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Continuous mixing of solids. A review
Powder Technology. 15 (1976) 237 - 243 @ Elsevier Sequoia S-A_, Lausanne - P-inted
237
in the Netherlands
Continuous Mixing of Solids. A Review*
J. C. WILLIAMS Postgraduate School (Received
January
of Studies in Powder-Technology.
University
of Bradford.
Bradford 507
IDP
(Gt_ Britain)
30, 1976)
INTRODUCTION
In designing industrial plant for the handling and processing of liquids, it is taken for granted that a continuous system will be chosen unless there are compelling reasons for using a batch process_ The economic advantages of using a continuous process are now widely accepted and do not need to be argued here. Processes involving the handling and processing of particulate solids are, however, often designed as batch processes. Even when other parts of the process are continuous, it is common for solids to be mixed in batches and stored in a hopper until required_ Such systems should be examined critically to see if it would not be bettez to use continuous mixing_ Apart from economic advantages, there may be good technical reasons for preferring continuous mixing_ For materials which tend to segregate it is often very difficult to obtain good mixing in a batch mixer, and even if satisfactory mixing is obtained the subsequent handling and storage of the mixture will lead to segregation, so that on arrival at the point at which the mixture is used it is almost inevitable that some loss of mixing quality wili have occurred_ In such a case good mixing may be more readily obtained and handling and storage of the mixture can be avoided by using a continuous mixer placed immediately before the next part of the process. A change to continuous mixing to some extent changes the problem, as it is now necessary to ensure satisfactory controlled feeding of the components. However, since the technical advantages of continuous *Paper presented at the 5th CHISA Congress, Symposium on Mixing, Segregation and Sampling pf Particulate Solids, Prague, August 24 - 29.1975. <, \
mixing are most marked with components which tend to segregate, these will be freeflowing and the problem of controlling feed rat& is generally then more tractable than obtaining good batch mixing_ In addition a continuous mixer can be used to smooth out fluctuations in the feed rates, so that the requirements of the feeders can be more easily met.
ADVANTAGES CONTINUOUS
AND DISADVANTAGES MIXING
OF
The advantages and disadvantages of continuous mking and the conditions under which it is most likely to be the preferred method of mixing are summarised below. Aduantages
(1) It is easier to obtain good mixing of segregating materials with continuous mixing. (2) The mixer can be placed immediately before the next part of the process, reducing the risk of segregation in handling and storing the mixture. (3) Labour costs involved in filling and emptying the mixer are reduced_ (4) The cost of materials held up in the process can be reduced_ (5) The plant will generally require less space. Disadvcntages a.-zd limitations (1) The capital cost of the equipment
may
be higher, although this will be offset by the fact that the continuous mixer will be smaller than the batch mixer which it replaces and provision need not be made for storing the mixture_ (2) The most difficult problem in running a continuous mixing system is likely to be
238
the maintenance of the continuous feeders. (3) Continuous mixing may be unacceptable when a large number of components have to be mised. (4) For feeding small amounts of trace elements, commercial weigh feeders may not be avaiIabIe to give the required delivery rates. (5) A continuous mixing system is generally less flexible in accommodating changes in the materials handled.
EQUIPMENT
FOR
CONTIW_IOUS
MIXING
The simplest type of continuous mixer is one which can be used when the materials to be mixed are free-flowing and very prone to segregation_ The components are fed, one on top of the other, onto a conveyor belt which then carries a ribbon of material like a striped toothpaste_ If any short length of this ribbon couId be cut out from the stream it v:ouId contain the correct proportions of the components, subject to the fluctuations of the delivery rates of the feeders. To obtain a mixture it is therefore necessary only to cause transverse mixing of the components in the stream- This can be done as the material faIIs off the conveyor belt by arranging that it passes through a set of rotating blades The mixture then goes directly, without storage, to the next part of the process, which might be a bagging machine, tableting machine, glass melting tank or other equipment. Since there is a negligibly small residence time in the mixer, no appreciable amount of back-mixing takes place, and the quality of the mixture obtained is therefore only as good as the feeding systems will allow_ For very segregating materials this may be the best method to use, and it is used in at least one place to my knowiedge with satisfactory rest&s. In general, however. it is possible to use a continuous mixer with an appreciable residence time in which back-mixing takes pIace and this smooths out some of the fluctuations in the composition of the stream entering the mixer. There is very Little information avaiIabIe about the performance of industriaI continuous mixers. Some general conclusions can, however, be drawn from the results for batch mixers_ Mixers which rely predominantly on
the tumbling of the ingredients are known to be unsuitable for segregating materials in batch mixing, and it is reasonable to suppose that this will also apply to continuous mixing. If the materiaIs are not Iiable to segregation, generally because they are very fine or cohesive, any type of mixer shouId give satisfactory results, provided the mean residence time of the mixer (capacity divided by volume feed rate) is large enough. The best operating conditions for a given type of mixer can be found by experimenting using the components to be mixed.
