Inflow concentration

The experiments were performed at a constant inflow concentration of ascorbic acid ([H2A]) in the CSTR. Oscillations were found by changing the flow rate and the inflow concentration of the copper(II) ion systematically. At constant Cu(II) inflow concentration, the electrode potential measured on the Pt electrode showed hysteresis between two stable steady-states when first the flow-rate was increased, and then decreased to its original starting value. The results of the CSTR experiments were summarized in a phase diagram (Fig. 6). 

We will assume that and X2 indicate the positive and negative signs, respectively, before the square root in equation (6.19). The final solution for P i) will depend on the inflow concentration, and the determination of Pi and P2 will depend on the boundary conditions. 

Equations (6.16) to (6.18) are still applicable to the pulse input. The boundary conditions and the inflow concentration, however, are different. The inflow concentration at t = 0+ will be zero. Thus, in equation (6.17), fP, = 0. The new boundary conditions are ... 

Now, consider an air-water system with a given inflow concentration and an outflow concentration that results from the mass transfer in the system and the inflow. If the system were to run for a long time, steady state would be achieved, although the system is nowhere near equilibrium, as shown in Figure 8.6. 

Figure 8.7. An infinitely long reactor with time-dependent inflow concentrations used to show the case when equilibrium is approached, but not steady state.

Finally, let us look at a system that approaches equilibrium, but is not at steady state. If we had an input concentration that is a function of time, and an inhnitely long reactor, the system would never be at steady state because of the temporal variations in the inflow. The outflow, however, would always approach equilibrium, regardless of the inflow concentrations because of the reactor length. This is shown in Figure 8.7. 

In open, or flow, reactors chemical equilibrium need never be approached. The reaction is kept away from that state by the continuous inflow of fresh reactants and a matching outflow of product/reactant mixture. The reaction achieves a stationary state , where the rates at which all the participating species are being produced are exactly matched by their net inflow or outflow. This stationary-state composition will depend on the reaction rate constants, the inflow concentrations of all the species, and the average time a molecule spends in the reactor—the mean residence time or its inverse, the flow rate. Any oscillatory behaviour may now, under appropriate operating conditions, be sustained indefinitely, becoming a stable response even in the strictest mathematical sense. 

The concentrations of A and B are related by the reaction stoichiometry to the inflow concentrations, so... 

A response showing multiple stationary states requires that the inflow concentration of B be significantly less than that of A. Multiple intersections and tangencies are only possible if... 

For larger inflow concentrations of the autocatalyst B, i.e. with b0 > a0/8, the curves R and L can only intersect once, whatever the flow rate. As shown in Fig. 1.13, we then have a monotonic dependence of ass on kf. 

Fig. 1.13. Cubic autocatalysis with relatively high inflow concentration of the catalyst, b0 =

Our aim is to determine the concentration of A in the reactor as a function of time and in terms of the experimental conditions (inflow concentrations, pumping rates, etc.). We need to obtain the equation which governs the rate at which the concentration of A is changing within the reactor. This mass-balance equation will have contributions from the reaction kinetics (the rate equation) and from the inflow and outflow terms. In the simplest case the reactor is fed by a stream of liquid with a volume flow rate of q dm3 s 1 in which the concentration of A is a0. If the volume of the reactor is V dm3, then the average time spent by a molecule in the reactor is V/q s. This is called the mean residence time, tres. The inverse of fres has units of s-1 we will call this the flow rate kf, and see that it plays the role of a pseudo-first-order rate constant. We denote the concentration of A in the reactor itself by a. 

Diagrams such as Fig. 6.3, which show the dependence of the stationary-state composition on a particular parameter, are known as bifurcation diagrams. It is customary, when trying to judge the efficiency of various processes for instance, to discuss the extent of reaction rather than the concentration of the reactant. The former is the fractional conversion of A into products and is given by the difference between the inflow concentration of A and its stationary-state value, i.e. how much A has reacted, divided by the original (inflow) concentration. For the extent of reaction we use the symbol y, and under stationary-state conditions... 

In the present case, however, we can use the stoichiometry of the reaction so the concentrations of A and B are linked to the (constant) inflow concentrations ... 

The process of obtaining suitable dimensionless forms for the present model is very similar to those followed in the previous chapters, 2.8 and 3.4 we need first to choose a reference concehtration and a reference timescale. For the first of these, the concentration, there is a natural choice, as we have seen already in eqn (6.6), i.e. the inflow concentration of the reactant a0. Again, using Greek letters for dimensionless quantities, we have... 

