Oxygen Diffusion in Animal Cells Slab Model

Transcription

1 1 Oxygen Diffusion in Animal Cells Slab Model Chemical Engineering Department Student Reg. No Computational Method from the Simulation Naim Hasolli

2 Oxygen Diffusion in Animal Cells 2 Statement: Slab Model The system is modeled by taking small segments divided in n increments. The slab model is, as shown in Fig. 1, divided in segments. The volume of a single segment (Fig. 2) is then as follows: (1) DVn = Dx A Dx Solid Bulk liquid DV Figure 2. nth Segment of the Slab Figure 1. Slab Model A

3 3 Statement: Slab Model The system is modeled by taking small segments divided in n increments. The volume of a single segment is then as follows: a) (1) DV = Dx A Figure 3. a) Slab Model and b) a DV segment Solid Bulk liquid LSla Dx b b) The segment thickness derived from VSphere / ASphere to LSlab = RP / 3 (2) DV A

4 4 Mass Balance: Mathematical Model Oxygen balance: Accumulation rate within the nth element of the cell with the volume DV is equal to S n1 Sn Sn ds n +1 rsn jn-1 Dx A Dx dt jn A jn n DV A Figure 3. Oxygen balance - jn -1 A = diffusion rate entering the DV diffusion rate leaving the DV + rsn A Dx reaction rate within the DV ds n A Dx = jn A - jn -1 A + rsn A Dx dt Reaction rate is expressed as: rsn = -OURmax X Sn K S + Sn (2) (3)

5 5 Solving ODEs in MATLAB: Initial Values Symbol Description Value Unit Increment length, LSlab/6 LSlab/6 m DS Diffusion coefficient 7.0E-6 m2/h KS Saturation constant 1 kg/m3 OURmax Oxygen uptake rate 0.01 g/(kg h) Length of the slab 3.3E-5 m S0 Substrate concentration in Bulk Liquid 10 g/m3 Sn Initial Substrate concentration of S1-6 1 g/m3 X Biomass concentration 1 kg/m3 Deltax (Dx) LSlab

6 6 Solving ODEs in MATLAB: Function File The Oxygen balance ODEs are slightly modified to fit the condition for the slab model according to Fig. 2. Code 4. Matlab Function File for Slab Model.

7 7 Command Window File Code 5. Matlab Command Window File for Slab Model.

8 8 Sphere vs. Slab: Substrate Concentration b) a) Figure 4. Concentration vs. Time for a) Slab and b) Spherical Model

9 9 Sphere vs. Slab: Substrate Concentration b) a) Figure 5. Concentration vs. Radius/Length for a) Slab and b) Spherical Model

10 Oxygen Diffusion in Animal Cells 10 For given conditions the oxygen concentration within the aggregate of the animal cell the most noticeable effect of the parameters is shown for variation of diffusion coefficient DS and radius RP in case the conditions are considered unchanged concerning the saturation constant KS and the bulk liquid concentration S0. The bulk liquid concentration S0 shows the effect on the level outer substrate concentration but no significant effect on the diffusion behavior inside the cell. The change of the DS by one order of magnitude effects the penetration distance for given reaction end time. For order (-9), oxygen reaches the center of the cell soon after reaction is started. For saturation constant KS values the effect is hardly noticeable. The smaller the radius RP of cell the faster the reaction is proceeded and the oxygen reaches the center of the cell for given reaction end time.

11 Oxygen Diffusion in Animal Cells Reference [1] I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil, Biological Reaction Engineering Dynamic Modeling Fundamentals with Simulation Examples, 2nd Edition 2003, WILEY-VCH Verlag, Weinheim, pp [2] [3] 11