Approach to the Steady State

In reactor startup it is often very important /tow temperature and concentrations approach their steady-state values. For example, a significant overshoot in temperature may cause a reactant or product to degrade, or the overshoot may be imacceptable for safe operation. If either case were to occur, we would say that the system exceeded its practical stability limit. Although we can solve the imsteady temperature-time and concentration-time equations numerically to see if such a limit is exceeded, it is often more insightful to study the approach to steady state by using the temperature-concentration phase plane. To illustrate these concepts we shall confine our analysis to a liquid-phase reaction carried out in a CSTR. 

Neglecting AC, because it changes the heat of reaction insignificanth over the temperature range of the reaction, the heat of reaction is assumed constant at  

Unsteady-State Nonisotheimal Reactor Design Chap. 9 

Figufe E9-4.3 Concentration-temperature pliase-plane trajectory. 

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If, as it is usually done, the interaction of enzyme with glycal is studied in the presence of substrate S having Michaelis constant K , the observed rate constant k pp, for the approach to the steady-state inhibition has to be corrected for the competition of substrate for the free enzyme, in order to calculate the rate constants kp , kp, and k yj, from the experimental data. 

Usually, although not always, the approach to the steady state following commencement of the illumination or the decay during a subsequent dark period is too rapid for satisfactory measurements. In such cases it is nevertheless possible to observe average rates of polymeriza-... 

The results in Fig. 5.157 show the approach to the steady-state for a setpoint of 80°C. The influence of control parameter changes is shown in Fig. 5.158. 

A radically different approach to the steady-state problem was investigated by Hsing (H6). In this approach the steady-state flow problem was formulated as the following constrained minimization problem ... 

This value is an order of magnitude larger than the transition time following a concentration or current step at the plate approach to the steady-state flux following a concentration step is complete to within 1 % at t = 1.25 (SI 7c). 

Figure 3. Plot of fluxes (solving integral equation (7) numerically) for parameters in Figure 2, but Fmax 2 = 10 8 mol m 2. The individual uptake fluxes for sites of type 1 (/u,i, + ) and sites of type 2 (/U)2 x ) are added to provide the total flux /u(0). The inflexions in Jm arise from the saturation of sites of type 2 and from the approach to the steady-state...

Figure 28 shows an example of the way these surface concentrations change in the approach to the steady state. The time that it takes for the surface concentrations to reach within 1% of their final steady-state values can be calculated from eqn. (117) as 0.45 L2 DX. 

Voorhees s experimental study of low-volume-fraction-solid liquid+solid Pb-Sn mixtures carried out under microgravity conditions during a space shuttle flight enabled a wider range of solid-phase volume fractions to be studied without significant influence of buoyancy (flotation and sedimentation) effects [13]. The rate of approach to the steady-state particle-size distribution in 0.1-0.2 volume-fraction... 

Paper [109] determined the value of Uo upon approach to the steady state from above. One-dimensional crystals were simulated of length from 8 x 103 to 2 x 104ao (ao is a lattice constant the spatial correlation in genetic pairs is neglected). The limiting values Uo = 3.5-3.6 for 500 and 700 sites in the recombination sphere (Table 7.3, third column) are close to the value 3.43 obtained in the continuum approximation by an approximate method [22] and considerably exceed the estimate 1.36 implied by the approach based on many-point densities in the linear approximation [31, 111] remember that... 

Steady state kinetic measurements on an enzyme usually give only two pieces of kinetic data, the KM value, which may or may not be the dissociation constant of the enzyme-substrate complex, and the kcM value, which may be a microscopic rate constant but may also be a combination of the rate constants for several steps. The kineticist does have a few tricks that may be used on occasion to detect intermediates and even measure individual rate constants, but these are not general and depend on mechanistic interpretations. (Some examples of these methods will be discussed in Chapter 7.) In order to measure the rate constants of the individual steps on the reaction pathway and detect transient intermediates, it is necessary to measure the rate of approach to the steady state. It is during the time period in which the steady state is set up that the individual rate constants may be observed. 

The approach to the steady state for Da = 2 and various initial conditions. 

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

The Smith-Ewart equations can be solved using a single numerical eigenvalue determination under all conditions. Analytical solutions can also be obtained if n is not too large (n< 0.7)(8,9,10). These solutions encompass both the steady state and the approach to the steady state. Thus the particle number concentrations N, N, . .are known once P, k and c have been determined experimentally. As will be seen, these populations are the starting point for the MWD analysis. 

Physical Complications (1) Kinetic Behaviour During the Approach to the Steady State. ... 

Just like the approach to isomerization equilibrium in a system without conversion to products, the approach to the steady-state isomer distribution is a first-order process whose characteristic rate coefficient is the sum of the forward and reverse coefficients (see Section 5.1) ... 

To estimate the isomerization rate coefficients, eqn 5.8 is applied to the time required for close approach to the straight-line behavior of the first-order curves. Judging this time to be about 25 minutes for a 90% approach to the steady-state isomer distribution, eqn 5.8 yields k 0.1 min-1. With this value and an isomerization equilibrium constant Kn = 20 at 150°C calculated from thermochemical data [5,6] (with 2-cis and 2-trans pentene lumped into a single pseudo-component), eqns 5.40 give as rough estimates... 

A qualitative discussion of how a CSTR approaches steady state is given in the CD-ROM. This analysis, summarized in Figure S-1 in the Summary, is developed to show the four different regions into which the phase plane is divided and how they allow one to sketch the approach to the steady state. 

Startup of a CSTR (Figure S-1) and the approach to the steady state (CD-ROM). By mapping out regions of the concentration-temperature phase plane, one can view the approach to steady state and learn if the practical stability limit is exceeded. 

Approach to the Steady-state Phase-Plane Plots and Trajectories of Concentration versus Temperature... 

Fig. 1. Approach to the steady state in the pyrolysis of acetaldehyde time-dependence of (Me) for three temperatures the dotted curve represents a 50-fold expansion of the 400 data calculations of W. J. Probst. From ref. 9.

Figure 2. Approach to the steady-state for a set point of 80 with instant heater (TAUQ goes to zero) and inlet random disturbances. DT=0.1, Kp=300, TAUsens=2, and TAUI=9.

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