Although there are no a priori limits on po, which is simply the ratio of inflow concentrations, (other than P0 > 0) we might expect this to be less than unity under most circumstances. 

The reaction rate curve R is zero at complete conversion and also has low (but non-zero) values close to 1 — a = 0, with a maximum close to two-thirds conversion (actually at 1 — a = — / 0). Importantly, R does not depend on the residence time rres, although it does vary if / 0 is changed. The flow line L is zero when 1 — a = 0 since the inflow and outflow have the same composition (no conversion of A to B). The gradient of the flow line (Fig. 6.7(b)) is given by 1 /Tres, so it is steep for short residence times (fast flow rates) and relatively flat for long rres. (Note how tres actually compares fres and lch, so short residence times are those that are much less than the chemical timescale etc.) The flow line is, however, unaffected by the inflow concentration of the autocatalyst f 0. 

Fig. 6.8. (a) Schematic three-dimensional representation of the stationary-state surface (1 — ass)-tres- 0 showing the folding at low autocatalyst inflow concentrations which gives rise to ignition, extinction, and multiplicity, (b) The projection onto the P0-t , parameter plane of the two lines of fold points in the stationary-state surface, forming a typical cusp at P0 = gT,e> = 7 inside this cusp region the system has multiple stationary states outside, there is... 

The final term in eqn (6.22) involves the discriminant (1 — 8/30)1/2. For real roots, we therefore require that the dimensionless inflow concentration of the autocatalyst should be less than , so b0 < ja0 in real terms. 

When P0 = i, the two roots of eqn (6.22) are exactly equal. The ignition and extinction points are coincident at ires = 64/27 multistability is lost. For larger inflow concentrations of B the stationary-state extent of reaction increases smoothly with the residence time and the distinction between the flow and thermodynamic branch is lost. 

The tangency condition can only have real roots so long as (8 + 9K 1)/ 0 < 1. This provides limits on either the autocatalyst inflow concentration... 

Thus, reversibility decreases the range of inflow concentrations over which multiple stationary states can exist. If the reactor has no autocatalyst in the inflow, multistability exists over some range of residence times, no matter how small the equilibrium constant becomes. Otherwise, increasing the inflow concentration decreases the extent of reversibility (i.e. raises the minimum value for Kc) over which multistability can be found. 

The variation of Tr s and t s with k2 given by (6.60) can be drawn out as a projection in the parameter plane, in a similar way to that portrayed for the residence time-inflow concentration plane in Fig. 6.8 for the model without decay. Again we see the characteristic cusp shape as k2 - yg. 

The full expressions for R and L have been given in eqns (6.52) and (6.53). The only difference between these forms and those of the previous subsection is that the reaction rate curve R now involves the inflow concentration of the autocatalyst. The flow line L, its dependence on the residence time, and its minimum gradient are all independent of po. 

For small values of the autocatalyst inflow concentration, such that fi0 1, the roots of eqn (6.64) are given approximately by... 

Let us consider a few specific examples. We will take a system with the inflow concentrations fixed so that / 0 = 32. The tangencies for this case occur for... 

There is another type of time dependence possible in this system. If the inflow concentration of the autocatalyst is adjusted so that b0 - a0, then the ignition and extinction points merge at trcs = (k1ao) 1, with ass = Iu0 Under these special conditions, the coefficient of the term in (Aa)2 in the rate equation, and hence in the denominator of eqn (8.21), becomes zero as well as those of the lower powers in A a. Thus the inverse time dependence disappears, and the only non-zero term governing the decay of perturbation is that in (Aa)3 ... 

In the course of any given experiment we may vary the residence time. In between experiments there are now two parameters which we can alter the decay rate constant k2 and the inflow concentration of autocatalyst fi0. We thus wish to divide up the parameter plane into different regions, within each of which our experiments will reveal qualitatively different responses. We have already achieved this for the stationary-state behaviour, yielding regions of unique, isola, and mushroom patterns (see Fig. 6.18). We will now add the... 

Figure 19.5 Physiological pharmacokinetic model for hepatic uptake of drug constantly infused in the isolated rat liver perfusion system. Q, flow rate (mL/min) Cb, inflow concentration (pg/mL) Cs, sinusoidal concentration (pg/mL) Vs, sinusoidal volume (mL) X, binding constant (pg) Xm, maximum binding amount (pg) K, binding constant (mL/pg) kmt, internalization rate constant (min-1).

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