Fixed-time integral methods

In Example 13.1 the initial concentration of analyte is determined by measuring the amount of unreacted analyte at a fixed time. Sometimes it is more convenient to measure the concentration of a reagent reacting with the analyte or the concentration of one of the reaction s products. The one-point fixed-time integral method can still be applied if the stoichiometry is known between the analyte and the species being monitored. For example, if the concentration of the product in the reaction... 

The one-point fixed-time integral method has the advantage of simplicity since only a single measurement is needed to determine the analyte s initial concentration. As with any method relying on a single determination, however, a... 

Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times ti or f2, because the relevant concentration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer s law... 

Sensitivity The sensitivity for a one-point fixed-time integral method of analysis is improved by making measurements under conditions in which the concentration of the monitored species is larger rather than smaller. When the analyte s concentration, or the concentration of any other reactant, is monitored, measurements are best made early in the reaction before its concentration has substantially decreased. On the other hand, when a product is used to monitor the reaction, measurements are more appropriately made at longer times. For a two-point fixed-time integral method, sensitivity is improved by increasing the difference between times t and f2. As discussed earlier, the sensitivity of a rate method improves when using the initial rate. 

Equation 13.14 shows how [A]o is determined for a two-point fixed-time integral method in which the concentration of A for the pseudo-first-order reaction... 

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. 

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

Other established attempts on heat integration of batch plants are based on the concept of pinch analysis (Linnhoff et al., 1979 Umeda et al., 1979), which was initially developed for continuous processes at steady-state. As such, these methods assume a pseudo-continuous behaviour in batch operations either by averaging time over a fixed time horizon of interest (Linnhoff et al., 1988) or assuming fixed production schedule within which opportunities for heat integration are explored (Kemp and MacDonald, 1987, 1988 Obeng and Ashton, 1988 Kemp and Deakin, 1989). These methods cannot be applied in situations where the optimum schedule has to be determined simultaneously with the heat exchanger network that minimises external energy use. 

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 Constant time In the fixed-time method of measurement the change in concentration of the indicator substance I [which could be [R] or [P] in Equation (21-2)] is measured twice to cover a preselected time interval (Figure 21-2). 

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... 

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... 

Two types of techniques are employed for analyzing a single-component system. The most straightforward is the derivative or slope method in which one obtains the derivative of the electrical signal by electronically differentiating the signal from the transducer. The second approach uses the integral forms of the rate equations, and one of two possible types of measurement the fixed-time or constant-time method... 

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 differential and the integral method are compared in Figure 4.11.4 for a fixed bed reactor where, usually, the modified residence time (ratio of catalyst mass to total feed rate) is used. 

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