Residence-time distributions normalized function

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. 

E E(t) E(tr) f. Fit) Activation energy Residence time distribution Normalized residence time distribution Fraction of A remaining unconverted, Ca /Ca0 or nja0 Age function of tracer kJ/(kgmol) Btu/(lb-mol)... 

Figure 9 displays the probability distribution function (p r) and the effectiveness factor r] k), which have been calculated via Eqs. 36 and 34 from the tracer exchange curves in the limiting cases of single-hle diffusion, normal diffusion and barrier confinement. The fact that in all cases the residence time distribution function is found to decrease monotonically may be easily rationalized as a quite general property. Due to the assumed stationarity of the residence time distribution function, the number of molecules with a residence time r is clearly the same at any instant of time. The number of molecules with a residence time r + At may therefore be considered as the number of molecules with a residence time r minus the number of molecules which will leave the system in the subsequent time interval At. Therefore, (p x) must quite generally be a monotonically decaying function. 

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-... 

We will use the NormalDistribution to make the representations of the residence time distribution. The Probability Density Function (PDF) is made up of the Normal Distribution and the variable 9. This can be integrated in closed form ... 

In evaluating the residence time distributions in a continuous system oper> ating without recycling, for the case of the ideal discontinuous stirred vessel the curve with an exponential decay of the dilution process normally appears. But it appears in such a way that the pulse functions do not overlap (see Fig. 3.6 and also Blenke, 1979). This means that mixing and dilution processes are superimposed, and that in reactors that deviate from the ideal continuous... 

In the method known as pulse, an amount of tracer is injected into the feed entering the reactor over a period of time approaching zero. The discharge concentration (or equivalent) is then measured as a function of time. Typical concentration curves at the outlet of the reactor can take the form of any of the residence time distribution plots discussed earlier. The most usual response takes the form similar of that in Figure 14.13. Generally, the response approaches a normal or log-normal distribution curve. 

Extensions of residence time distributions to systems with multiple inlets and outlets have been described (27-29). If the system contains M inlets and N outlets one can define a conditional density function E. (t) as the normalized tracer impulse response in outlet j to input in inlet i as shown schematically in Figure 2. 

It is normally called the differential distribution function (of residence times). It is also known as the density function or frequency function. It is the analog for a continuous variable (e.g., residence time i) of the probabiUty distribution for a discrete variable (e.g., chain length /). The fraction that appears in Equations (15.2), (15.3), and (15.6) can be interpreted as a probability, but now it is the probability that t will fall within a specified range rather than the probability that t will have some specific value. Compare Equations (13.8) and (15.5). 

For any quantity that is a function of time we can describe its properties in terms of its distribution function and the moments of this function. We first define the probability distribution function p(t) as the probability that a molecule entering the reactor wih reside there for a time t. This function must be normalized... 

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... 

A good distribution function to examine in this context is the Normal or Gaussian distribution. Using this function, we would take the residence times 0 to be normal distributed around some mean value 0m and with a standard deviation or spread of SO ... 

Because the Normal distribution PDF requires a mean value and a variance, we supply these and then Plot the result in order to visualize the distribution. We have purposefully chosen a very narrow distribution for the first case. Next the parameter values are assigned and we Integrate the products of the concentration functions and the PDF in d over the range of d values. As shown in the following graph, these are the residence-time-averaged values of the concentrations and the conversion of A and B ... 

If the tail is truncated, then the tracer impulse response should be normalized based on the area under the curve to obtain a proper density function that approximately describes the distribution of residence times in regions through which there is active flow. 

The ideal cases are the piston flow reactor (PFR), also known as a plug flow reactor, and the continuous flow stirred tank reactor (CSTR). A third kind of ideal reactor, the completely segregated CSTR, has the same distribution of residence times as a normal, perfectly mixed CSTR. The washout function for a CSTR has the simple exponential form... 

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