Tracer Tests in Flow Systems K. B. Bischoff
E. A. McCracken
Exact reactor output is crucial since it determines the necessary separation equipment, usually the major item of investment. The authors present wme generally useful methods for describing flow patterns in chemical reactors that have been found useful in industrial situations. Flow pattern description is necessary for the accurate prediction of reactor output he description of flow patterns in continuous flow T p r o c e s vegsel~has been greatly improved during the last decade. Previously, the two ideal extremes of plug flow or perfect mixing were about the only general models available. Often, one of these extremes of no mixing or complete mixing is sufficient to characterize many common tvpes of equipment. Heat exchangers,
workable design to be developed. The situation is not so simple in the case of chemical reactors. Any mixiig occurring in the above mentioned types of reactors can usually be treated as a perturbation on the basic plug-flow predictions. However, other types of reactors such as fluidized beds, two-phase flows, etc., cannot be dismissed so lightly. These have
E"" \
single phase flow i long tubular and packed bed chenucal reactors, packed gas absorbers and extractors, for example, can often be designed on the basis of plug flow, leading to the familiar log mean AT, NTU, and HTU. There is some question about the necessity of consideration of mixing for the latter two-phase flow operations since the mass transfer eoe5cient correlations often show quite a bit of data scatter which may be caused by complete neglect by the plug-flow model of an important variable-the mixing. A clear answer to this is not yet available, but the vast experience of the procars industries with this type ofseparation equipment still enables a 18
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
the possibility oflarger scale mixiig patterns which can be far from the two ideal extremes and these require more detailed descriptions. The reaction pmcegs is also much more sensitive to the concentration (on temperature) fields, leadii to further complications. For these reasons, the major application of more precise flow pattern descriptions has been to chemical reactors, although other procurres have also been worked upon. Finally, there are two economic reamns for the primary emphasis on chemical reactors: -The exact output of the reactor is extremely crucial since it determines the required separations equip
ment, etc., which is usually the major plant investment -The reactor itself is often a small fraction of the total investment, and so the custom designs that often result from a detailed analysis are feasible The purpose of this article is to present some generally useful methods for the description of flow patterns and then illustrate them with examples that have been found useful in actual industrial situations. Types of Flow Patterns
I t is useful to break the possible cases into two main divisions: uniform flow ; large scale, nonuniform flow. The first refers to situations which can be reasonably well represented by the ideal models and any mixing occurring is of the second -order in magnitude. In other words, it is visualized that the mixing is caused by a large number of small disturbances existing uniformly throughout the vessel. This process is somewhat analogous to molecular diffusion and, in fact, the model most commonly used to represent this situation is similar to the classical diffusion equation. The second case is the more difficult and is the one
upon which we will concentrate most of our attention. Here, there are large regions in the vessel that have mixing characteristics quite different from other regions. Some common examples are channeling in packed beds, dead regions around baffles, or intense mixing close to an impeller with little fluid motion elsewhere. Thesc situations usually cause undesirable performance, and so it is important that they be recognizable. This can often be accomplished by introducing a tracer of some sort into the flow through the vessel and then inspecting its behavior in the exit stream. The next section will introduce some concepts that are useful in this context.
Age Distribution Functions
Danckwerts (3) first introduced the general idea of age distribution functions. These give information about the fraction of fluid that resides a certain time in the vessel. The details of exactly where the fluid was during its stay are not considered and so information about point-to-point changes of the variables is not available from this type of treatment. The method thus does not yield complete information about the behavior, but many of the important problems mentioned above can be discovered. Detailed discussions of these concepts and the published applications can be found in Levenspiel (6) and Levenspiel and Bischoff (7). We will here briefly summarize this information, putting more emphasis on newer material that is not as thoroughly covered in the above references. Deflnitions
The most useful functions will be given with their physical meanings. The first two were introduced by Danckwerts (3) and the third by Naor and Shinnar (72). Their applications will then be discussed.
f
1. Residence time distribution (RTD)
= E(t) or Exit (fluid) age distribution E(t) dt = fraction of material in exit stream with ages between (t, t dt)
+
Internal age distribution = I(t) I(t) dt = fraction of material in vessel with ages between (t, t dt) 3. Intensity function = A(t) A ( t ) dt = fraction of fluid in vessel of age I that will leave at time between (t, t dt)
2.
