Days26n27

Days26n27 - CE 561 Lecture Notes Fall 2009 Days 26 and 27...

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CE 561 Lecture Notes Fall 2009 p. 1 of 21 Days 26 and 27: Residence Time Distributions and Non-Ideal Flow Patterns So far, we have considered only two idealized types of continuous reactors. We analyzed the PFTR, where there was assumed to be no mixing inside the reactor, then we considered the CSTR, in which there was assumed to be complete mixing, so that the concentrations and temperature were uniform throughout the reactor. These represent two limiting cases for any real system, where there must be a finite amount of mixing within the reactor. While reactors can often be configured to closely approximate one extreme or the other, many reactors are not well described by either idealized case. Any reactor that is neither unmixed nor perfectly mixed is partially mixed . An example of a partially mixed reactor is the fluidized bed reactor, which is widely used in many forms. The degree of mixing within a reactor can be characterized by a residence time distribution function . We will now investigate the determination and use of the residence time distribution function for reactor analysis and design. Residence Time Distributions in Reactors: We can imagine a fluid element as a tiny bit of fluid that is infinitesimally small compared to the reactor volume, but large enough that it contains a large number of molecules and can be treated as a continuum. Then we can (in our imaginations) track individual fluid elements and see how long they stay in the reactor. The ‘age’ of a fluid element is the amount of time it has been in the reactor. We can imagine sitting at the reactor outlet and asking individual fluid elements their ‘age’. In a PFTR, where there is no mixing, all of the fluid elements at the outlet would have the same age – the mean residence time of the reactor, τ = L/v x . In a CSTR, where there is perfect mixing, the fluid elements would have a range of ages, from zero to infinity, with the largest number of them having an age of 0. The average age would still equal the residence time. We can quantify this idea of the age of fluid elements using the residence time distribution function . This is defined such that ( ) the fraction of the fluid exiting the reactor that has an age between and . Ed d θθ θ θ θ = + Because all fluid elements have some age, this function must be normalized. That is ( ) 0 1 = The fraction of the fluid with residence times between 1 and 2 is given by ( ) 2 1 The mean residence time is given by ( ) 0 θ θθ = It is often convenient to use a dimensionless time, defined by ´ = / , in the residence time distribution. The fraction of fluid with a dimensionless age between ´ and ´ + d ´ is the same as the fraction of fluid with an age between and + d (they are the same fluid). So

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CE 561 Lecture Notes Fall 2009 p. 2 of 21 ( ) ( ) ( ) , so ( ) ( ) d Ed E d E E E θ θθ θ θ θ τθ τ ′′ = = = This is consistent with the normalization condition ( ) ( ) ( ) 00 0 1 d E d E Ed τ θ ∞∞ = = = ∫∫ Experimentally, we can measure the residence time distribution function (RTD) by performing a tracer experiment.
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Days26n27 - CE 561 Lecture Notes Fall 2009 Days 26 and 27...

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