Assessment and Future Directions of Nonlinear Model Predictive ...

306 H. Arellano-Garcia et al.application. Thus, we propose alternatively a dynamic adaptive back-off strategyfor a NMPC scheme embedded in an online re-optimization framework.The performance of both proposed approaches is assessed via application to arunaway-safe semi-batch reactor under safety constraints.2 Problem FormulationA strongly exothermic series reaction conducted in a non-isothermal fed-batchreactor is considered. The reaction kinetics are second-order for the first reactionproducing B from A, and an undesirable consecutive first-order reactionconverting B to C. The intermediate product B is the desired product.2A K 1−→ B K 2−→ C (1)A detailed first-principles model of the process is given by a set of DAEs basedon mass balances:n 2 Aṅ A = −ν ak E A101V e− RT + feed ; ṅB = −k 02 n B e − E A2 n 2 RT +AE A1k01V e− RT ;ṅ C =+k 02 n B e − E A2RT ,(2)the energy balance:˙QHTcool˙¯T cool = ˙V cool · ρ cool · c p,cool · (T cool,in − ¯T cool ) −; T ˙ =V cool · ρ cool · c p,cool˙Q reac + ˙Q feed + ˙Q cooln SÈi (cpixi) (3)and constitutive algebraic equations:˙Q reac = − (h iṅ i)=(h 0A + c pA(T − T 0))ṅ A +(h 0B + c pB(T − T 0))ṅ B+(h 0C + c pC(T − T 0))ṅ C˙Q feed =(h 0A + c pA(T − T 0)) · feed˙Q HTcool = −k HTA(T − ¯T cool )=−k HT(0.25πd 2 +4Vd −1 )(T − ¯T cool )n S = n A + n B + n C; n S(c pix i)=c pAn A + c pBn B + c pCn Ci(4)V = nA ˜M A + n B ˜M B + n C ˜M Cn A ¯˜ρ A + n B ¯˜ρ B + n C ¯˜ρ Cn S.In these equations V denotes the varying volume, n i the molar amount ofcomponent i, T , T F , ¯T cool , T cool , the reactor, dosing, jacket and cooling mediumtemperatures, respectively. h 0i are the specific standard enthalpies, k HT theheat transfer coefficient, d the scaled reactor diameter, A the heat exchangesurface, ˜Mi molecular weights, ρ i densities and c pi are heat capacities. Besides,since the heat removal is limited, the temperature is controlled by the feedrate of the reactant A(feed), and the flow rate of the cooling liquid ˙V cool inthe nominal operation. The reactor is equipped with a jacket cooling system.