Chemical Reaction Engineering II
Notes 3
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Choice of Models
Models are useful for representing flows in real
vessels, for scale up, and for diagnosing poor
flow. There are different kinds of models
depending on whether the flow is close to plug,
mixed, or somewhere in between.
In this Section we shall deal with models with
small deviation from plug flow. There are two
models for this: the dispersion model and the
tanks-in-series model. Use the one you are
comfortable with. They are almost equivalent.
These models apply to turbulent flow in pipes,
laminar flow in very long tubes, flow in packed
beds, shaft kilns, long channels, screw
conveyers, etc.
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These models may not apply to laminar
flows in short tubes or laminar flows of
viscous materials; it may be that the
parabolic velocity profile is the main
cause of deviation from plug flow. This
situation is called pure convection model
which shall be treated later.
If you are unsure which model to use
there are charts in literature which can
tell you which model should be used to
represent your setup to be seen later.
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Suppose an ideal pulse of tracer is
introduced into the fluid entering a
vessel, the pulse spreads as it passes
through the vessel, we assume a
diffusion-like process superimposed on
plug flow. This is called dispersion or
longitudinal dispersion to distinguish it
from molecular diffusion. The dispersion
coefficient D (m2/s) represents this
spreading process.
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large D means rapid spreading of the tracer
curve
small D means slow spreading
D = 0 means no spreading, hence plug flow
And
is a dimensionless group that
characterizes the spread in the whole
vessel.
Figure 1. denotes the spreading of tracer
according to the dispersion model.
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We evaluate D or D/L recording to the
shape of the tracer curve as it passes the exit
of the vessel.
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In particular we measure;
These measured
are directly linked
by theory to D and D/L. The mean, for
continuous or discrete data, is defined as;
(1)
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The variance is defined as:
(2)
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Or
(3)
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The variance represents the square of the
spread of the distribution as it passes the
vessel exit and has units of (time)2. It is
particularly useful for matching
experimental curves to one of a family of
theoretical curves.
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Consider plug flow of a fluid, on top of which is
superimposed some degree of back-mixing, the
magnitude of which is independent of position
within the vessel. This condition implies that there
exist no stagnant pockets and no gross bypassing
or short-circuiting of fluid in the vessel.
This is called the dispersed plug flow model, or
simply the dispersion model. Note that with varying
intensities of turbulence or intermixing the
predictions of this model should range from plug
flow at one extreme to mixed flow at the other. As
a result the reactor volume for this model will lie
between those calculated for plug and mixed flow.
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Since the mixing process involves a
shuffling or redistribution of material either
by slippage or eddies, and it is repeated
many times during the flow of fluid through
the vessel, we can consider these
disturbances to be statistical in nature like
in molecular diffusion. For molecular
diffusion in the x-direction the governing
differential equation is given by Fick's law:
(4)
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Where
, the coefficient of molecular
diffusion is a parameter that uniquely
characterizes the process.
Analogously we may consider all the
contributions to intermixing of fluids
flowing in the x-direction to be
described by a similar expression;
(5)
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Where D, the longitudinal or axial
dispersion
coefficient
uniquely
characterizes the degree of back-mixing
during flow.
The term longitudinal or axial are used to
distinguish the mixing in the direction of
the flow from mixing in the lateral or radial
direction which is not of our primary
concern here. In streamline flow of fluid
through pipes, axial mixing is mainly due to
fluid velocity gradient while radial mixing is
due to molecular diffusion.
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The basic differential equation
representing this dispersion model is;
(6)
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Where;
The dimensionless group
, called
the vessel dispersion number is the
parameter that measures the extent of
axial dispersion, therefore
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This model usually represents quite
satisfactorily flows that do not deviate too
greatly from plug flow, thus real packed
beds and long tubes with streamline flows.
Fitting Dispersion model for small
dispersion,
< 0.01
If we impose an idealized pulse onto the
flowing fluid then dispersion modifies this
pulse as shown in the figure 2.
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Figure 2: Relationship between D/L
and EΘ
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(7)
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This represents a family of Gaussian
curves also called error or normal curves.
The equations representing this family
are:
(8)
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Figure 2 shows various ways to evaluate
the
parameter
D/L
from
an
experimental curve by calculating its
variance, by measuring its maximum
height or width at the point of inflection
or by finding that width which includes
68% of the area. Also note how the tracer
spreads as it moves down the vessel.
From the variance expression of eq. (8),
we find that
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For small extents of dispersion, numerous
simplifications and approximations in the
analysis of tracer curves are possible.
First, the shape of the tracer curve is
insensitive to the boundary condition
imposed on the vessel, whether closed or
open. So for both closed and open vessels
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For a series of vessels the
of the
individual vessels are additive, as in Figure
3. Thus;
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(9)
And
(10)
This additive property of variance allows us to
treat any one shot tracer input (no matter the
shape) and to extract the variance of the E
curve of the vessel from it. So we can have;
(11)
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According to the figure, the increase of
variance is the same in both cases or
Aris (1959) showed that for small extent
dispersion;
(12)
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Meaning that the
value for a vessel
can be found no matter what the shape of
the input curve is
The goodness of fit can only be evaluated
by comparison with the more exact
solution.
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From such a comparison we find that the
maximum error in estimating
is
given by;
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Large deviation from Plug Flow; D/L >
0.01
Here the pulse response is broad and it passes
the measurement point slowly enough that it
changes shape-it spreads-as it is being
measured. This gives a
non-symmetrical E curve.