DEFINITION
V u B r
= = = =
OF TERMS
AND
SYMBOLS
volume of material in mixer (L3) volume feed rate to mixer ( L3 T ’ ) mean residence time = V/u (T) reduced time = time/O (-)
Both the input and the output streams of a continuous mixer can be characterised by a composition variance and a correlation coefficient_ The variance (of, af) is a measure of the magnitude of the fluctuations from the mean value. The function of a continuous mixer is to smooth out fluctuations in the composition of the input stream. A measure of its ability to do this is the variance reduction ratio (V-R-R.), which is the ratio of the variances of the input and output streams (~$/a~). This ratio is used as a measure of the effectiveness of the mixer. The correlation coefficient of a stream is a description of the order in which the fIuctuations are arranged; for points separated by a time r the correlation coefficient R(r) is given by R(r)
=
cov 6%
Xt*r)
var et ) where X~ represents the fraction of one component in the stream at time t. Smce the correlation coefficient varies with the separation time, the effect of correlation over aU separaticm times is represented by a correlation index, where
1, = f 0
R(r)
emrle dr
239
THE
IDEAL
MIXER
The best that can be expected from a continuous mixer is that at any time its contents shall be randomly mixed. Such a mixer is referred to as an “ideal” mixer. The concept is useful in that it gives the highest value of the variance reduction ratio that can be expected from a mixer. In design it gives the smallest capacity of mixer to perform a given duty. Danckwerts and Sellers [l J have shown that for an ideaI mixer the variance reduction ratio is given by the equation 1 p=V.R.R_
0
V
-
os
--cr'v R(r) dr e
(1)
To inte,ate this espression it is required to know how the correlation coefficient varies with the interval r. Goldsmith [ 2 J suggested that it could be assumed that as the interval r between the points examined increases, R falls off in geometrical progression, that is,
R(r) = d
(2)
where ial & Z_ Substituting this into eqn. (2) and integrating gives V.R.R.=l--ina
(3)
0
Equation (3) can be used to calculate the variance reduction ratio for an ideal mixer or the least capacity of a miver required to a given dutyEquations (1) and (3) apply when the input and output are continuous. Beaudry [ 3 3 and Goidsmith [ 2 J have discussed the effect of various types of intermittent feeding and discharging of an ideal mixerThe work of JKramers and Alberda [4 J and Walker and Cholette [ 5 J suggests that it is sometimes possibIe to obtain better mixing by using a number of small mixers in series rather than one large mixer. These results were obtained for liquid mixing systems and have not yet been applied to solid particle mixing.
