Implicit algebraic loop

The determination of the steam density, pj, therefore requires the simultaneous solution of two algebraic equations. This represents an IMPLICIT algebraic loop and cannot be solved within a simulation program without the incorporation of a trial and error convergence procedure. 

Under normal circumstances, the use of a characteristic velocity equation of the type shown above can cause difficulties in computation, owing to the existence of an implicit algebraic loop, which must be solved, at every integration step length. In this the appropriate value of L or G satisfying the value of h generated in the differential mass balance equation, must be found as shown in the information flow diagram of Fig. 3.54. 

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

As discussed by Franks (1972), in order to solve this system of equations, a value of temperature T must be found to satisfy the condition that the difference term 6 = P - Zpj is very small, i.e., that the equilibrium condition is satisfied. This is known as a bubble point calculation. The above system of defining equations, however represent, an implicit algebraic loop and the trial and error solution procedure can be very time consuming, especially when incorporated into a dynamic simulation program. 

Ideal Gas Law 535 Immiscible liquid phases 167, 180 Implicit algebraic loop 200, 557... 

The highly interactive nature of the balance and equilibria equations for the distillation period are depicted in Fig. 3.66. An implicit, iterative algebraic loop is involved in the calculation of the boiling point temperature at each time interval. This involves guessing the temperature and calculating the sum of the partial pressures, or mole fractions. The condition required is that Zyi + yw = 1. The iterative loop for the bubble point calculation is represented by the five interconnected blocks in the lower right hand corner of Fig. 3.66. The model of Prenosil (1976) also included an efficiency term E for the steam heating, dependent on liquid depth L and bubble diameter D. 

In this example, it is assumed that the holdup relationship is given as a direct empirical function of the continuous and dispersed phase flow rates. This avoids any difficulties due to a possible algebraic implicit loop in the solution... 

The algebraic equations for temperatures and volumes are selected so as to be solved directly by the DAE solver, since they are implicit equations and we prefer to use a common numerical solver instead of tuning tolerances for internal and external loop solvers. 

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