Lecture 2 Non Ideal Flow

Lecture 2: Non ideal
flow
Review of lecture 1
oReviewed Ideal flow
oFactors that make up the flow pattern:
RTD, state of aggregation, earliness or
lateness of mixing
oExit age distribution
oExperimental methods for finding exit
age distribution
2
Relationship between the F and
E Curves
The dimensionless form of the curve is called
the F curve.
It is found by having the tracer concentration
rise from zero to unity
To relate E with F imagine a steady flow of

white fluid. Then at time t = 0 switch to red
and record the rising concentration of red fluid
in the exit stream.
3
At any time t > 0 only red fluid in the exit
stream is younger than age t. Thus we have
But the first term is simply the F value, while the

second is given by Eq. 1. So we have, at time t,
and on differentiating
4
𝑑𝐹
= E (8)
𝑑𝑡
In graphical form, this relationship is

shown in the figure below.
These relationships show how stimulus-

response experiments, using either step or
pulse inputs can conveniently give the
RTD and mean flow rate of fluid in the
vessel.
5
6
Relationship between the E and F
7
At any time these curves are related as
follows:
8
EXAMPLE FINDING THE RTD BY
EXPERIMENT
The concentration readings in Table 1

represent a continuous response to a pulse
input into a closed vessel which is to be
used as a chemical reactor. Calculate the
mean residence time of fluid in the vessel
t, and tabulate and plot the exit age
distribution E.
9
Table 1
Time t, min Tracer output
concentration,
(g/L)
0 0
5 3
10 5
15 5
20 4
25 2
30 1
35 0
10
The mean residence time, from pulse
equations is
11
12
This area gives the total amount of tracer
introduced.
To find E, the area under this curve must

be unity; hence, the concentration readings
must each be divided by the total area,
giving,
13 We thus have’
14
EXAMPLE 2. FINDING THE E CURVE
FOR LIQUID FLOWING THROUGH A
VESSEL
15
(d) The vessel has a strong recirculation of liquid, probably
induced by the rising bubbles.
The Convolution Integral
• A convolution is an integral that
expresses the amount of overlap
of one function g as it is shifted
over another function f.
• It therefore "blends" one function
with another.
• Suppose we introduce into a
vessel a one-shot tracer signal Cin
versus t as shown in the Fig. below
18
• In passing through the vessel the signal will
be modified to give an output signal Cout
versus t.
• Since the flow with its particular RTD is
responsible for this modification let us
relate Cin, E, and C,.
• Focus attention on tracer leaving at time
about t.
• This is shown as the narrow rectangle B in
the figure above.
• We may then write
20
the tracer which enters t' seconds earlier than t as the narrow
rectangle A. In terms of this rectangle the above equation may be
written
In symbols and taking limits (shrinking the rectangles) we obtain the

desired relationship, which is called the convolution integral
We say that Coutis the convolution of E with Cin and we write concisely
Modification of an input tracer signal Cin on passing
through three successive regions.
Application of These Tools.
To illustrate the uses of these mathematical tools consider three
independent flow units a, b, and c, which are closed and
connected in series
Problem 1. If the input signal Cin is measured and the exit age
distribution functions Ea, Eb, and Ec are known, then C1 is the
convolution of Ea with Cin and so on, thus
Thus we can determine the output from a multiregion flow

unit.
23
• Example 3: Let us illustrate the use of the convolution equation,
with a very simple example in which we want to find Cout given Cin
and the E curve for the vessel, as shown in Fig.
The Cin , E, and Cout curves, in discrete and in continuous form, are shown in Fig.
Note that the area under the Cout curve equals the area under the Cin curve.
CONVERSION IN NON-IDEAL
FLOW REACTORS
• To evaluate reactor behavior in general we have to know four factors:
1. the kinetics of the reaction
2. the RTD of fluid in the reactor
3. the earliness or lateness of fluid mixing in the reactor
4. whether the fluid is a micro or macro fluid
To consider the early and late mixing of a microfluid, consider the two flow
patterns shown in Fig. for a reactor processing a second-order reaction.
(a) the reactant starts at high concentration and reacts away rapidly because.
(b) the fluid drops immediately to a low concentration.
Since the rate of reaction drops more rapidly than does the concentration you will end
up with a lower conversion.
For macrofluids, imagine little clumps of fluid staying for different lengths
of time in the reactor (given by the E function). Each clump reacts away
as a little batch reactor, thus fluid elements will have different
compositions.
So the mean composition in the exit stream will have to account for these
two factors, the kinetics and the RTD. In words, then
which says that the pulse occurs at t = to, as seen in Fig. The
two
properties of this function which we need to know are
EXAMPLE 4: CONVERSION IN REACTORS HAVING
NON-IDEAL FLOW
Find the fraction of reactant unconverted in the real reactor and compare
this with the fraction unconverted in a plug flow reactor of the same
size. Using a reaction time of 15 minutes.
For the plug flow reactor with negligible density change we have
and with
Thus the fraction of reactant unconverted in a plug flow

30
reactor equals 1.0%.
For real system
For the real reactor the fraction unconverted, given by above Eq. for macrofluids,
is found in Table. Hence the fraction of reactant unconverted in the real reactor
From the table we see that the unconverted
material comes mostly from the early
portion of the E curve.
This suggests that channeling and short-

circuiting can seriously hinder attempts to
achieve high conversion in reactors.
32
• Note that since this is a first-order reaction we can treat it as a
microfluid, or a macrofluid, whatever we wish.
• In this problem we solved the plug flow case as a microfluid, and
we solved the non-ideal case as a macrofluid.
EXAMPLE 5: REACTION OF A MACROFLUID
is given by the curve in Fig

Equation is the pertinent performance equation. Evaluate terms in
this expression. For
E= CAo/Area
Area = 2x2 =4
E = 2/4 = 0.5
Next Lecture:
Heterogeneous processes
35

Lecture 2 Non Ideal Flow

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Lecture 2: Non ideal

To relate E with F imagine a steady flow of

But the first term is simply the F value, while the

In graphical form, this relationship is

These relationships show how stimulus-

The concentration readings in Table 1

To find E, the area under this curve must

In symbols and taking limits (shrinking the rectangles) we obtain the

Thus we can determine the output from a multiregion flow

Thus the fraction of reactant unconverted in a plug flow

This suggests that channeling and short-

EXAMPLE 5: REACTION OF A MACROFLUID

is given by the curve in Fig

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