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cascade can backtrack to the previous vessel any number of times from zero to infinity, and there is a term in the series for each number of backtrack...
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D I S T R I B U T I O N OF R E S I D E N C E T I M E S I N A CASCADE OF M I X E D V E S S E L S W I T H BACKMIXING When there i s no backmixing, the probability density for any residence time is given b y a single term, but in the case of backmixing, this density i s the summation of an infinite series. A particle passing through the cascade can backtrack to the previous vessel any number of times from zero to infinity, and there i s a term in the series for each number of backtracks. Described i s a computer calculation for generating the coefficients for the terms in the series.

for the distribution without backmixing was published over 40 years ago, a n d is \vel1 knoivn. H e r e the equation is extended to include backmixing. LVhen a stream of particles (or molecules) fl0u.s through a cascade of perfectly mixing vessels. such as in Figure 1, the probability that a pdrticle \vi11 remain in the cascade between t and t dt units of time is ( 7 . 2) :

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This equation applies only \\-hen there is no backmixingthat is, \