Steady state diagrams

From the above simple steady-state diagram, point B is unstable because for any concentration change slightly to the right of B, the rate of substrate consumption is smaller than the rate of substrate supply. Therefore the concentration continues to increase and the system does not go back to the steady state B. Similarly, as the concentration is lowered slightly to put the system to the left of its steady-state point B, the rate of substrate supply is smaller than the rate of consumption and therefore the concentration continues to decrease and never returns to the steady state B. This makes the steady state B unstable. 

Let us illustrate this by the example of critical effects. For CO oxidation these effects were observed quite often at normal pressures but extremely rarely in the high-vacuum region (see Sect. 1). Meanwhile the above calculated data show that the critical effect can also be observed in this region. Steady-state diagrams (Figs. 2 and 3) demonstrate intervals for the conditions under which critical effects do exist. This contradiction has existed until recently. 

Figure 6.28. Comparison of CSTR and CPFR operation in a steady-state diagram of...

Figure 1. The steady-state diagrams for several values of the parameter d. Curve 1 corresponds to the dependence of the steady-state temperature 9° on parameter V for fixed values of 9 ", p and g. Curves 2-7 are lines 6° — Sj + d v — Vo) corresponding to different values of the stabilization parameter d.

Let now the system be started in the region of parameter values left to the ignition point 6 and at a value of temperature close to the value of T on the turning point of the lower branch of Fig. 4. One will observe then a thermal explosion, whereby after an induction period temperature increases suddenly and tends to the combustion branch of the steady state diagram, in a way similar to Fig. 3. Because of the unrealistically high value of the latter, we discard the part of the evolution referring to the final saturation, since for that stage the assumption of no consumption of the reactant breaks down. Fig. 5 describes the induction... 

Fig. 4. Steady-state diagram for the exothermic reaction model of eq. (16), Parameter value = 0.08... 

Figure A3.14.3. Example bifurcation diagrams, showing dependence of steady-state concentration in an open system on some experimental parameter such as residence time (inverse flow rate) (a) monotonic dependence (b) bistability (c) tristability (d) isola and (e) musliroom.

The distributions of excess, or injected, carriers are indicated in band diagrams by so-called quasi-Fenni levels for electrons or holes (Afp). These functions describe steady state concentrations of excess carriers in the same fonn as the equilibrium concentration. In equilibrium we have... 

The example spreadsheet covers a three-day test. Tests over a period of days provide an opportunity to ensure that the tower operated at steady state for a period of time. Three sets of compositions were measured, recorded, normalized, and averaged. The daily compositions can be compared graphically to the averages to show drift. Scatter-diagram graphs, such as those in the reconciliation section, are developed for this analysis. If no drift is identified, the scatter in the measurements with time can give an estimate of the random error (measurement and fluc tuations) in the measurements. 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

Figure 4.15 eombines equations (4.18), (4.20) and (4.22) in bloek diagram form. Under steady-state eonditions, the torque developed by the DC servo-motor is... 

The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the. v-plane at positions that produce a transient response that meets a given performance specification. It should be noted that a root locus diagram does not provide information relating to steady-state response, so that steady-state errors may go undetected, unless checked by other means, i.e. time response. 

At a location MRT and T are often measured with a sphere or ellipsoid representing the person, as shown in Fig. 5.10. In the diagram the energy balance on the globe at steady state is q q, or... 

Referring to Figure 2-35 (process diagram), the first law for this steady-state flow system becomes... 

FIGURE 2.23 Schematic diagram showing the routes of possible removal of drug from the receptor compartment. Upon diffusion into the compartment, the drug may be removed by passive adsorption en route. This will cause a constant decrease in the steady-state concentration of the drag at the site of the receptor until the adsorption process is saturated. 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

Fig. 10.9 Diagram of steady states I and III are domains of existence of single solution, II is a domain of existence of three solutions (two stable and one unstable). Lines A and A2 correspond to two stable solutions. Reprinted from Yarin et al. (2002) with permission...

Figure 3.17. Phase-plane representations of reactor stability. In the above diagrams the point -I- represents a possible steady-state solution, which (a) may be stable, (b) may be unstable or (c) about which the reactor produces sustained oscillations in temperature and concentration.

Figure 4. Evolution of the (N2/N1) ratio in a reservoir in the two cases of closed system evolution (as a function of t/T2, where t is the time since fractionation), or in an open-system, steady-state reservoir (the steady-state (N2/N1) ratio is plotted as a function of x/ T2, where x is the residence time of the magma in the reservoir). Initial fractionation results in an arbitrarily chosen ratio of 2, which is kept constant for the iirfluent magma in the continuously replenished reservoir. The diagram shows that radioactive equilibrium is reached sooner in a closed system evolution. It also illustrates the fact that the radioactive parent-daughter pair should be chosen such as T2 is commensmate with the residence time of the magma in the reservoir (e.g., x/ T2 between 0.1 and 10). If T2 is much longer than the residence time x, then the (N2/N1) ratio will remain close to the initial value (here 2). If T2 is much shorter than x, equilibrium will be nearly established in the reservoir.

Fig. 20. Schematic diagram showing the estimation of the time-average rate of S02 oxidation under periodic flow interruption or reduction employing steady-state oxidation rate vs liquid loading data (Figure from Haure etal., 1989, with permission, 1989, American Institute of Chemical Engineers.)...

For the rest of the control loop, Gc is obviously the controller transfer function. The measuring device (or transducer) function is Gm. While it is not shown in the block diagram, the steady state gain of Gm is Km. The key is that the summing point can only compare quantities with the same units. Hence we need to introduce Km on the reference signal, which should have the same units as C. The use of Km, in a way, performs unit conversion between what we dial in and what the controller actually uses in comparative tests. 2... 

Let s take another look at the algebra of evaluating the steady state error. The error that we have derived in the example is really the difference between the change in controlled variable and the change in set point in the block diagram (Fig. 5.6). Thus we can write ... 

Figure 4 shows the application (6) of potentials to the Pt and Au electrodes of the sandwich (vs. a reference electrode elsewhere in the contacting electrolyte solution) so that they span the E° of the poly-[Co(II/I)TPP] couple (Fig. 4B). There is a consequent redistribution of the concentrations of the sites in the two oxidation states to achieve the steady state linear gradients shown in the inset. Figure 4C represents surface profilometry of a different film sample in order to determine the film thickness from that the actual porphyrin site concentration (0.85M). The flow of self exchange-supported current is experimentally parameterized by applying Fick s first law to the concentration-distance diagram in Fig. 4B ...

---