One-at-a-time procedure

In the one-at-a-time procedure a change is made in a single variable and the results are evaluated. As this procedure is continued, a change in only one variable is made at each step along the way. This is well suited to plant design. 

Once the basic unit operations and their sequence have been decided upon, each one should be investigated to see how it can be improved. Finally, the conditions at which each step is run should be scrutinized to determine whether they are optimal. 

Courtesy of Baasel, W.D. Exploring Response Surfaces to Establish Optimum Conditions, Chemical Engineering, Oct. 25, 196.5, p. 147. 

One way to cut down on the number of tests is to approximate the response surface by a quadratic equation and from it to predict where the maximum will occur. The equation at constant T would be 

Since there are three constants, we must have the results of three tests to evaluate them. The yield is known at point A (P=29 psia, Y = 15%). If the second pressure were chosen as 35 psia the yield would be better (Y =20%). This would indicate that the third point should be taken at a higher pressure, since the yield appears to increase with pressure. It might be taken at 41 psia (Y=25%). If the yield for the second test had been less than for the first, the third experiment point should have been taken at a pressure less than that for the first test, such as 23 psia. 

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For this last stage, the one-at-a-time procedure may be a very poor choice. At Union Carbide, use of the one-at-a-time method increased the yield in one plant from 80 to 83% in 3 years. When one of the techniques, to be discussed later, was used in just 15 runs the yield was increased to 94%. To see why this might happen, consider a plug flow reactor where the only variables that can be manipulated are temperature and pressure. A possible response surface for this reactor is given in Figure 14-1. The response is the yield, which is also the objective function. It is plotted as a function of the two independent variables, temperature and pressure. The designer does not know the response surface. Often all he knows is the yield at point A. He wants to determine the optimum yield. The only way he usually has to obtain more information is to pick some combinations of temperature and pressure and then have a laboratory or pilot plant experimentally determine the yields at those conditions. 

While there is always this possibility of a blind spot, for discrete variables the one-at-a-time procedure is still frequently used. For continuous variables other procedures, which follow, should be used. 

Figure 14-1 gives the yield of a chemical reactor versus temperature and pressure. Starting at point A and using the one-at-a-time procedure, obtain using a quadratic... 

Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitaole dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linear-regression techniques such as least-sqiiares methods. However, details concerning the procedures utihzed in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the apphcation. 

Various procedures for solving Eqs. (13-149) to (13-161), ranging from a complete tearing method to solve the equations one at a time, as shown by Distefano, to a complete simultaneous method, have been studied. Regardless of the method used, the following considerations generally apply ... 

To check for soft foot, place a dial indicator onto the machinery foot, and loosening the base bolt. If the indicator moves more than 0.002 inches, the foot is soft and it should be corrected. Go through the same procedure on the remaining feet one at a time. 

In some instances it may be possible, though it is usually very difficult, to undertake laboratory corrosion tests under conditions that will be the same as those encountered in some practical application, and thus to secure some directly applicable data. More often, the conditions of service are so variable or so difficult to appraise accurately and duplicate in the laboratory that it is impractical and probably unwise to attempt to do so. A better procedure is to examine the individual effects of the several controlling factors by varying them one at a time so as to provide a picture of their influence on the... 

These methods move one step at a time. A strategy that delineates the particular method dictates how to proceed once the results of the most recent tests are available. The one-at-a-time method is an example of an algebraic sequential procedure. The goal of the algebraic sequential methods is to find and follow a path to the summit. The geometric sequential methods attempt to isolate the area in which the maximum may exist. 

A well-known class of techniques for reducing the number of iterates is the use of tearing (L4). We shall illustrate this procedure by way of an example taken from Carnahan and Christensen (C3). Let us consider the two-loop network shown in Fig. 5 and assume that formulation A is used. To abbreviate the notation let us denote the material balance around vertex i [Eq. (35)] by fi = 0 and the model of the element [Eq. (36)] by fu — 0. Then assuming all external flows and one vertex pressure, p, say, are specified, we have a set of 12 equations that must be solved simultaneously. But if we now assume a value for ql2, the remaining equations may be solved sequentially one at a time to yield the variables in the following... 

For a problem for which we cannot obtain an analytical solution, you need to determine sensitivities numerically. You compute (1) the cost for the base case, that is, for a specified value of a parameter (2) change each parameter separately (one at a time) by some arbitrarily small value, such as plus 1 percent or 10 percent, and then calculate the new cost. You might repeat the procedure for minus 1 percent or 10 percent. The variation of the parameter, of course, can be made arbitrarily small to approximate a differential however, when the change approaches an infinitesimal value, the numerical error engendered may confound the calculations. 

We can call this one state at a time procedure as one-state one-root (SC) dressing. 

Procedure. Wear PPE - rubber gloves, face shield and lab coat when handling the permanganate-buffer solution. Place the weighed sintered glass crucibles containing the ADF residue one at a time into a stainless steel or polythene tray containing 2-3 cm cold water (a 400 x 320 x 50 mm tray will hold 48... 

In a full factorial design all combinations between the different factors and the different levels are made. Suppose one has three factors (A,B,C) which will be tested at two levels (- and +). The possible combinations of these factor levels are shown in Table 3.5. Eight combinations can be made. In general, the total number of experiments in a two-level full factorial design is equal to 2 with /being the number of factors. The advantage of the full factorial design compared to the one-factor-at-a-time procedure is that not only the effect of the factors A, B and C (main effects) on the response can be calculated but also the interaction effects of the factors. The interaction effects that can be considered here are three two-factor interactions (AB,... 

In traditional medicine there are two major therapeutic approaches to the treatment of human disease surgical and medical. Surgical procedures are labour intensive and time demanding they help a limited number of individuals, one at a time, mostly in rich or developed nations. Medical therapy, on the other hand, is based on drug molecules and thus has the capacity to positively influence the lives of more people, often over a shorter time frame. Medical therapeutics offer hope in both developed and developing parts of the world—hopefully to rich and poor alike. 

The second method, the leave-one-at-a-time or jackknife procedure, repeats the whole LDA procedure as many times as there are objects, and each time one object alone is the evaluation set. 

Here, (k + Ak) represents the original parameter vector k to whose ith element, M is added. A separate calculation must be performed for each element of k. In other words, the derivatives with respect to the elements in k must be calculated one at a time. It is probably most instructive to study the MATLAB code in MATLAB Box 7.5b, where this procedure is defined precisely. 

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