THE
NON-IDEAL
MIXER
Generally, it cannot be assumed that a continuous mixer will be ideal. To predict the
performance of a non-ideal mixer it is necessary to obtain information about the amount of back-mixing occurring in it. This can be found by determining the residence time distribution which is obtained by a stimulus response technique. The methods most convenient for studying particulate solids mixers are the introduction of a step change, which gives the internal age distribution 1, or the introduction of a pu!se, which gives the exit age distribution E_ If either distribution is obtained by experiment, the other can be obtained from the relation
ELd$ Most of the pubhshed papers on the performance of continuous systems are concerned with fluids in connection with the design of continuous reactors 16, 7 J , distillation columns 18 J , heat eschangers [9 J , etc. Levenspiel 110 J has discussed the interpretation of residence time distributions_ In the past few years, although many papers dealing with the batch mixing of particulate solids have appeared, there seems to be little movement towards the study of continuous mixing systems. Poole et al_ [ 11,12 J considered the continuous mixing of cohesive powders in a ribbon mixer. In their experiments they used very accurately controlled feeders, so that there were no appreciable fluctuations in the input. The problem was thus reduced to the mixing of the components in the direction perpendicular to that of the flow. They investigated the homogeneity of the mixture by extracting samples from random positions in the mixer and in the output. They found that the degree of homogeneity obtained was as good as in batch mixing. They also found, in mixing urania and thoria, that if the minor component was fed intermittently, maintaining the same overall rate as in continuous mixing, the homogeneity of the mixture at the outlet was the same as in the case of continuous feeding, provided the interval between feed batches was less than one-fifth of the mean residence time. They obtained residence time distributions for the mixer but did not investigate the relation between residence time distribution and mixer performance. Sugimoto and his workers 113 - 151 attempted to explain the performance of a
continuous horizontal drum mixer in terms of the mechanisms of flow and particle segregation inside the mixer. They found that segregating zones occur in the same way as had previously been reported in batch mixing. In the case of a binary mixture of particles of different size, they studied the situation where the smaller particles are f-me enough to fit into the voids between the larger particles without disturbing their packing arrangements and the proportion of fine particles present is higher than that which can be accommodated in the voids. They then found that two zones were formed, one consisting of coarse particles with fines filling the voids and the othe; consisting of fines only_ They argued that the bulk density of the former zone would be higher than that of the latter, and that this would account for fluctuations in the discharge rate from the mixer. Predictions of the variations of composition and discharge rate agree only moderatoIy with experimental results. The same workers [ 141 argued that a continuous drum mixer may be divided into two parts, one where the segregating zones had already been formed and no further changes were taking place in the asial direction, and the other where the zones were not yet completely formed. From a residence time distribution they obtained an axial dispersion coefficient, and showed esperimentally that asial dispersion increased with increasing proportion ofthe smallerparticlesand stoppedwhenthesegregatingzones hadbeen formed; the rate of axial dispersion also increased with drum speed within the range 10 - 30% of the c&i&l speed. Sugimoto 1153 attempted to predict the residence time distribution of a continuous horizontal drum mixer by considering the paths followed by different particles_ The model he proposed was that each particle follows a path in which it alternately moves upwards in a circular path and downwards in a straight line in a direction parallel to the maximum slope of the surface of the particle bed- During each drum rotation the particle moves rorward a distance which depends on its position within the bed_ By assuming that each particle remains at the same relative position in the bed in passing through the mixer (e-g- a particle which is in the outer surface of the bed continues to move in the oY.ttersurface), the residence time distribution oL^the particles
001
J
’ 0
05
10
15
20
25
30
Fig. 1. Effect of drum speed on the performance of a drum mixer. Internal age distribution, I_ 30, 40, 50, 130, 60, 100, SO r.p.m. can be predicted_ In spite of the simplicity of the model, reasonable agreement was obtained between predicted and experimental results. The papers referred to above approach the problem by attempting to predict the performance of a mixer by using a model which describes the mechanisms of flow and mixing Such attempts are valuable in that they increase our understanding of what is going on inside the miser, but it is unlikely that they will lead to sufficiently reliable predictions of mixer performance and of the effect of operating conditions on the quality of mixing produced for use in the design or selection of mixers for particular duties. Even in the comparatively simple case of a rotating drum, only partial success can be claimed, and for mixers with more complex mechanisms (e-g_ ribbon blenders), there would be little hope of producing a satisfactory working model. In studying continuous systems involving the mixing of liquids, it has been found that the most successful approach has been to use residence time distributions to give information about the effectiveness of the mixer and to examin e the effect of operating variables_ This approach to the continuous mixing of solid particles was taken by WiIIiams and
211 TABLE
1
Effect of drum speed on mixer performance Drum speed. r.p.m.
20
40
50
60
80
100
120
Variance reduction ratio
2.0
2.9
5.5
10.6
13.6
12.1
8.8
For ideal mixing,
V.R.R.
= 11.5.
