2. ON THE MODELING OF AN AIRLIFT PHOTOBIOREACTOR Christo Boyadjiev, Jose Merchuk Introduction Mathematical model Average concentration model Hierarchical approach Conclusions INTRODUCTION Photobioprocesses include dissolution of an active gas component (CO 2, O 2 ) in liquid and its reaction with a photoactive material (cells). These two processes may take place in one environment (mixed bioreactors ; bubble columns) or in different environments (air lift photobioreactor). The comparison of these systems shows apparent advantages in the use of airlift photo – bioreactors, because the possibility of manipulation of the light - darkness history of the photosynthetic cells. The hydrodynamic behavior of the gas and the liquid in airlift reactors is very complicated, but in all cases the process includes convective transport, diffusion transport and volume reactions. That is why convection - diffusion equation with volume reaction may be use as a mathematical structure of the model.

4. MATHEMATHICAL MODEL Let’s consider an airlift reactor with a horizontal cross-sectional area F 0 for the riser zone and F 1 for the downcomer zone. The length of the working zones is l. The gas flow rate is Q 0 and the liquid flow rate (water) - Q 1. The gas and liquid hold - up in the riser are ε and (l – ε). The concentrations of the active gas component (CO 2 ) in the gas phase is c(x,r,t) and in the liquid phase – c 0 (x, r, t) for the riser and c 1 (x 1,r,t) – for the downcomer, where x 1 = l - x. The concentration of the photoactive substance in the downcomer is c 2 (x 1,r,t) and in the riser - c 3 (x 1,r,t). The average velocities in gas and liquid phases are: The interphase mass transfer rate in the riser is: The photoreaction rates in the downcomer and the riser are taken, respectively, as:, J = J (x 1, r, t), J 1 = J 1 (x 1, r, t).. Let consider cylindrical surface with radius R 0 and length 1 m, which is regularly illuminated with a photon flux density J 0. The photon flux densities over the cylindrical surfaces with radiuses r  R 0 is:.

6. r RoRo rr J The increasing of the photon flux density between r and r -  r is:. The volume between the cylindrical surfaces with radiuses r and r -  r is:, and the decreasing of the photon flux density as a result of the light absorption in this volume is: The different between photon flux densities for r and r -  r is:,. As a result where J(R 0 ) = J 0. The solution is:.

8. The equations for the distribution of the active gas component in the gas and liquid phases in the riser are:, ε = const. The equations for the distribution of the active gas component in the gas and liquid phases in the downcomer are: x 1 = l – x. Photochemical reaction may take place in riser too, and the equation for the cell concentration is:.

10. The initial conditions will be formulated for the case, of thermodynamic equilibrium between gas and liquid phases, i. e. a full liquid saturation with the active gas component and the process starts with the starting of the illumination : where c (0) and c 2 (0) are initial concentrations of the active gas component in the gas phase and the photoactive substance in the liquid phase. The boundary conditions are equalities of the concentrations and mass fluxes at the two ends of the working zones - x = 0 (x 1 = l) and x = l (x 1 = 0). The boundary conditions for c (x, r, t) and c 0 (x, r, t) are:

12. The boundary conditions for c 1 (x 1, r, t), c 2 (x 1, r, t) and c 3 (x, r, t) are: The radial non - uniformity of the velocity in the column is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter) in the column scale-up. In the specific case of photo-reactions, an additional factor is the local variations of light availability. Here an average velocity and concentration in any cross - section is used. This approach has a sensible advantage in the collection of experimental data for the parameter identification because measurement the average concentrations is very simple in comparison with local concentration measurements.

