Reciprocal residence time

Reciprocal Residence Time Figure 1. Observed sequence of oscillations. 

F i g. 11.7 Two-dimensional bifurcation diagrams in the parameter plane reciprocal residence time versus inflow concentration of essential species for categories 2C (a, b) and 2B (c, d). (Adapted from [9].)... 

Fig. 11.1 5 The experimentally determined bifnrcation structure of the chlorite-iodide reaction. The two constraints used here are the ratio of input concentrations, [CIO2 ]o/[I ]o> and the logarithm of the reciprocal residence time, log k(,. Notation filled triangles, supercritical Hopf bifurcations filled circles, saddle-node infinite period bifurcations in the region between these two kind of bifurcations, stable oscillations were observed. Open circles, excitable dynamics the smallest circles correspond to perturbations of 2 x 10 M in NaC102 the next smallest, 6 X 10 M in NaC102 the next smallest, 1.25 x 10 X x oMn - o io-2 1...

A typical experimental study of bistability requires monitoring the steady state concentration of a particular species as a function of a bifurcation parameter such as reactant flow rate. (Bihircation parameters are described in more detail in a following section.) A convenient species to monitor in the iodate-arsenite reaction is iodide, the autocatalyst. Figure 4 shows the steady state iodide concentration as a function of the reciprocal residence time, kQ. As the flow rate is increased, displacing the system from equilibrium (where the extent of reaction, and iodide concentration, is high), the iodide concentration gradu-... 

Figure 4 Experimental steady state iodide concentrations as a function of reciprocal residence time. Arrows indicate transitions from one steady state to another (Reprinted from Ref. 14 with permission of the American Institute of Physics.)...

Ganapathisubramian Showalter (1984b) measured the steady state iodide concentration in the iodate-arsenous acid system as a function of reciprocal residence time and found multistationarity. 

In a closed system, the (globally stable) asymptotic solution is obviously a = 0, b = Oq, where Uq is the initial concentration of A. Not very interesting Now we want to study this reaction in a CSTR. Assume that we let flow in a solution with concentration Uq at a flow rate of F mLs and that the reactor has a volume VmL. In order to include the effects of the flow in a fashion that does not explicitly depend on the reactor geometry, we define the reciprocal residence time ko by... 

Figure 4.9 Bistability in the chlorite-iodide reaction as measured by (a) 460-nm absorbance (proportional to [I2]) and (b) an iodide-selective electrode. Dashed arrows indicate spontaneous transitions between states. Reciprocal residence time in CSTR = 5.4 X 10-- s-, [ClOjjo = 2.4 X 10-" M, pH = 3.35. (Adapted from Citri and Epstein, 1987.)...

Figure 15.5 Two-reactor model of imperfect mixing in a CSTR. (a) Idealized CSTR in which mixing is perfect and concentrations are homogeneous throughout the reactor, (b) Stirred tank with cross flow between active (well-mixed) and dead (poorly mixed) zones. Q represents volume flow, C is concentration, k is flow rate (reciprocal residence time), and V is volume. Subscripts i specifies chemical species, 0 signifies input from reservoirs, r is homogeneous reactor, and a and d are active and dead zones, respectively. (Adapted from Kumpinsky and Epstein, 1985.)...

Fig. 7.1. Measured hysteresis in our reaction system [1]. Plot of the redox potential of Br, Fpt, as a function of the flow rate coefficient fct (in units of reciprocal residence times, the time spent by a volume element in the laminar flow reactor (LFR)). Filled dots represent one of the stable stationary states (the oxidized state) and empty dots the other stable state, the reduced state. Prom [1]...

Fig. 8 Birhythmicity in the chlorite-bromate-iodide system. Potential is that of Pt electrode vs. Hg/Hg SO reference electrode. At times indicated by the arrows, flow rate Is changed. Flow rate in each time segment (measured as reciprocal residence time k ) is shown at top. Note that A...

Fig. 1. Low iodide, unstable, and high iodide steady states on composition line. Combined reactant feed stream concentrations [lOg lQ = 1.00 X 10-3 M, [HgAsOglo = 5.00 x 10-3 M, [I ]o = 4.41 X 10-5 m (indicated by vertical dashed line). Acidity maintained constant with sulfate/bisulfate buffer at pH = 2.122 ([H+] = 7.55 X 10-3 M). Reciprocal residence time ko = 8.20 X 10 3 s-1. Temperature 25.0 + 0.2°C. From Ref. [1].