+
+
VOL. 5 8
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19
These can easily be related as follows by using their physical meaning:
has age 2. The function representing this behavior is the impulse or Dirac delta function :
1. Amount of material leaving in time less than t equals : Q
sot
E ( t ‘) dt
Q = volumetric flow rate
where
Thus, the amount of fluid not leaving (remaining in vessel) = Pi
V I ( t ) -. Q - Q
J
E ( t ’ ) dt’
0
and
tZ(t) where
t
=
1 -
Li
E ( t ’ ) dt’
= V / Q = mean residence time
t
For the internal age distribution, Z(t), some fluid in the vessel has age of O + (‘just entered), some has age t (just leaving), some has age V 2 , etc., but none has age > t since fluid of this type must have already left. For steady input flow, the amount of fluid of all ages (less than i) is-the same and could be found by taking slices of the vessel all along its length. Therefore, the shape of the I(t) curve for plug flow is:
Also,
2.
+ +
Amount leaving between ( t , t d t ) = (amount not leaving before t ) [fraction of fluid of age t that will leave between (t, t dt) ] or in symbols, Q E ( t ) dt = [VI(t)][A(t) d t ]
or
The same basic information is contained in any one of the functions since they are all related by the above. However, some aspects of nonideal flow are often more easily seen in some functions than others. In the original work, Danckwerts (3) (see also 6, 7) discussed how one would experimentally measure these functions. The response at the vessel exit to a unit impulse at the entrance was termed the C-curve, and under perfect experimental conditions this is equivalent to the RTD, E(t). The response to an up-step was termed the F-curve, which leads directly to I ( t ) . Further physical meaning of the E ( t ) and I ( t ) functions will now be developed by deriving them for the ideal cases of plug flow and perfect mixing ( 3 ) . The newer, and somewhat more complicated, A ( t ) will be covered later. For plug flow, all fluid that enters at a certain time, say t = 0, passes through the vessel with no intermixing with other fluid and emerges in exactly one mean holding time, t = V/Q. In other words, the age of all fluid in the exit stream is the same since it has all spent the same length of time in the vessel before emerging: t . I n mathematical terms, the fraction of exit fluid with age greater or less than t is zero and it all Kenneth B. Bischof is Assistant Professor of Chemical Engineering at the University of Texas. E. A. McCracken is on the staf of the Esso Research Laboratories, Baton liouge, L a .
AUTHORS
20
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
t
In a perfectly mixed vessel, any fluid element has an equal probability of leaving, even if it just entered. Thus, of any fluid that enters at say t = 0, the amount that leaves will be proportional to the amount in the vessel. This, then, leads to the familiar first-order exponential decay law. Another way of saying this is that it is most probable that the fluid that enters at t = 0 leaves immediately, although a small amount will stay until (theoretically) t + m . Also, the exit stream is identical to the remainder of the vessel contents and so E ( t ) = Z(t):
-t Mathematical derivations of all this are given in (7). Finally, all of the variables can be normalized with respect to the mean residence time, t. The mean residence time, 1, is defined as the vessel volume divided by the volumetric flow rate.
