Steady state profiles

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. 

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

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. 

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.

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.

Comparison of steady-state profiles shows that neglecting axial mass diffusion has very little effect on the temperature and concentration profiles even though the axial gradients are significant. However, Figure 16 shows that neglecting the axial thermal dispersion in the gas does affect the solution... 

Fig. 16. Steady-state profiles neglecting axial thermal dispersion, type I conditions.

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

The evolution of the dimensionless density profile across the soot layer is shown in Fig. 23. The initial gradual replenishment of the soot in the catalytic layer (at t = 140 s) is followed by sudden penetration events (t — 262 and 326 s) before the establishment of a steady state profile (at =531 and 778 s). Regarding the non-catalytic (thermal) layer only a gradual reduction of its thickness, accompanied by a very small reduction of its uniform density is observed. This simple microstructural model exhibits a rich dynamic behavior, however we have also established an experimental program to study the soot cake microstructure under reactive conditions. 

Due to the spherical geometry of the surface, the concentration profile across the boundary layer is no longer a straight line as was the case for the flat bottleneck boundary (Fig. 19.4). We can calculate the steady-state profile by assuming that CF and CFq = Cs/Fs/F are constant. Then, the integrated flux, ZF, across all concentric shells with radius r inside the boundary layer (r0 < r < r0 + 8) must be equal ... 

Calculate the vertical steady-state profile of benzene as well as the total vertical flux of benzene, XFbenzene (expressed as mass per unit time), if the following conditions hold ... 

The conclusion could be drawn from Fig. 4.2 that the steady-state profile depends essentially on the defect mobility or temperature - unlike the black sphere model, equation (4.1.70). The steady-state solution y(r) defines the stationary reaction rate K(00) through the effective radius of reaction R n-... 

Another distinctive feature of strong tunnelling recombination could be seen after a step-like (sudden) increase (decrease) of temperature (or diffusion coefficient - see equation (4.2.20)) when the steady-state profile has already been reached. Such mobility stimulation leads to the prolonged transient stage from one steady-state y(r,T ) to another y(r,T2), corresponding to the diffusion coefficients D(T ) and >(72) respectively. This process is shown schematicaly in Fig. 4.2 by a broken curve. It should be stressed that if tunnelling recombination is not involved, there is no transient stage at all since the relevant steady state profile y(r) — 1 - R/r, equation (4.1.62), doesn t depend on >( ). 

If the catalyst were not to decay but for some other reason, perhaps temperature control, the particles were taken out and recycled, then each might be supposed to be in pristine condition on entering the bed. Each particle would then undergo a transition during which the steady state profile of reactant within the particle would be built up. The analysis of Amundson and Aris (1962 this part is not tainted with the error mentioned above in fn. 13) may be used. We assume spherical particles of radius R, and call the profile of concentration at time a, c r, a). If D is the diffusivity of the reactant and k the rate constant per unit volume of catalyst,... 

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