Capacity noninteracting

After all this analysis, can we say that the Stem model is consistent with experimental results In other words, is the Stem model able to reproduce the differential capacity curves Under certain conditions, it is. So, to some extent, the Stem model was successful. However, what are the restrictions the model imposes Recall that in the Helmholtz-Perrin model the ions lay close to the electrode on the OHP. The condition for the Stem model to succeed is that ions not be in close proximity to the electrode they are not to be adsorbed. Thus the model proved to be valid only for electrolytes such as NaF (Graliame, 1947).45 Both of these ions, Na+ and F, are known to have a hydration layer strongly attached to them in such a way that even in the proximity of the electrode they are almost not interacting with the electrode surface. The Stem model works well representing noninteracting ions. 

Rgure 9.6 Heat capacity per atom, in units of Boltzmann s constant, as a function of T for a noninteracting gas of He atoms within a tube with R = 0.5 nm The low T limiting behavior is that of a ID gas, while the high T limit is that of a 2D gas. (Adapted from Ref [24].)... 

Let us now see how multicapacity processes result in second-order systems. We start with noninteracting capacities. 

When a system is composed of two noninteracting capacities, it is described by a set of two differential equations of the general form ... 

For the case of N noninteracting capacities (Figure 11.5b), it is easy to show that the overall transfer function is given by... 

Equation (11.22) indicates that the relationship between the external input, F,(t), and the final output, /12U), is that of an overdamped second-order system. Using eq. (11.20) for the response of two noninteracting capacities with rP1 tP2 we find that... 

How do you understand the interaction or noninteraction of several capacities in multicapacity processes Give the general set of two differential equations describing (a) two noninteracting capacities, and (b) two interacting capacities. 

Explain why two interacting capacities have more sluggish response than two equivalent but noninteracting capacities. 

In Section 11.3 we found that two capacities in series, interacting or noninteracting, give rise to a second-order system. If we extend the same procedure to N capacities (first-order systems) in series, we find that the overall response is of Nth order that is, the denominator of the overall transfer function is an iVth-order polynomial,... 

Show that as the number of noninteracting or interacting capacities in series increases, the response of the system becomes more sluggish. 

Consider N identical noninteracting capacities in series, with gain K and time constant xp for each capacity. Show that as N - oo, the response of the system approaches the response of a system with dead time t and overall gain Kt. 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

If the time constants xPI and xP2 are equal, we have two equal poles. Therefore, noninteracting capacities always result in an overdamped or critically damped second-order system and never in an underdamped system. The response of two noninteracting capacities to a unit step change in the input will be given by eq. (11.7) for the overdamped case, or eq. (11.8) for the critically damped. Instead of eq. (11.7), we can use the following equivalent form for the response ... 

Thus we see that the effect of interaction is to change the ratio of the effective time constants for the two tanks (i.e., one tank becomes faster in its response and the other slower). Since the overall response of h2(t) is affected by both tanks, the slower tank becomes the controlling and the overall response becomes more sluggish due to the interaction. Therefore, interacting capacities are more sluggish than the noninteracting. 

Example 17.2 Frequency Response of N Noninteracting Capacities in Series... 

Figwe 12.7 Deteimination of the protein binding capacity. One measures the difference between the elution volume of a noninteracting species and the protein used for the binding study by calculating the difference at the 50% point of the step height. The amount of protein iMund is calculated from the concentration of the protein in the feed and this volume. 

The principal distinction to be made in multicapatnty processes is the manner in which the capacities are joined. If they are said to be i.solatcd or noninteracting, the capacities behave exactly as they would alone. Rut if coupled, they interact with one another, in which case the contribution of each is altered by the interaction. Figure 2.1 compares the two forms. 

FIG 2.1. Noninteracting (above) compared to interacting (below) capacities. 

An important point to grasp is that interaetion makes control easier. Recall that the proportional band required to regulate a two-eapacity process varies with t, Vijvith the most difficult case being t = ti. Where capacities interact, however, it is impossible to make rj = ri. The ratio of two equal interacting time constants is 0.382 - 2.0)18 = 0.146. By this standard the noninteracting process is nearly seven times more difficult to control ... 

Now let us examine the case of multiple capacities in series. Consider the two noninteracting tanks in series shown in Figure 3.28. 

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