Variable-time integral method

An alternative to a fixed-time method is a variable-time method, in which we measure the time required for a reaction to proceed by a fixed amount. In this case the analyte s initial concentration is determined by the elapsed time, Af, with a higher concentration of analyte producing a smaller Af. For this reason variabletime integral methods are appropriate when the relationship between the detector s response and the concentration of analyte is not linear or is unknown. In the one-point variable-time integral method, the time needed to cause a desired change in concentration is measured from the start of the reaction. With the two-point variable-time integral method, the time required to effect a change in concentration is measured. 

One important application of the variable-time integral method is the quantitative analysis of catalysts, which is based on the catalyst s ability to increase the rate of a reaction. As the initial concentration of catalyst is increased, the time needed to reach the desired extent of reaction decreases. For many catalytic systems the relationship between the elapsed time, Af, and the initial concentration of analyte is... 

Application of the fixed-time and variable-time integral methods relies on eqn [6] in the form... 

Since the integral method described above is based on the premise that some rate function exists that will lead to a value of i/ Cj) that is linear in time, deviations from linearity (or curvature) indicate that further evaluation or interpretation of the rate data is necessary. Many mathematical functions are roughly linear over sufficiently small ranges of variables. In order to provide a challenging test of the linearity of the data, one should perform at least one experimental run in which data are... 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

This assumption can be tested using the integral method. Integrating Eq. 13.12 from the starting concentration [A]o at time 0 to the concentration [A] at time t, we get by separation of the variables... 

The numerical solution is accomplished with a method-of-lines approach, using a control-volume spatial discretization. The time integration can be done using Dassl, which implements an implicit, variable-order, variable-step, method based on the BDF method [46],... 

The distributed nature of the tubular plug flow reactor means that variables change with both axial position and time. Therefore the mathematical models consist of several simultaneous nonlinear partial differential equations in time t and axial position z. There are several numerical integration methods for solving these equations. The method of lines is used in this chapter.1... 

The state variable profiles of the model are assumed to be continuous and are obtained by integration of the DAEs over the entire length of the time. Also efficient integration methods (as available in the literature) are based on variable step size methods and not on fixed step size method where the step sizes are dynamically adjusted depending on the accuracy of the integration required. Therefore, the discrete values of the state variables are obtained using linear interpolation... 

FIGURE 21-2 Fixed-time (teft) and variable-time (right) integral methods of measurement of reaction rates. 

Integral methods Variable time In the variable-time method of measurement of the initial slope, the concentration of the indicator substance I is measured twice, and the time interval At required to bring about a preselected change in concentration A[I] is the important quantity (Figure 21-2, right). Since the change in concentration is a fixed preselected value, it can be incorporated with the constant in Equation (21-5) to give... 

Implicit time integration schemes are not as efficient as the corresponding explicit schemes due to the computational time required on the iterative process. With larger time steps the accuracy of implicit schemes decrease rapidly. The widespread use of the implicit schemes with Courant numbers ten- or even hundredfold the magnitude of what is used in an explicit method, is not justifiable in the presence of gradients or steps in the convected variable. 

Equation 18.12 is the basis for the derivative approach to rate-based analysis, which involves directly measuring the reaction rate at a specific time or times and relating this to [A]fl. Equation 18.11 is the basis for the two different integral approaches to kinetic analysis. In one case, the amount of A reacted during a fixed time is measured and is directly proportional to [A]o ( fixed-time method) in the other case, the time required for a fixed amount of A to react is measured and is also proportional to [A]o variable-time method). Details of these methods will be discussed in Section... 

The variable-time method, like the fixed-time method, is an integral method which, for short measurement times and small changes in concentration, also gives results approaching the instantaneous reaction-rate. 

The variable-time method, as employed for first- or pseudo-first-order reactions, also uses the integral form of the first-order rate equation (Eqn. 18.15). Solving for [A]o yields... 

Batch Analyzers. The American Monitor Programachem 1040 does one test at a time on up to 89 samples at up to 15 results per minute. A prepunched program card automatically sets virtually all of the system variables for each method on insertion into the instrument card-reader. A second-generation instrument, the KDA, was shown in 1975. This provides an integrated system from request slip to report form, with a design heavily dependent on the dedicated minicomputer. Another feature offered is graphics, which allows an oscilloscopic display of calibration curves, kinetic reaction-curves, quality-control points, etc. 

The algorithms most fi equently used for calculation of fractal coefficients from the AFM results are [58] Fourier spectrum integral method, surface-perimeter method, structural function method and variable method. To determine the surface dimension by the Fourier spectrum integral method it is necessary to obtain the picture of the surface 2D FFT generating amplitude and time of the matrix (more detail s are given in paper [58]. Assuming the surface function as f(x,y), the Fourier transform in two-dimensional space can be expressed as [58] ... 

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