14. AVERAGE CONCENTRATION MODELS For the velocity and concentration of the gas phase in the riser: After introducing of average velocity and concentration in the equation of the gas phase in riser and integrating over r in the interval [0, r 0 ] is obtained: Let’s use a property of integral average functions: where

16. As a result is obtained: with boundary condition: must be obtained from the continuity equation after integrating over r in the interval [0, r 0 ] :

18. The parameters in the model are of two types - specific model parameters ( D, k, ,  ) and scale model parameters ( A, B, G ). The last ones (scale parameters) are functions of the column radius r 0. They are a result of the radial non-uniformity of the velocity and the concentration and show the influence of the scale - up on the equations of the model. The parameter  may be obtained beforehand from thermodynamic measurements. From the model follows that the average radial velocity component influences the transfer process in the cases, i. e. when the gas hold - up in not constant over the column height. As a result for many cases of practical interest and the radial velocity component will not taken account in this case. The hold - up  can be obtained using: where l and l 0 are liquid level in the riser without gas motion. The values of the parameters D, k, A, B, G must be obtained using experimental data for measured on a laboratory column. In the cases of scale - up A, B and G must be specified only (because they are functions of the column radius and radial non-uniformity of the velocity and concentration), using a column with real diameter, but with small height ( D and k do not change at scale – up).

20. The model for the liquid phase in the riser is : where A 0, B 0 and are obtained in the same way as A, B and G. The concrete expressions of A, B and G are not relevant because those values must be obtained, using experimental data in any case. In the equations of the downcomer must put the average velocity, concentrations and photon flux density:

22. After integration over r in the interval [ r 0, R 0 ] the problems has the form and are obtained in a similar way as, but taking into account that the limits of the integrals are [ r 0, R 0 ]. may be obtained after integration of the photon flux density model :,.

24. The equation for and A 3, B 3 and G 3 are obtained in a similar way. may be obtained after integration: For many cases of practical interest and the number of parameters of the model decreases, i.e. The photo-chemical reaction rates equation shown in are acceptable when J and c 1 are very small.

26. A more general form of the photo-chemical reaction rate equation is (written here for the downcomer): Another possible form for these equations could be: where the kinetic parameters k 0, γ, γ 1, γ 2 must be obtained wsing experimental data. Applying the last expressions in the model equations, the photo-chemical reaction rate equations could be:

28. The parameters c (0), c 2 (0), J 0, ε, χ, α, β, k 0, γ, γ 1, γ 2, k, D, D 0, D 1, D 2, D 3 are related with the process (gas absorption with photobioreaction in liquid phase), but the parameters R 0, r 0, A, A 0, A 1, A 2, A 3, B, B 0, B 1, B 2, B 3, M, M 3, P 1, P 2 are related with the apparatus (column radius and radial non-uniformity of the velocities and concentrations). The equations allow to obtain ( k, D, D 0, A, B, A 0, B 0 ) without photo-bioreaction if it can be assumed that. The equations allow to obtain ( k, D, D 0, A, B, A 0, B 0 ) without photo-bioreaction if it can be assumed that. HIERARCHICAL APPROACH The obtained equations are the mathematical model of an airlift photobioreactor. The model parameters are different types: The obtained equations are the mathematical model of an airlift photobioreactor. The model parameters are different types: – beforehand known ( c (0), c 2 (0), R 0, J 0, r 0 ); – beforehand obtained ( ε, χ, α, β, k 0, γ, γ 1, γ 2 ); – obtained without photo-bioreaction ( k, D, D 0, A, B, A 0, B 0 ); – obtained with photo-bioreaction ( D 1, D 2, D 3 ), because diffusion of the gas and photoactive substance is result of the photobioreaction; – obtained in the modelling and specified in the scale - up (A, A 0, A 1, A 2, A 3, B, B 0, B 1, B 2, B 3, M, M 3, P 1, P 2 ), because they are functions of the column radius and radial non-uniformity of the velocity concentration. CONCLUSIONS The results obtained show a possibility of formulation airlift photobioreactor models using average velocities and concentrations. These models have two type parameters, related to the process and to the apparatus (scale - up). This approach permit to solve the scale - up problem concerning the radial nonuniformity of the velocity and concentration, using radius dependent parameters. The model parameter identification on the bases of average concentration experimental data is much simpler than considering the local concentration measurements. The results obtained show a possibility of formulation airlift photobioreactor models using average velocities and concentrations. These models have two type parameters, related to the process and to the apparatus (scale - up). This approach permit to solve the scale - up problem concerning the radial nonuniformity of the velocity and concentration, using radius dependent parameters. The model parameter identification on the bases of average concentration experimental data is much simpler than considering the local concentration measurements.