In Fig. 28, the abscissa kt is the product of the reaction rate constant and the reactor residence time, which is proportional to the reciprocal of the space velocity. The parameter k co is the product of the CO inhibition parameter and inlet concentration. Since k is approximately 5 at 600°F these three curves represent c = 1, 2, and 4%. The conversion for a first-order kinetics is independent of the inlet concentration, but the conversion for the kinetics of Eq. (48) is highly dependent on inlet concentration. As the space velocity increases, kt decreases in a reciprocal manner and the conversion for a first-order reaction gradually declines. For the kinetics of Eq. (48), the conversion is 100% at low space velocities, and does not vary as the space velocity is increased until a threshold is reached with precipitous conversion decline. The conversion for the same kinetics in a stirred tank reactor is shown in Fig. 29. For the kinetics of Eq. (48), multiple solutions may be encountered when the inlet concentration is sufficiently high. Given two reactors of the same volume, and given the same kinetics and inlet concentrations, the conversions are compared in Fig. 30. The piston flow reactor has an advantage over the stirred tank... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The easiest reactor to analyze is a steady-state CSTR. Biochemists call it a chemostat because the chemistry within a CSTR is maintained in a static condition. Biochemists use the dilution rate to characterize the flow through a CSTR. The dilution rate is the reciprocal of the mean residence time. 

This equation has two parameters t, the mean residence time (z = V/F) with dimensions of time and k, the reaction rate constant with dimensions of reciprocal time, applying for a first-order reaction. The concentration of reactant A in the reactor cannot, under normal circumstances, exceed the inlet feed value, Cao and thus a new dimensionless concentration, Cai, can be defined as... 

The negative reciprocal of phosphorus residence time in each reservoir is found on the diagonal entries of matrix A (Table 7.3). A is factored giving six eigenvalues and six characteristic times of the system as the negative reciprocal of the eigenvalues... 

These results fall fairly close to chondritic values, 0.51264 and 0.1967, respectively. The Nd kinetic factors (the reciprocal of residence times) can be computed from equations (7.4.6) as... 

The reciprocal of this term is called the process opeidoop time constant and we use the symbol r,. The bigger the time constant, the slower the dynamic response will be. The solution [Eq. (6.48)] predicts that a small value oflc or a big value of r will give a large process time constant. Again, this makes good physical sense. If there is no reaction, the time constant is just equal to t = V/F, the residence time. 

Another useful measure of biogeochemical processing is the fractional residence time or turnover time of a material in a reservoir. Computation of this time is similar to that of a residence time except that some subset of the input or output processes is substituted into the denominator of Eq. 1.2. The resulting turnover time represents how long it would take for that subset of processes by itself to either supply or remove all of the material from the reservoir. Turnover times can be calculated for reservoirs that are not in steady state. As will be shown in Chapter 21, the residence time can be computed by summing the reciprocals of the turnover times. 

Using the rock cycle as an example, we can compute the turnover time of marine sediments with respect to river input of solid particles from (1) the mass of solids in the marine sediment reservoir (1.0 x 10 g) and (2) the annual rate of river input of particles (1.4 X lO g/y). This yields a turnover time of (1.0 x 10 " g)/(14 x lO g/y) = 71 X lo y. On a global basis, riverine input is the major source of solids buried in marine sediments lesser inputs are contributed by atmospheric feUout, glacial ice debris, hydrothermal processes, and in situ production, primarily by marine plankton. As shown in Figure 1.2, sediments are removed from the ocean by deep burial into the seafloor. The resulting sedimentary rock is either uplifted onto land or subducted into the mantle so the ocean basins never fill up with sediment. As discussed in Chapter 21, if all of the fractional residence times of a substance are known, the sum of their reciprocals provides an estimate of the residence time (Equation 21.17). 

Note that E(t) has dimensions of (time) and that E(0) is dimensionless. Intuitively, it is obvious that if a system has a mean residence time of, say, 2.3 min, then when the system RTD time scale is compressed by a factor of 1/2.3 (0 — t/2.3), the ordinate must be expanded by the reciprocal of this factor [E(0) = 2.3 E(f)] in order to maintain the unit area property. This is illustrated in Fig. 4. 

Finally, we ignore expansion effects in this chapter thus, we take s = 0 throughout. This means that we may use the terms mean residence time, reactor holding time, space time, and reciprocal space velocity interchangeably. 

The first term is the ratio of maximum loading of solid for a specific inlet concentration to that concentration, whereas the second term is the space velocity (the reciprocal of the residence time), and the third term is the slope of the breakthrough curve. Thus, the constant pattern condition is achieved for... 

---