e
=
tv‘t
E(e) = i ~ ( t )
I&>
=
Zz(l(t)
A ( 8 ) = tA(t) d E ( e ) = - - z(e> de
I t is worth repeating once again that all of the functions describe what is called macromixing. This means that they tell how long various parts of the fluid stream spend in the vessel but not the detailed location of every element within the age group. This latter information involves what is called micromixing and is much more difficult to obtain experimentally. Thus, the age distribution functions give only part of the details of the mixing; but since any further details are so difficult to obtain, in practice we must usually be satisfied with the macromixing information. Direct Use of Age Distribution Information
The various distribution functions can reveal many features of the flow situation. One method of visualizing what is occurring is to postulate a mathematical model, fit it to the data, and examine the parameters. This will be discussed in more detail later, First, however, we will see what information can be directly obtained from the E and I functions. The intensity function, h(t),will be discussed separately since its properties are not so well known. Again, it is useful to break the mixing phenomena into two main categories :
For this simple case, the other functions give no new information although they could be used equally as well to measure the amount of mixing. The second, and more important, case of gross mixing is naturally somewhat more complicated. The three features of dead space, bypassing, and nonuniform regions are particularly serious in applications. The mixing aside from these factors can be handled as for the first case above. Dead Space
There is no true dead space in a real system, since even in a completely nonmoving region, transport of matter would eventually occur by molecular diffusion. Often, however, a region of the vessel will have fluid whose holding time is 5-10 or more times the holding time of the rest of the fluid. For all practical purposes, this region is dead and is wasted space in the vessel. The existence of dead space is most easily seen from the E-fcn from the following characteristics :
-Relatively small degree of mixing uniformly throughout vessel, and -Gross mixing problems such as dead space, bypassing, nonuniform regions in vessel The first of these is, of course, relatively easy to handle. Probably the best approach is to use a simple model whose parameters are then usually rather easily correlatable for broad classes of situations. Thus, for single phase flow in packed beds, flow in straight empty tubes, etc., an eddy diffusion or axial dispersion model can be used, and the parameters have been correlated in the literature (7). The E-function (E-fcn) will show a small amount of spreading depending on the degree of mixing :
The curve will have a very long tail corresponding to the fluid held in the dead space. As discussed above, the true mean will be equal to the vessel volume divided by the flow rate, as usual, t = V/Q
However, the data for times longer than 2-3 mean residence times are seldom of sufficient accuracy to use in calculations. This means that the various integrals will be truncated at 0 = 2 or 3, so that the calculated apparent mean will be
G, PLUG FLOW (NO MIXING)
SMALL AMOUKlO1 MIXING
=
J e ~ ( ed)e , n
2,3
MORE MIXING
The true mean would be The mean of the curve will be at 0 = 1.O and the peak will be almost at the mean holding time, 0 = t / t = 1, and there will be no excessively long tails on the curves. A convenient measure of the amount of curve spreading for these cases is the variance or second moment about the mean: Mean
= p1 =
Lrn
0E(0) d 0
co
Variance
=
62
=
J,
=
1
(e - p1)2 ~ ( 0 de )
1 =6 =
Ime ~ ( e ) + de
m ( 0 )de = Jm
eE(0)de
=
e, +
la
0 E ( 0 ) de
Since, in this situation, the curve has a long tail, the second integral is an appreciable fraction of the total and so,
G, < 1 or t, < t The apparent mean gives a measure of the dead volume : VOL. 5 8
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actual case and would not be as easy to distinguish as shown in the above sketch. The I-fcn would be:
One problem with using these relations is that they assume that the true mean holding time, 2, is known. Since it can’t be found from the tracer curve (because of inaccurate tail), it must be known independently, and for certain situations information on the fluid phase holdup is difficult to obtain. Again, similar information could be obtained from the I-fcn (6, 7). Bypasring
I n a physical system there is no true inshntaneous bypassing in that all fluid takes a certain length of time to move through the vessel. Again, if some of the fluid passes through in a time 0.1-0.2 of the holding time of the main fluid stream, it can be practically said to bypass the vessel. The E-fcn would then be:
IIL
e
1
The amount of the rapid initial drop is a measure of the fraction of fluid bypassing the major portion of the vessel. In this case, the I-fcn is probably easier to use than the E-fcn. If there is a large amount of dead space or bypassing, the distinction between the two is not clear-cut. It depends on which part of the fluid is considered as the major part, which is somewhat arbitrary in this situation. Nonuniform Regions
Anything more complicated than simple dead space or bypassing is difficult to visualize completely from inspection of the distribution curves. Certain characteristics of ,the curve, such as moments (4), have been proposed for this purpose but they have not proved to be very successful. For example, it was found (5) that the variance of a deliberately faulted gas-liquid packed reactor was not much different from the variance of a normal bed. These data will be used later in an example. Higher moments (third, etc.) are often somewhat more sensitive but because of the heavy weighting they give to the tail, calculation from experimental data is not very certain. Also (73), many common theoretical models have very similar third moments which would increase the difficulties. In these situations it is probably most useful to utilize some type of flow model for the total vessel made up of small regions with the simpler types of mixing: uniform mixing, dead space, bypassing. These combined Models (6, 7) have been found useful for fluidized beds (7, 8)T: ..:.;..F 1
