Normalized RTD function

In using the normalized distribution function, it is possible to directly compare the flow performance inside different reactors. If the normalized function E(6) is used, all perfectly mixed CSTRs have numerically the same RTD. If E(t) is used, its numerical values can change for different CSTRs. 

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

Since tracer experiments are used to obtain RTD functions, we wish to establish that the response to a pulse-tracer input is related to (r) or E(0). For this purpose, c(t) must be normalized appropriately. We call c(t), in arbitrary units, the nonnormalized response, and define a normalized response C(t) by... 

In all the just-mentioned examples, quantitative prediction and design require the detailed knowledge of the residence time distribution functions. Moreover, in normal operation, the time needed to purge a system, or to switch materials, is also determined by the nature of this function. Therefore the calculation and measurement of RTD functions in processing equipment have an important role in design and operation. 

Frequently, a normalized RTD is used instead of the function E t). If the parameter is defined as... 

Figure 3.6 Normalized distribution functions for the two ideal reactors. The spread of the RTD (denoted by a) is zero for the PFR.

Since only positive values of are of concern in RTD work, this function is normalized by dividing by the integral from 0 to oo with the result... 

Two mathematical models have been described in the literature for characterizing normal flow.4" Each of these defines the RTD curve, as a function of a single parameter. The dispersion model uses the dimensionless Pec let number for this parameter since this is a ratio between convective and dispersive forces. 

The F curve is another function that has been defined as the normalized response to a particular input. Alternatively, Equation (13-12) has been used as a definition of F(t), and it has been stated that as a result it can be obtained as the response to a positive-step tracer test. Sometimes the F curve is used in the same manner as the RTD in the modeling of chemical reactors. An excellent example is the study of Wolf and White, who investigated the behavior of screw extruders in polymerization processes. 

The RTD is normally considered a steady-state property of a flow system, but material leaving a reactor at some time 0 will have a distribution of residence times regardless of whether the reactor is at steady state. The washout function for an unsteady reactor is defined as... 

Unfortunately, the RTD of a CSTR does not provide the lower limit on reactor performance. Equation (30) is worthless unless one can guarantee that there is no bypassing and stagnancy in the reactor. To prove that, one needs an experimental RTD and the design is not truly predictive any more. We consider here the RTD based on the generalized tanks in series model, i.e. based on the gamma probability density function, which allows the following representation of the normalized exit age density functions ... 

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