Rahman [16 - 18]. Their aim was to develop a method for predicting the performance of a mixer from the result of a stimulus response test. The work described was the first phase of a more extensive programme, being confined to the study of non-segregating systems_ The mixer used was an inclined rotating
drum
which
was
fed
10
15 seconds_
-
_______________-
_ &
The
_ -o&+&~z~~_~%~~_
_
= 2 /
I
0
ci
_ _ _
intermittently
outlet stream was examined by taking E-second samples, the whole of the stream being collected. Under these conditions the results are the same as if the feeding were continuous_ The effects of feed rate, drum inclination and drum speed were first examined to establish the optimum conditions for future tests, using a pulse input test to determine the residence time distribution. The effect of varying the drum speed is shown in Fig. 1, where the internal age distribution is plotted in the form log I against reduced time_ For an ideal mixer the internal age distribution is of the form I = e-‘, SO that a plot of log I against r will be a straight line of slope -l/2.3. This line is shown for comparison in Fig. 1, which shows that at low drum speeds the graph departs every
very much from that for-an ideal mixer. As the speed increases, the graph approaches a straight line of slope -l/2.3. At 80 r_p.m., the graph is very nearly a straight line parallel to the ideal miuing line but displaced to the right. Further increase in speed leads to lines of greater slope, indicating a fall-off in mixing performance. Two methods were developed for predicting the performance of the mixer, as characterised by the variance reduction ratio, from the residence time distribution_ In the first, the residence time distribution is plotted as in Fig. 1 (log I against reduced time) and is approximated by one or more straight lines. Using a theorem proved by Danckwerts: E(t) E(c f r) R(r) dr dt
V.R.R. can be evaluated provided the correlogram R(r) can be represented in a mathematical form. If it is assumed that the correlation coefficient falls off geometrically with time. R(r) = ,r where [al < 1. the integration can be performed and the value of the variance reduction ratio obtained. For the experimental results shown in Fig- 1 these the
calculations
have
been
carried
out,
assuming
a = 0.11,
and the values of the variance reduction ratio are shown in Table 1. This method has the disadvantage that it requires the assumption of a mathematical form for the correlogram of the input stream. A second method was developed which did not have this limitation. It provides a set of equations which give the composition of any sample in the output streams in terms of the
Meosumxf
_
123
6 t
Fig. 2. Comparison of predicted and measured output sample compositions. (Input serial correlation index = 0.1.) % Composition_ Variance: input 1155; output-predicted 1.19, measured 1.23.
OutXow Predicted 405
Measured
,o
4 32
6
Fig_ 3. Comparison of predicted and measured output sample‘compositions. (Input serial correlation OS6.) ‘Z Composition. variance: input 11.56; outflow - predicted -I.O5. measured 1.33. TABLE
3
Effect of input serial correlation reduction ratio
index on variance
Input serial correlation index
i-1
-to_66
+O.lO
V_R.R_
1.3
2.9
9.9
Predicted Measured
l-3
2.7
9.6
-
0.19
-O.-X4
5@0
16.3 15.6
1200
(In the experiment at correlation index = - 0.44, the amplitude of composition fluctuations in the output is so small that more accurate results cannot be expected_)
composition variations in the input stream and the residence time distribution_ Physically, the method assumes that the input can be divided into a number of short time-elements and that for each such element the fraction of one of the components acts as a puke whose distribution in the discharge is known from a pulse input test. Experimental measurements were made to determine the form of composition fluctuations leaving the mixer for various types of input, the results being compared with the predicted values. ‘fwo typical results are shown in Figs. 2 and 3, which demonstrate the close agreement_ It was shown that the V_R.R. was independent of the magnitude of the input fluctuations but was heavily dependent on the serial correlation index, which can vary over the range -t- to -O-5The results are shown in Table 2. The method therefore permits the prediction of the composition of each sample in the output provided the residence time distribution and the form of the input are known In more recent work by WiRiams and Richardson (not yet published), the effect of segregation on the performance of a
index =
continuous mixer has been investigated and the above method of predicting the form of the output has been extended. For segregating materials the inclined rotating drum previously used was found to give very poor mixing, even under the most favourable conditions. Experiments were therefore carried out on a continuously operated fluidised bed mixer. It was shown that the composition fluctuations in the discharge from the mixer were the results of two effects: (i) input fluctuations and (ii) segregation within the mixer. The fiit effect was evaluated, as with non-segregating materials, by predicting the composition of each sample in the output and thus predicting the output variance. The second effect was evaluated by feeding to the mixer a stream of constant composition, measuring the composition of samples in the output stream and calculating the variance u& . The two variances were then added to find the total variance of the output stream. Experiments with a large number of inputs of different serial correlation index confirmed that the predicted output variance agreed closely with the measured value. Typical results are shown in Fig. 4. The dotted line shows the variance predicted for a non-segregating system and the full line gives values of the predicted total output variance, which agrees well with measured variances-
FUTURE
WORK
There is a need for considerably more research into the continuous mixing of solid particles for both cohesive and free-flowing materials_ Preferably, the work should be directed towards the problems of selecting xuitable mixers for different types of
243
0 -0.