mk
The first hump wouldI correspond to the bypassing fluid
and the second to the main portion of fluid. Because of transfer between the bypassing and the other fluid, the two peaks would probably be greatly smeared in an
$p;&
..,..k.$
1.o
90
z
E
s= e= z
0.5
0
0.0 0
I
2 IOIMALIZfD TIME
3
‘-el@
Figurc 1. R e i d m e
timc dis!rib&m w w s for dispersion trampon model with expm~mintollynormoliud srolcs
22
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
VOLUMmlt MUN RfSID11Cf TlMf
e
Figure 2. Residmci time distribution curvcs for dispersion h m f i m t model With hn volumchic holding time scales
real stirred tanks (2, 74,tubular reactor with return bends (7), and other problems. lnbnsity Function
This section will consider the appearance of A@) for the above discussed cases. The original proposal (72) of A@) was for its use to identify what was termed stagnancy. This term included dead space and bypassing, both of which involve one part of the fluid being stagnant with respect to the other. From the physical definition of A(@, one can reason the qualitative shape of the curve for the various cases. If all of the fluid passes through the vessel in a fairly regular fashion, the longer an element of fluid has been in the vessel, the more probable it w ill be that it will leave. Also, most of the fluid will leave after a time equal to about the mean holding time and very little will remain for a long time. Thus, the A-fcn will have the form
A special case is the perfectly mixed vessel where the probability is the same for fluid of all ages, as shown on the sketch. When stagnancy exists, the A-fcn has a decidedly different shape. For dead space, the main portion of the fluid will have a A-fcn similar to those shown above. After this passes through the vessel, the remaining stagnant fluid has a low probability of leaving until a time, about equal to its log-mean holding time, is reached. Therefore, the A-fcn will no longer be monotonically increasing and should have a decreasing portion somewhere past the vicinity of 2. Eventually, the fluid will all leave and so A will be increasing again for these very long times. For bypassing, which is in a sense symmetric to dead space, about the same type of argument is used. Here, for small time, the small amount of bypassing fluid will have an increasing A. After this fluid leaves, the remainihg fluid will have a low probability of leaving until the vicinity of the mean holding time at which A will increase again. Thus, for those two cases the intensity function will have the form:
I;
Again, for large dead space or bypassing, the two effects are not distinct, and the A-curves would have about the same shape for either designation. When the E- and/or I-fcn can be used to check for dead space or bypassing, A really gives no further in-
> 1110.1
(21 0.2 131 1.0 141 5.0 I51 10.0 I61 0.1 171 0.1
0.05 0.05 0.05
15
0.05
0.05 0.10
15
15
0.05
5
wmmn WE Figurc 3. htmity function curuc~for d i p m i o n tnmsport model
Figure 4. RcridmM time disiribuh'on m w s fm pmollel dined tanks model lvith smnll mixing 'OL 5 8
NO. 7 J U L Y 1 9 6 6
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formation. Actually, it is not even as good a method since quantitative estimates of dead volume or bypassing fluid cannot easily be obtained from the curve as was the case with the E- and I-fcn. However, as was discussed above, if the true holding time is not known, the E-curve cannot be used to determine dead space. The intensity function, on the other hand, always has the same shape as regards the existence of maxima no matter what the time scale. Also, the intensity function is sometimes more sensitive to different flow situations, especially dead space. Therefore, it would seem that the greatest utility of the intensity function ia in the determination of the existence of dead space from experimental data, especially if the true mean holding time is unknown. Often, inspection of the tracer C U N ~ S using the above concepts is all that is required. If large amounts of dead space or bypassing exist, it may not be worthwhile to try to construct a detailed mathematical model in that the unit is certain to give poor performance. Some modifications might be called for to first eliminate the large mixing problems. After this, the unit will probably still have a certain amount of almost unavoidable uniform mixing for which a mathematical model can then be developed. The final section will discuss the most commonly used model. The Axial Dispersion-Tranrporl Model for Uniform Mixing
We can also include, along with the diffusive type of mixing, the effects of transfer to a stationary solid packing. Here we will consider the case of reversible adsorption. Components of the flowing phase transfer to the solid and residue for finite times and then transfer back to the flowing phase. The mathematical representation of this model is found (70) by making the usual material balances on the flowing phase
aY 1 a v + = - - - HiY + HgU + (a@) ae az N , ~ P
ay
-
-
a@))
and for the stationary phase
aU/&
HiY - HzU
+ (r(V,)
where the dimensionless variables are
Y
flowing phase concentration (dimensionless) U = stationary phase concentration (dimensionless) e = time (= t/t, as usual) Z = position N9 = axial dispersion Hi, H2 = mass transfer coefficients a( ) = Dirac delta function representing any tracer injection r(U) = rate of any reaction occurring =
This model is based on analogy with molecular diffusion as discussed above. Thus, it is most useful for representing situations in which a very large number of small, random, mixing events take place, since this would provide the closest analogy to molecular diffusion. For flow in packed beds (especially single phase flow), in empty pipes and other relatively uniform flow environments, the model is extremely useful and many correlations for these situations are available. See the literature (7)for a discussion of these and also the more complex radial gradients, etc.
