Step-change

Virtually every chemical production unit will improve its productivity and reduce costs to some extent over time. Typically in such cases, a number of more or less obvious improvement ideas are generated and implemented by a highly motivated but small part of the workforce. In addition, targets are mostly low, and the level of improvement that is typically reached is insufficient to compensate for the price-cost squeeze. Furthermore, the creeping improvement in performance is overlaid by segment-specific price cycles, making consistent tracking of the improvement activities almost impossible. 

Breaking out of this situation requires an internal discontinuity - a real step change in performance, characterized by improvement targets of inconceivable proportions. The concept of core costs has proved to be highly successful here over the last 10 to 15 years, particularly in production environments. Core costs are the minimum production costs that would be required if all types of costs 

The difference between the present cost base and the core cost is described as the fully compressible cosf. A target - or ideal hurdle - of reducing compressible costs by about 30 to 50 percent has proved to be both stretching and also achievable in all process industries, and has been attained by many chemical companies with very different performance histories (Fig. 12.5). 

Finally, by applying the appropriate indicators from the value driver trees mentioned in the previous section, management can lay the groundwork for comprehensive and practical monitoring of performance improvements during the implementation phase. 

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For more complex examples, total cost profiles return step changes such as shown in Fig. 7.12 (due to changes in Nu ts and /Vshklls)-These step changes are easily located, prior to design, through simple software. Most important, extensive experience has shown that predicted overall costs are typically accurate within 5 percent or better. ... 

For a step change in sorbate concentration at the particle surface (r = R) at time 2ero, assuming isothermal conditions and diffusion control, the expression for the uptake curve maybe derived from the appropriate solution of this differential equation ... 

The equations of combiaed diffusion and reaction, and their solutions, are analogous to those for gas absorption (qv) (47). It has been shown how the concentration profiles and rate-controlling steps change as the rate constant iacreases (48). When the reaction is very slow and the B-rich phase is essentially saturated with C, the mass-transfer rate is governed by the kinetics within the bulk of the B-rich phase. This is defined as regime 1. 

Fig. 7. Residence time distributions where U = velocity, V = reactor volume, t = time, = UtjV, Cj = tracer concentration to initial concentration and Q = reactor volume (a) output responses to step changes (b) output responses to pulse inputs.

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

Solid Heat Capacity Solid heat edacity increases with increasing temperature, with steep rises near the triple point for many compounds. When experimental data are available, a simple polynomial equation in temperature is often used to correlate the data. It should be noted that step changes in heat capacity occur if the compound undergoes crystalline state changes at mfferent temperatures. 

Equations (8-23) and (8-24) can be multiphed together to give the final transfer function relating changes in ho to changes in as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R Ro and two time constants, R A and RoAo. For two equal tanks, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. 

FIG. 8 26 Resp onse for a step change in disturbance with tuned P, PI, and PID controllers and with no control. 

The performance of a controller (and its tuning) must be based on what is achievable for a given process. The concept of best practical IE (lEb) for a step change in load Aq can be estimated (Shinskey, Feedback Controllers foi the Process Industries, McGraw-Hill, New York, 1994) ... 

Foxboro developed a self-tuning PID controller that is based on a so-called expert system approach for adjustment of the controller parameters. The on-line tuning of K, Xi, and Xo is based on the closed-loop transient response to a step change in set point. By evaluating the salient characteristics of the response (e.g., the decay ratio, overshoot, and closed-loop period), the controller parameters can be updated without actually finding a new process model. The details of the algorithm, however, are proprietary... 

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

FIG. 8-43 Step response for u, a unit step change in the input. 

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

In this section, we consider the transient adsorption of a solute from a dilute solution in a constant-volume, well-mixed batch system or, equivalently, adsorption of a pure gas. The solutions provided can approximate the response of a stirred vessel containing suspended adsorbent particles, or that of a very short adsorption bed. Uniform, spherical particles of radius are assumed. These particles, initially of uniform adsorbate concentration, are assumed to be exposed to a step change in concentration of the external fluid. 

In addition to the drift with time, step changes due to operating decisions, atmospheric changes, or other conditions result in additional time dependence. Not only is there a sudden change due to the actual decision, but also the plant changes due to the time constants. For example. Fig. 30-13 shows measurements with step changes in the operation. 

FIG. 30-13 Plant measurements exhibiting step changes, drift, and random fluctuations. 

Fault detection is a monitoring procedure intended to identify deteriorating unit performance. The unit can be monitored by focusing on values of important unit measurements or on values of model parameters. Step changes or drift in these values are used to identify that a fault (deteriorated performance in unit functioning or effectiveness) has occurred in the unit. Fault detection should be an ongoing procedure for unit monitoring. However, it is also used to compare performance from one formal unit test to another. 

The unit was built in a loop because the needed 85 standard m /hour gas exceeded the laboratory capabilities. In addition, by controlling the recycle loop-to-makeup ratio, various quantities of product could be fed for the experiments. The adiabatic reactor was a 1.8 m long, 7.5 cm diameter stainless steel pipe (3 sch. 40 pipe) with thermocouples at every 5 centimeter distance. After a SS was reached at the desired condition, the bypass valve around the preheater was suddenly closed, forcing all the gas through the preheater. This generated a step change increase in the feed temperature that started the runaway. The 20 thermocouples were displayed on an oscilloscope to see the transient changes. This was also recorded on a videotape to play back later for detailed observation. 

In Chapter 7.4, empty reactor volume determination of a perfect CSTR is described by following the discharge concentration from the sudden step-change injection of a non-adsorbing inert gas (solid line in the picture.) Next the same experiment is discussed if made with a chemisorbing gas and shown on the previous picture with a dotted line. In this second case, the reactor... 

Lag time The time interval between a step change in input concentration and the first observable corresponding change in response. 

Dynamic load response time This is the amount of time it requires the power supply to recover to within load regulation limits in response to a step change in the load. 

Quasi-resonant converters utilize an T-C tank circuit, which rings at its natural resonance frequency in response to a step change in its terminal voltage or current. The tank circuit is placed between the power switch and the transformer and/or the transformer and the output filter. 

Fig. 4.28 Response of the PI controlled liquid-level system shown in Figure 4.26 to a step change in ha t) from 0 to 4 m.

The response to a step change in the desired temperature of 0-20 °C for the closed-loop transfer function given by equation (4.134) is shown in Figure 4.35. 

As with simple imines, the identity of the rate-limiting step changes with solution pH.. s the pH decreases, the rate of the addition decreases because protonation of the amino compound reduces the concentration of the nucleophilic unprotonated form. Thus, whereas the dehydration step is normalfy rate-determining in neutral and basic solution, addition becomes rate-determining in acidic solutions. 

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