RESIDENCE TIME FUNCTIONS

The quality of the packed bed may also be determined by frontal analysis where the sample is applied until it reaches a plateau to give the residence time function and then the solution is momentarily switched to wash to give the washout function. The latter is used to calculate the plate height of the column... 

Solution The first step in the solution is to find a residence time function for the axial dispersion model. Either W t) or f(t) would do. The function has Pe as a parameter. The methods of Section 15.1.2 could then be used to determine which will give the desired relationship between Pe and... 

Solution Equations (15.27) and (15.28) give the residence time functions for the tanks-in-series model. For A =2,... 

An alternative way of interpreting the residence-time function is in its integral form ... 

Here X(t) is the drying curve, corrected as before to the new scale and new operating conditions, and E(t) is the residence time function, which must be known. This approach has been used successfully for well-mixed flujdized beds. For pure plug flow, E(t) is a spike (Green s function) and X = X t). 

Theorem 16.10 The MRT in a kinetic space is equal to the total integral of the residence time function of the kinetic space ... 

The residence time function of a kinetic space of a linear system is the UIR of the kinetic space expressed in terms of the dose-normalized amount vs. time function with respect to drug input into the kinetic space. For example, let the kinetic space be the unchanged drug molecules in the general systemic blood circulation, and let c(f) denote the systemic drug concentration following an IV bolus injection then. 

Theorem 16.11. The MTT of a kinetic space is related to the initial derivative of the residence time function. 

From the resulting residence-time function [F(t) from displacement labeling or w(t) from injection labeling], the mean residence time r can be calculated (Equation 2.2-24). 

Fig. 2.2-4 Cumulative residence-time function F t/z) for an ideal continuous stirred tank (1), an ideal tubular reactor with plug flow (2), and laminar flow in a tubular reactor (3).

The cumulative residence-time function for a real tubular reactor can be derived from the general mass balance by taking into account the dispersion term (Equation 2.2-34) ... 

Differentiation of the cumulative residence-time function F t/r) (Equations 2.2-23 and 2.2-37) gives the corresponding residence-time distribution function (Equation 2.2-39) ... 

Fig. 2.2-6 Cumulative residence-time function (Equation 2.2-37) for various Bo values Bo = 0.1, 1, 5, 10, 20, 50, 100, 200 from left to right).

The functions discussed in this section form the theoretical foundation of RTDs. The fundamental concept of RTD functions was presented by RV. Danckwerts in a classical paper in the 1950s [ 1 ]. The derivation of the residence time functions starts on a microscale. 

In the following, the most important residence time functions are described. 

TABLE 4.1 Relationships between Residence Time Functions... 

In the treatment of the theoretically calculated and experimentally obtained residence time functions, the concept of variance is useful, particularly for the comparison of different theories of RTD. Variance is defined as, in mathematical statistics, that is, the quadratic value of standard deviation ... 

The residence time functions for a tube reactor with plug flow (a PFR) are trivial. Let us assume that we suddenly inject an amount of inert tracer into the inlet of the reactor. The whole amount of the tracer resides in the reactor tube for a period of time t = L/w, in which L and w denote the reactor length and the flow velocity, respectively. We can even see that the time t can be expressed as the reactor volume and the volumetric flow rate (t = V/V). 

A tube reactor, in which turbulent flow characteristics prevail, the radial velocity profile, is flattened so that, as a good approximation, it can be assumed to be flat. Function (f) becomes extremely narrow and infinitely high at the residence time of the fluid. This function is called a Dirac S function. The residence time functions (0) and T(0) for the plug flow and backmix models are presented in Figure 4.15. In this figure, the differences between these extreme flow models are dramatically emphasized. 

It is necessary to know the density function, E t), if the segregation model is applied. E(t) can be determined either experimentally or theoretically. Since it is possible to derive the residence time functions for many flow models, the segregation model is very flexible. According to the segregation model, the average concentration at the reactor outlet is obtained from the relation... 

Even for the maximum mixedness model, the calculation of an extreme concentration value is possible. This extreme value, in comparison with the previous one for the segregation model, represents the opposite end of the scale. The maximum mixedness model is relevant to microfluids and yields, even in this case, results that do not differ considerably from those that can be obtained by means of direct utilization of separate flow models. The model is, however, of importance in cases in which experimentally determined residence time functions are available for a reactor system. The maximum mixedness model is more difficult to visualize than the segregation model. However, its underlying philosophy can be described as follows ... 

Equations 4.64 and 4.65 yield the final value of y, at 0 = 0, as the solution is obtained backwards, by starting with large values of 0 (in addition to the X values valid at these points) and ending up with 0 = 0. Similar to the segregation model, a precondition for the maximum mixedness model is that the residence time functions—or more precisely the intensity function—are available, either from theory or from experiments. 

Some common guidelines are presented here on how the derived residence time functions can be utilized for calculating the conversion for a maximum mixedness tanks-in-series model. It is necessary, in this context, to state that if each tank in a series is completely backmixed, the model provides the same result as the tanks-in-series model. In this case, no special treatment is required. If, on the contrary, the tanks in series as an entity are considered to be completely backmixed, one has to turn to an earlier equation. Equation 4.64, together with the boundary condition. Equation 4.65. Instead of the original expression for the intensity function, a new one that is valid for the tanks-in-series model. Equation 4.77, is utilized. 

The concept of dispersion is used to describe the degree of backmixing in continuous flow systems. Dispersion models have been developed to correct experimentally recorded deviations from the ideal plug flow model. As described in previous sections, the residence time functions E(t) mdF(t) can be used to establish whether a real reactor can be described by the ideal flow models (CSTR, PFR, or laminar flow) or not. In cases where none of the models fits the residence time distribution (RTD), the tanks-in-series model can be used, as discussed in Section 4.4. However, the use of a tanks-in-series model might be somewhat artificial for cases in which tanks do not exists in reality but only form a mathematical abstraction. The concept of a dispersion model thus becomes actual. 

Here, ajn and aout are the inlet and outlet concentrations of a reactive component, A, that reacts according to A products. Use the version of eq. (1-6) that contains the residence time function actually measured, W(t) or f(t). 

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