Accuracy and Stability

The number of radial increments can be picked arbitrarily. A good approach is to begin with a small number, 7=4, for debugging purposes. When the program is debugged, the value for 7 is successively doubled until a reasonable degree of accuracy is achieved or until computational times become excessive. If the latter occurs first, find a more sophisticated solution method or a faster computer. 

Given a value for 7 and the corresponding value for Ar, it remains to determine Az. The choice for Az is not arbitrary but is constrained by stability considerations. One requirement is that the coefficients on the a iJi) and a,M (0) terms in Equations (8.25) and (8.26) cannot be negative. Thus, the numerical (or discretization) stability criterion is 

This stability requirement is quite demanding. Superficially, it appears that AZmax decreases as Ar2, but VZ R — Ar) is also decreasing, in approximate proportion to Ar. The net effect is that Azmax varies as Ar3. Doubling the number of 

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example, A a, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself. Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

Regarding accuracy, the finite difference approximations for the radial derivatives converge 0(Ar2). The approximation for the axial derivative converges O(Az), but the stability criterion forces Az to decrease at least as fast as Ar2. Thus, the entire computation should converge 0( Ar2). The proof of convergence requires that the computations be repeated for a series of successively smaller grid sizes. 

Numerical solutions to PDEs must be tested for convergence as Ar and Az both approach zero. The flnite-difference approximations for radial derivatives converge O(Ar ) and those for the axial derivative used in Euler s method converge 0(Az). In principle, just keep decreasing Ar and Az until results with the desired accuracy are achieved, but it turns out that Ar and Az cannot be chosen independently when using the method of lines. Instead, values for Ar and Az are linked through a stability requirement that the overall coefficient on the central dependent variable cannot be negative  

As shown in Table 8.2, the value of Az depends on Ar, i, and V (i). Pick a value for Ar. Then apply Equation 8.36 to each of the Co(t) to calculate an acceptable value for Az for each value of /.Then, choose the smallest value among these. The smallest value usually occurs at / = / - 1, and the corresponding, most restrictive value for Az is usually given by 

The packed-bed models in Section 9.1 assume a flat velocity profile and are an exception. For them, the smallest value for Az occurs at the centerline. 

Equations 8.37 and 8.38 impose a very severe restriction on the axial increment. If Ar is halved, Az must be decreased by a factor of more than 8. The net effect on convergence is that Az will be far smaller than necessary for convergence in the z direction while Ar is still too large for accurate results. The situation becomes worse when the radial diffusion rate, 0J/R, is large. The method of false transients discussed in Chapter 16 can be used, but there is no perfect method for solving PDEs. All methods are computationally intensive. 

The stability requirement for the temperature equation is identical to Equations 

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It is rcconiincnded that relays beyond 100 A be CT operated for better accuracy and stability. 

A one-dimensional mesh through time (temporal mesh) is constructed as the calculation proceeds. The new time step is calculated from the solution at the end of the old time step. The size of the time step is governed by both accuracy and stability. Imprecisely speaking, the time step in an explicit code must be smaller than the minimum time it takes for a disturbance to travel across any element in the calculation by physical processes, such as shock propagation, material motion, or radiation transport [18], [19]. Additional limits based on accuracy may be added. For example, many codes limit the volume change of an element to prevent over-compressions or over-expansions. 

Control systems for process compressors become an absolute necese whenever deviations from a single operating point occur in the process system. These systems consist of a mechanism that can sense performance and, by making a calculation, adjust the process controls with satisfactory speed, accuracy, and stability. 

Many high molecular weight synthetic polymers, such as polyethylene and polypropylene, have a large percentage of their molecules in the crystalline state. Prior to dissolution, these polymers must usually be heated almost to their melting points to break up the crystalline forces. Orthodichlorobenzene (ODCB) is a typical mobile phase for these polymers at 150°C. The accuracy and stability of the Zorbax PSM columns under such harsh conditions make them ideal for these analyses (Fig. 3.8). 

Devices with a measuring function Accuracy and stability of results... 

A marching-ahead solution to a parabolic partial differential equation is conceptually straightforward and directly analogous to the marching-ahead method we have used for solving ordinary differential equations. The difficulties associated with the numerical solution are the familiar ones of accuracy and stability. 

One should perform both the temperature accuracy and stability tests as described in the OQ section. These may be modified to meet the user s requirements, within the functional specifications of the oven. 