An additional complication enters the picture
for large D/L; What happens right at the
entrance and exit of the vessel strongly affects
the shape of the tracer curve as well as the
relationship between the parameters of the
curve and D/L.
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We consider two types of boundary
conditions;
(i) either the flow is undisturbed as it passes
the entrance and the exit boundaries (this is
called the open boundary condition) or
(ii) you have plug flow outside the vessel up
to the boundaries (this is closed boundary
condition). This leads to 4 combinations of
b.c.; close-close, open-open and mixed.
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The figure below illustrates the closed and
open extremes whose RTD curves are
designated as Ecc and Eoo
Only one boundary condition gives a tracer
curve identical to the E function; that is the
closed vessel.
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For all other boundary conditions you do
not get a proper RTD. In all cases you
can evaluate D/L from the parameters
of the tracer curves but each curve has
its own mathematics. Let us look at
tracer curve for close vessel: Here an
analytical expression for the E curve is
not available but we can construct the
curve by numerical method (fig. 5) or
evaluate its mean and variance exactly;
(13)
thus;
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Open Vessel
This represents a convenient and
commonly used section of a long pipe. It
is a physical situation where the
analytical expression for the E curve is
not too complex. The results are given in
the response curve shown in Figure 6 by
these equations derived by Levenspiel;
(14)
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Notes:
1)- For small D/L all curves for different(15)
boundary conditions approach the curve of
eq. (8). For “small deviation”. At large
deviations the curves differ more from each
other.
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2)- To evaluate D/L, either match the
measured tracer curve or the measured
2 to theory.
3)- If the flow deviates greatly from plug
(D/L large), chances are that the real
vessel does not meet the assumption of
the model. It becomes questionable
whether the model should even be used.
4)- Always ask whether the model should
be used. Match 2 values if the shape
looks wrong, don’t use the model. Wrong
shapes here
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5)- For large D/L literature is conflicting due
to unclear assumptions about what is
happening at the vessel boundaries. The
mathematical treatment is questionable. Due
to these facts one needs to be very careful in
using dispersion model where back-mixing is
large, particularly in open systems.
Step input of tracers
The output F curve is S-shaped and is obtained
by integrating the corresponding E curve
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(16)
Thus at any time t or Θ;
The shape of F curve depends on D/L
and the boundary conditions of the
vessel. Analytical expressions are not
available for any of the F curves; but
their graphs can be constructed. Two
typical cases are as displayed below;
For small deviation from plug flow
(D/L<0.01) from eqns. (8) and (16) we
can find the curves of Figure 7.
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For this small deviation, we can find
D/L directly by plotting the
experimental data on the probability
graph paper as seen of Fig. 8.
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Plot on probability graph paper from which
you can find D/L
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For large deviation, the problems of
boundary conditions must be
considered. The resulting S-shape
response curve are not symmetrical,
their equations are not available, and
they are best analyzed by first
differentiating them to give the
corresponding Cpulse curve. Figure 9 shows
an example of this family of curves.
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Step response for large deviation, D/L
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Notes:
1)- One direct commercial application of
the step experiment is to find the zone
of intermixing- the contaminated width
between two fluids of somewhat similar
properties flowing one after the other in
a long pipe. Given D/L we can find this
from the probability plot of fig. 8. Design
charts to ease calculations are given by
Levenspiel (1958a).
2)- Sometimes one type of experiment
is more convenient for many reasons,
when you have a choice, the pulse
experiment is preferred.
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On the assumption that the closed vessel
of example 1 of Note 1. is well
represented by the dispersion model,
calculate the vessel dispersion number
D/L. the C versus t tracer response of
this vessel is;
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Since the c curve for this vessel is broad
and unsymmetrical, let us guess that
dispersion is too large (we cannot use
small dispersion curves).
We start with the variance matching
procedure. The mean and variance of a
continuous distribution measured at a
finite number of equidistant locations is
given by;
And
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Using the original tracer concentrationtime data we find;
Therefore;
and
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For a closed vessel eq. (13) relates the
variance to D/L.
Ignoring the 2nd term on the right, our
first approximation becomes;
Now correcting for the term ignored we
find by trial and error;
This value of D/L is beyond the limit
where Gaussian approximation should be
used so the guess was right.
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Von Rosenberg(1956) studied the
displacement of benzene by n-butyrate in a
38 mm diameter packed column 1219 mm
long, measuring the fraction of n-butyrate
in the exit stream by refractive index
methods. When graphed, the fraction of nbutyrate versus time was found to be Sshaped. This is the F curve for von
Rosenberg’s run at the lowest flow rate
where u = 0.0067 mm/s which is about 0.5
m/day. Find the vessel dispersion number
of this system.
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The F curve is given below.
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Instead of using slopes of F curve to give
E curve and determining the spread of
this curve, we use the probability paper
method; plotting the data on this paper
gives close to a straight line
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To find the variance and D/L from the
probability chart, follow the procedure in
Figure 8. so our figure here shows that;
This time interval represents 2,
therefore the standard deviation is;
This standard deviation is needed in
dimensionless time unit in order to find
D.
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Therefore the variance is;
And from eq (8);
This is well below 0.01 which justifies
the use of Gaussian approximation.
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