Assessment and Future Directions of Nonlinear Model Predictive ...

A Minimum-Time Optimal Recharging Controller 4452.1 Equation of StateThe following two-coefficient virial equation of state for a binary mixture [1]Z =PRT ρ =1+B mixρ + C mix ρ 2 (1)B mix = b 1 (T )x 2 1 +2b 12 (T )x 1 x 2 + b 2 (T )x 2 2 (2)C mix = c 1 (T )x 3 1 +3c 112(T )x 2 1 x 2 +3c 122 (T )x 1 x 2 2 + c 2(T )x 3 2 (3)is used in this work where x 1 ,x 2 are the component mole fractions and B mix andC mix are, in general, functions of temperature. Because the composition of theinlet gas mixture is maintained at the desired target composition, x 1 and x 2 areconstant at this composition in stages C and D. The equation of state for thesingle component streams in stages A 1 and B 1 is obtained by setting x 1 =1andx 2 = 0 in Eqs. 2 and 3. The equation of state for stages A 2 and B 2 is handledin a similar manner by setting x 1 =0andx 2 =1.PressureP (initial)initial stageP (final)P (ideal gas)HRi∆HIGfinal stageH RfT (initial)TemperatureT (final)Fig. 2. Isoenthalpic stage transition model2.2 Isoenthalpic TransitionThe first two transitions are described by the path shown in Figure 2 for boththe individual pure components and the gas mixture. The resulting equation forthe transition from an initial stage i to the next stage f becomes∆H i→f =0=−H R i + ∆H IG + H R f (4)where HiR is the residual enthalpy between the gas at the initial stage and thegas at ideal conditions at the initial temperature, ∆H IG is the change in enthalpyof the gas at ideal conditions between the final and initial temperatures, and HfRis the residual enthalpy between the gas at the next stage and the gas at idealconditions at the final temperature. The residual enthalpy is