Re-qualification (RQ) is a combination of OQ and PQ. This can be written as a separate protocol or can be included as part of the PQ protocol. This protocol can be executed anytime to demonstrate that maintenance has been performed, or periodically to demonstrate that the system is within tolerances. The RQ protocol should be able to be executed as needed without getting new signatures. Suppose the power supply has been corrected. An RQ protocol could include the high voltage accuracy and stability tests along with system suitability for one of the methods run on the system. This would demonstrate that the system is performing appropriately after the maintenance, as the manufacturer and you intend. 

During the separation a direct current power supply is used to provide either a constant voltage or current across the capillary. The accuracy and stability of this applied voltage or current is essential to ensure reproducible migration times. 

The temperature of the sample compartment and capillary column should be tested for both accuracy and stability at the lowest and highest settings that are expected to be used. Typically these would be between 10 and 60°C with an accuracy of 1°C for the sample compartment and between 15 and 50°C with an accuracy of +2°C for the capillary column. 

For the systems that we have considered so far, the solutions behave smoothly in time and space. Often one can simply inspect the solution and decide if the mesh is sufficiently fine to represent it accurately. Refining mesh sizes and time steps is another simple method to assure oneself that a particular discretization was sufficient. Later we will be much more concerned about numerical accuracy and stability, especially when complex chemistry is considered. For now we take a somewhat cavalier approach, with the objective being mainly to explore the general numerical approaches to solving the conservation equations describing fluid flow. For relatively simple problems we can implement usable solutions with relative ease, for example, in a spreadsheet. 

Recall that both accuracy and stability are considerations in an explicit method. If the time step is too large, specifically if the coefficient of D20 in the difference formula given in cell D21 is less than zero, the the solution will be unstable. Try giving a too large time step and observe the solution. The instability will be unmistakenly obvious. 

Sophisticated differential-equation solution software is designed to take care of the accuracy and stability problems automatically. However, these simple spreadsheets can return very useful and effective results as long as the analyst takes a bit of care to make some common-sense judgments in choosing the discretization. 

The derivative, dT /dt, may be calculated by performing a least squares fit arouna N1 number of points. N should be chosen inversely proportional to the accuracy and stability of the temperature measurements. The higher the accuracy, the lower N can be. N was chosen to be 7 in these tests. 

Note that the above approximation is a first order approximation. If we were to use a central difference, we would increase the order, but contrary to what is expected, this choice will adversely affect the accuracy and stability of the solution due to the fact the information is forced to travel in a direction that is not supported by the physics of the problem. How convective problems are dealt will be discussed in more detail later in this chapter. The following sections will present steady state, transient and moving boundary problems with examples and applications. 

When the imposed light beam interacts with the sample, its response must then be detected by the instrument. Generally, for a given detector, when the incident light source is shorter in wavelength, the instrument is more sensitive to smaller particles. A combination of transmitted, forward-scatter and back-scatter detectors and black mirrors increase the accuracy and stability of the instrument and decrease the stray light (for instance in Instruments A and F). The source/de-tector combination defines the effective spectral characteristics of the instrument and the manner in which it responds to a sample. 

Although experimental verification is still lacking, it would appear that single ion optical frequency standards will eventually yield extremely high performance. Accuracies and stabilities better than 1 part in 10 seem quite feasible eventually they could exceed 1 part in 10. ... 

The method is one way to handle a stiff set of odes, and is an extension of fourth-order explicit Runge-Kutta. The function to be solved is approximated over the next time interval by a combination of a linear function of the dependent variable and a quadratic function of time (assuming that it is strongly time-dependent) and this increases the accuracy and stability of the fourth-order Runge-Kutta method considerably. Today, however, we have other methods of dealing with stiff sets of odes, so this method might be said to have outlived its usefulness. 

Scenario The Getcher Fish Company wants to offer a small hand-held turbidimeter that measures the amount of suspended solids in water. This new product will help sports fishermen detect high fish activity (indicated by low turbidity). A Variable MSA measures the accuracy and stability of the turbidimeter. 

Enzyme biosensors containing pol3mieric electron transfer systems have been studied for more than a decade. One of the earlier systems was first reported by Degani and Heller [1,2] using electron transfer relays to improve electrochemical assay of substrates. Soon after Okamoto, Skotheim, Hale and co-workers reported various flexible polymeric electron transfer systems appUed to amperometric enz5une biosensors [3-16], Heller and co-workers further developed a concept of wired amperometric enzyme electrodes [17—27] to increase sensor accuracy and stability. 

The accuracy and stability of the pump flow rate can be checked at 1 mL/min, with the column in place, by measuring the time required to fill a 10-mL volumetric flask... 

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