Dynamic modelling of aerobic bioprocesses in gel particles with immobilised cells Part 1: Growth-associated biotransformation V. Beschkov, S. Velizarov Abstract A dynamic model for aerobic growing cells im- mobilised into gel beads is developed and its operation is illustrated for the case of gluconic acid production by a strictly aerophilic strain of Gluconobacter oxydans. The model consists of both kinetic and mass transfer equations predicting the time course of bulk and intraparticle concentrations of substrates, products, and biomass. The model includes a product inhibition term. The parameter values are taken from own studies and from the literature. A sensitivity analysis of the model shows that the most signi®cant parameters for the process are the biotrans- formation rate constant, the speci®c cell growth rate in the bulk, and the Thiele modulus for glucose. The computer simulation reveals that depending on the parameter values the gel particles might perform as a source or a sink of the product, thus enhancing or retarding the net process. For a speci®c parameter selection, the biotransformation in the pellets can prevail compared with the bulk in the begin- ning of the process as long as the direction of the product diffusion ¯ux is from the beads toward the bulk. Since the process in the free culture dominates, the system is more sensitive to parameters associated with the bulk phase (aeration rate, speci®c microbial growth rate, oxygen up- take rate). The model can be applied for prediction and fast evaluation of the performance of aerobic processes accomplished by immobilised growing cells. List of symbols A Particles total interfacial area L 2 Bi kR=D Biot number [±] c Concentrations M L 3 D i Diffusivity of the ith species L 2 T 1 d p Particle diameter [L] k 1 Rate constant of the biotransformation, Eq. (1) [±] k Mass transfer coef®cient LT 1 k L a Volumetric oxygen transfer coef®cient in the bulk T 1 K O 2 Saturation constant for oxygen in Monod equation (7) M L 3 K S Saturation constant for glucose in Monod equation (7) M L 3 L 31 e Parameter taking into account the effect of speci®c particles area [±] N p Number of gel particles [±] P Product concentration M L 3 q O 2 Speci®c oxygen consumption rate T 1 R Radius of the particles [L] r Radial co-ordinate [L] R GA , R O 2 Ratio of diffusivities related to glucose's one [±] t Time [T] T D G t=R 2 Dimensionless time [±] v Rate of enzyme (microbial) reaction [moles L 3 T 1 V Total volume of the system L 3 X Living cells concentration M L 3 n Empirical constant in Eq. (2) [±] Greek symbols a Parameter, taking into account the bio- transformation by free cells, Eq. (9) [±] b Share of viable cells leaking from the parti- cles interface into the bulk, between 0 and 1 g c GA =c G Degree of conversion [±] n v eff =v Ef®ciency of enzyme (microbial) activity of immobilised catalyst [±] e Void fraction [±] l Speci®c microbial growth rate, Eq. (7) T 1 q r=R Dimensionless radial co-ordinate [±] U 2 G Thiele modulus for glucose oxidation, Eq. (3d) [±] U 2 O 2 Thiele modulus for oxygen consumption, Eq. (3d) [±] Subscripts crit Denotes critical product concentration, Eq.(2) eq Denotes equilibrium concentration G Denotes quantities related to glucose GA Denotes quantities related to gluconic acid i Denotes quantities related to certain species in the system im Denotes values, related to immobilised cells max Denotes to maximum values O 2 Denotes quantities, related to oxygen s Denotes quantities, related to particles surface 1 Denotes values, related to the bulk phase Bioprocess Engineering 22 (2000) 233±241 Ó Springer-Verlag 2000 233 Received: 22 March 1999 V. Beschkov (&), S. Velizarov Institute of Chemical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Bl. 103, 1113 So®a, Bulgaria