R _m ,R _m (t) Total membrane resistance per unit surface area of fiber ( cm²) - R _m (0),R _m () As above att=0 (excluding capacity transient) and att= during a current or voltage pulse - G _m ,G _m (t) Total membrane conductance (mho·cm^–2) per unit area of fiber surface - G _m (0),G _m () As above att=0 (excluding the capacity transient) and att= during a current or voltage pulse - R _sm ,G _sm Surface membrane resistance ( cm²) and conductance (mho·cm^–2), respectively, excluding the TTS - R _T ,G _T Input resistance ( cm²) and conductance (mho·cm^–2) of the TTS referred to unit area of fiber surface - f _T Fraction of the K⁺ conductance in the TTS to the total K⁺ conductance of the fiber [cf. Eq. (7)] - R _i Internal resistivity of the fiber ( cm) - r _s Electrical access resistance of the TTS [ cm²;cf. Fig. 3 and Eq. (24)] - h Diffusional access resistance of the TTS [cf. Eq. (27)] - I ₀ Total current entering fiber (amp) - I _m ,i _m Total current per unit area of fiber surface (amp·cm^–2; considered positive in the hyperpolarizing direction) - i _sm Current going through the surface membrane alone (amp·cm^–2;cf. Fig. 3) - i ₀,i ₀(t) Total current entering the TTS referred to unit area of surface membrane (amp·cm^–2;cf. Fig. 3) - I _K,I _K(r) K⁺ current density crossing the equivalent TTS disc at radial distancer [cf. Fig. 2A and Eq. (23)] - i, i(r, t) Radial current in the lumen of the TTS at radial distancer and timet (cf. Fig. 2B) - C, C(r, t) K⁺ concentration within the TTS at radial distancer and timet (mEquiv·liter^–1) - C _o ,C _K Both refer to external solution and initial TTS K⁺ concentration (mEquiv·liter^–1) - V, V(r, t) The potential at radial distancer in the lumen of the TTS with respect to the external solution at timet (cf. Figs. 2 and 3) - V(a), V(a, t) The p.d. across the access resistance (cf. Figs. 3B and 3C) - V ₀,V ₀(t) The potential of the sarcoplasm with respect to the external solution (cf. Figs. 2 and 3) - E _K The K⁺ equilibrium potential between the sarcoplasm and the externa solution or across the tubular wall - t _K ^m ,t _K ^s The transport number for K⁺ in the TTS membranes and in the solution of the tubular lumen, respectively - The fraction of fiber volume occupied by tubules, and not implicitly including branches - As above but always including branches - A dimensionless network factor for the TTS - G _W Conductance per unit area of tubular wall (mho·cm^–2) - G _L Conductance of tubular lumen (mho·cm^–1) - Volume-to-surface ratio of the TTS - Effective wall conductance of TTS membranes per unit volume of fiber [mho·cm^–3;cf. Eq. (14)] - Effective radial conductance of the lumen of the TTS per unit volume of fiber [cf. Eq. (20)] - d The thickness of the equivalent disc representing the TTS [cf. Eq. (15)] - _T Space constant of the TTS [cf. Eq. (37).cp. Eq. (11)] - D _K The diffusion coefficient of K⁺ ions in the lumen of the TTS (cm² sec^–1) - The effective radial K⁺ diffusion coefficient in the TTS [cf. Eq. (28)] - J ₀,J ₁ Bessel functions of order 0 and 1, respectively - I ₀,I ₁ Modified Bessel functions of order 0 and 1, respectively - Time constants of slow conductance changes - _vc Time constant of slow conductance changes during a constant-voltage pulse - _cc Time constant of slow conductance changes during a constant-current pulse - , _m Roots of various Bessel function equations - g ₁,g ₂,g ₃,g ₄ Constants used to fit cubic equation for conductance-voltage curves [cf. Eq. (71)]