CE 561 Lecture Notes
Fall 2009
p. 1 of 21
Days 26 and 27: Residence Time Distributions and NonIdeal 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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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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 Fall '09
 Fluid Dynamics, Chemical reactor, RTD, residence time

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