State profiles

FIGURE 2.10 TRIPOD Failure-State Profiles of Two Production Platforms (Wagenaar, 1992). 

Figure 5.68. With F = 30 and n = 1.7 this steady-state profile is obtained from TUBE.

Steady state vs. non-steady state profiles. Determination of " Th deficits is relatively simple. Production is balanced against decay and export, with the expectation that in the absence of the latter, " Th should be in radioactive equilibrium with and the flux is given by the integral term in Equation (9) ... 

It is straightforward to correct for non-steady state profiles if a station is occupied repeatedly over a period of weeks to months. However, this has been done in relatively few instances (e.g., Buesseler et al. 1992b Bacon et al. 1996 Cochran et al. 2000 Benitez-Nelson et al. 2001b Friedrich and Rutgers van der Loeff 2002). More often, oceanographic cruises attempt to cover large areas or occupy large numbers of stations, few of which are re-occupied in a systematic manner. 

In these equations, = 0 is the bottom of the catalyst bed and Xx is the conversion in the flow direction from bottom to top, while X2 is the conversion in the opposite flow direction. Bunimovich et al. (1990) suggest using Eqs. (52) to (54) for an initial estimate of the temperature profiles in order to speed up conversion on integration of the full model equations in Table X. This step would only be taken if it were the stationary cyclic state profiles that are wanted. 

Fig. 8-2. Steady-state profiles of the saturation index, omegadel = omega-1, the dissolution rate, and the respiration rate.

Another conceivable limiting case, though one less likely to be approached in practical cases, is that where the total hydrogen concentration always remains far below that of the traps, which continue to capture hydrogen irreversibly. For this case, as Corbett et al. (1986) have pointed out, the concentration of free monatomic hydrogen will approach a quasisteady-state profile that decays exponentially with the depth x. The concentration of trapped hydrogen, of course, will at any point of space approach a linear increase with time. 

Continuous binary distillation is illustrated by the simulation example CON-STILL. Here the dynamic simulation example is seen as a valuable adjunct to steady state design calculations, since with MADONNA the most important column design parameters (total column plate number, feed plate location and reflux ratio) come under the direct control of the simulator as facilitated by the use of sliders. Provided that sufficient simulation time is allowed for the column conditions to reach steady state, the resultant steady state profiles of composition versus plate number are easily obtained. In this way, the effects of changes in reflux ratio or choice of the optimum plate location on the resultant steady state profiles become almost immediately apparent. 

Fig. 2 These steady-state profiles were obtained from TUBE by setting three values of k.

DISCHARGE - Dissolved Oxygen and BOD Steady-State Profiles Along a River System... 

Stagewise and finite-differenced models involve changes with time and distance. When the model is written in array form the variable can be plotted as a function of the array index. This is done by choosing an index variable for the Y axis and the [ ] symbol for the X axis. The last value calculated is used in the plot, which means that if the steady-state has been reached then it is a steady-state profile with distance. An example is given in the Screenshot Guide in Section 2 of the Appendix and in many other simulation examples. 

The steady-state profile in the liquid is calculated in Section 9.6. 

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Equations (8.1)-(8.13) can be solved to provide transient- or steady-state profiles of O2 and CH4 concentration, reaction rates and surface fluxes for any combination of the controlling variables 9q,0], v,k, a, Vm,Vq and Vr. Where, as is usual, one or more of the controlling variables may be further simplified, approximated or neglected, process-based simulation of CH4 emission becomes possible using a relatively limited set of input data. 

Results. Figure 8.2 gives steady-state profiles of O2 and CH4 and the corresponding reaction rates calculated with the model for the fixed root system defined in Assumption 9. Net O2 consumption is 460 tLmolm h net CH4 emission is 480tLmolm h the fractions of the O2 and CH4 fluxes through the plant are 0.84 and 0.97, respectively, and the fraction of CH4 oxidized prior to emission is 0.13. These are all credible numbers. 

Figure 26. Predictions of the Adler model shown in Figure 25 assuming interfacial electrochemical kinetics are fast, (a) Predicted steady-state profile of the oxygen vacancy concentration ( ) in the mixed conductor as a function of distance from the electrode/electrolyte interface, (b) Predicted impedance, (c) Measured impedance of Lao.6Cao.4Feo.8-Coo.203-(5 electrodes on SDC at 700 °C in air, fit to the model shown in b using nonlinear complex least squares. Data are from ref 171.

Instead, the simultaneous method can be extended to select adaptively a sufficient number of finite elements. Here, we note that even if we set any element length to zero, the collocation equations and the continuity equations are still satisfied. Thus, any number of zero length (or dummy) elements can be added without changing the control or state profiles, or the solution to the NLP. Vasantharajan and Biegler (1990) take advantage of this important property and propose an adaptive element addition approach embedded within the simultaneous solution strategy. 

In order to illustrate this approach, we next consider the optimization of an ammonia synthesis reactor. Formulation of the reactor optimization problem includes the discretized modeling equations for a packed bed reactor, along with the set of knot placement constraints. The following case study illustrates how a differential-algebraic problem can be optimized efficiently using (27). In addition, suitable accuracy of the ODE model can be obtained at the optimum by directly enforcing error restrictions and adaptively adding elements. Finally, bounds on the continuous state profiles can be enforced directly in the optimization problem. 

Fig. 5. Constant V optimal state profile. Reprinted with permission from Comp. Chem. Eng., 14, No. 10,1083-1100, S. Vasantharajan and L. T. Biegler, Simultaneous Optimization of Differential/Algebraic Systems with Error Criterion Adjustment," Copyright 1990, Pergamon Press PLC.

For these time periods, the ODEs and active algebraic constraints influence the state and control variables. For these active sets, we therefore need to be able to analyze and implicitly solve the DAE system. To represent the control profiles at the same level of approximation as for the state profiles, approximation and stability properties for DAE (rather than ODE) solvers must be considered. Moreover, the variational conditions for problem (16), with different active constraint sets over time, lead to a multizone set of DAE systems. Consequently, the analogous Kuhn-Tucker conditions from (27) must have stability and approximation properties capable of handling ail of these DAE systems. 

The mass transfer coefficient is usually much lower in the Hquid phase, and therefore is a function of R, the distance from the wall to the interface. One would have to solve for the steady-state profile C iR), and find its average CX(z) to insert into the PFTR mass-balance equations simultaneously to find Ca(L) in each phase. 

Fig. 10. Adiabatic and nonadiabatic steady-state Profiles, type I conditions.

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