Assessment and Future Directions of Nonlinear Model Predictive ...

NMPC of the Hashimoto Simulated Moving Bed Process 477around an infinitely small cross-section of the column (TB- is referred to as A,while TB+ is denoted as B):SeparatorReactor∂c b,i∂t+ (1 − ɛ b)3k l,iɛ b R p∂ 2 c b,i(c b,i − c p,i | r=Rp )=D ax∂z 2 − u ∂c b,i(1)∂z(∂r 2 ∂c )]p,i=0, (2)∂r ∂r(1 − ɛ p ) ∂q [i∂t + ɛ ∂c p,i1p − ɛ p D p,i∂tr 2∂c b,i+ r liqkin,i∂t= D ∂ 2 c b,iax∂z 2 − u ∂c b,i∂zwith appropriate initial and boundary conditions(3)c b,i | t=0= c b,i (t =0,z), c p,i | t=0= c p,i (t =0,z,r), (4)∂c b,i∂z ∣ =u(c b,i − c in ∂c b,ii ),z=0D ax ∂z ∣ =0, (5)z=L∂c p,i∂c p,i∂r ∣ =0,r=0∂r ∣ = k l,i(c b,i − c p,i |r=Rpɛ p D r=Rp) . (6)p,iThe adsorption equilibrium and the reaction kinetics have been determined experimentallyin [11]. The adsorptive behaviour can be modelled best by an asymmetricmulti-component Langmuir isothermq i =H i c i1+ ∑ i = A, B, (7)b i,j c jjwhere H i denotes the Henry coefficient which dominates the adsorption. Theracemization of Tröger’s base is regarded as homogeneous reaction described byfirst order kinetics:r liqkin,i = ν ik m (c b,i − c b,j ) i, j = A, B i ≠ j. (8)From mass and concentration balances, the relations at the inlet and the outletnodes result asDesorbent node Q IV + Q De = Q I (9)c outi,IV Q IV = c ini,I Q I i = A, B (10)Extract node Q I − Q Ex = Q II (11)Feed node Q II + Q Fe = Q III (12)c outi,II Q II + c i,F e Q Fe = c ini,III Q III i = A, B (13)Raffinate node Q III − Q Ra = Q IV , (14)