@
; 1.0
ill
131
EB
151
0.1 1.0 10.0
0.05 0.05
0.05
15
15 15
15 15
15
'J
E
* 0.5
zr
RBIDfNCf TIM€ Figure 5. hksidence time distribution curves for parollel sliwed 1ank.1 model wifh moll mixing 24
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
F i p e 6. InfnMl age dish'bntim curuc~for parallel stirred tad5 model wifh small mixing
The solution of these differential equations (with r = 0) for a finite vessel with no dispersion occurring in the entrance and exit sections has been given (6), and for the pulse input is equivalent to E @ ) , as in Equation 1 where
residence time is larger than the volumetric mean m i dence time because some of the tracer would be held back in the stationary phase. In order to have a time variable with unity mean (which is what is done when normalizing experimental data for comparison with a model), introduce, =
.$(A,)
= [sin (XJ
+ (2 XJNJ
e/e
E(r) = 6 E ( e )
cos X,1rN*"
Then we obtain Equation 2. The variance about r = 1 is
-J
m
(r
- 1)'
E(r) d r =
The mean of this function is I ~ I=
Jo-f3E(B)de = 1
+ H,/Ha
Figure 1 shows E(T) for a few selected cases. When plotted in this fashion, they all have similar shapes, and it is seen that the mass transfer does not radically
Thus, with mass transfer occurring, the tracer mean
141 5.0
0.05
15
15
n,
Figure 7. Logm'lhmic plot of th internal age dish'bution curw for parallcl stirred tanks mo&i with small mixi.g
Figwe 8. Contows of cquol u o r i m c for porollcl stirred tanks model with small mi+ VOL 58
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7 JULY 1 9 6 6
25
alter the type of curve. Figure 2 shows the same curves plotted using volumetric mean residence time. Here, the mass transfer greatly modifies the curves. Notice that on Figure 1 all curves have peaks at the same location as the pure dispersion curve (A', = 10, Hi - 0 = Hn). Thus, it does not appear that the stagnancy phenomenon would be capable of representation by this model. Figure 3 shows intensity functions for a wide range of values, and it is seen that stagnancy never exists. Some of the curves where mass transfer into is greater than that out of stationary phase (Hi/Hn > 1) are fairly flat, but they never decrease. Therefore, in conclusion, this model is very useful for situations involving fairly uniform mixing but isn't capable of correlating stagnancy. It does represent well the many applications of uniform mixing and is the most extensively used model (7). For those cases where it is useful, it would be extremely beneficial to know independently the volumetric mean residence time in order to interpret the mass transfer effects. We have previously discussed the concepts of age distribution functions and their appearance for various types of flow situations. M4or emphasis was put on those cases deviating significantly from ideal plug flow or perfect mixing. This was done since situations close to ideal flow patterns usually do not require the more sophisticated design methods and thus are not of as much interest. We now illustrate these concepts with some definite calculations. Three examples will be used. The first consists of an extensive set of computed curves for a mathematical model in which the amount of bypassing, dead space, and mixing can be set beforehand. This will enable us to see the characteristics of the curves corresponding to these cases. The second example will consist of actual pilot plant data for gas-liquid flow through packed beds. For one run the bed was deliberately faulted in order to see the ability of the methods to pick up problems of this type. The third example will be a plant gas-liquid packed bed reactor known not to be giving proper performance. In this case, the problem was to decide whether poor flow patterns were the cause of the malfunction.
The normalized R TD for this case is
where
8 = t / t = reduced time f = Q1/Q = fraction of flow to branch @ a =
B
&/ti
=f
= ratio of residence times, branch @/ branch @ (1 -flu
+
Considering branch @ as the main fluid stream, it is seen that a lgives dead space and a 2 1 gives bypassing (for f 5 0.5). It should be emphasized that this model is not being proposed for use with actual vessels. In