-
PREDICTED(WITHOuT
x
EXPEFUMENTAL
-
PREDICTED~WITH x
o;;*
REFERENCES
1
O-&f*)
6.
INPUT
VARIANCE
Fig. 4. Predicted mixer performance materials)_
(segregating
industrial materials, the two approaches to the subject, mechanistic and stimulus response, being complementary_ Following the mechanistic approach, it would be valuable to have a better understanding of the patterns of flow and mixing in different types of mixer. The stimulus response method should be used to examine the performance of industrial type mixers, giving information which would lead to optimisation of their design and operation. The effect of scale of scrutiny on the quality of the mixing produced is being investigated at Bradford; such experiments are easier to perform with a continuous mixer than with a batch miuer.
P. V. Danckwerts and S. IM. Sellers, Ind_ Chem., (1951) 395. T. L. Goldsmith, Statistician, 16 (1966) 1. J. P. Beaudry, Chem. Met. Eng., (July 1918) 112. H. Kramers and G. Aiberda. Chem. Eng. Sci.. 2 (1953) 173. 0. S_ Waiker and A_ Cbolette. Pulp Pap. Mae. Can., (March 1958) 113. _ _ P, V. Danckwerts, Chem. Eng. Sci., 2 (1953) 1. M. Sharif, MI. SC. Thesis, Univ. of Bradford, 1966P. H. Hammond and D. L. A. Barber, Trans. Sot. Instrum. Technol., 1’7 (1965) 59. J. D. Cummins, Convention on Advance in Automatic Control, Inst. Mech. Eng., Nottingham, 5 - 9 April, 1965. 10 0. Levenspiel, Chemical Reaction Engineering, Wiley, New York, 1962. 11 K. R. Poole, R. F. Taylor and G_ P. Wall, Trans. Inst. Chem. Eng.. 13 (1965) T262. 12 K. R. Poole, R_ F. Taylor and G. P. Wail, Tram_ Inst. Chem. Eng.. 42 (1964) T305. 13 M_ Sugimoto. K Endoh and K. Tanaka, Kagaku Kogaku, 1(1966) 316. 14 M_ Sugimoto, K. Endoh and K. Tanaka, Kagaku Kogaku. 31 (1967) 145. 15 hl. Sugimoto, Kagaku Kogaku, 32 (1968) 291. 16 J. C. Williams and BI. A_ Rahman, J. Sot. Cosmet. Chem., 21(1970) 3_ 1’7 J. C. Williams and M. A Rahman, Powder Technol., 5 (1971f72) 87. 18 J. C. Williams and M. A. Rahman, Powder Technol., 5 (1971/72) 30'7.
237
in the Netherlands
Continuous Mixing of Solids. A Review*
J. C. WILLIAMS Postgraduate School (Received
January
of Studies in Powder-Technology.
University
of Bradford.
Bradford 507
IDP
(Gt_ Britain)
30, 1976)
INTRODUCTION
In designing industrial plant for the handling and processing of liquids, it is taken for granted that a continuous system will be chosen unless there are compelling reasons for using a batch process_ The economic advantages of using a continuous process are now widely accepted and do not need to be argued here. Processes involving the handling and processing of particulate solids are, however, often designed as batch processes. Even when other parts of the process are continuous, it is common for solids to be mixed in batches and stored in a hopper until required_ Such systems should be examined critically to see if it would not be bettez to use continuous mixing_ Apart from economic advantages, there may be good technical reasons for preferring continuous mixing_ For materials which tend to segregate it is often very difficult to obtain good mixing in a batch mixer, and even if satisfactory mixing is obtained the subsequent handling and storage of the mixture will lead to segregation, so that on arrival at the point at which the mixture is used it is almost inevitable that some loss of mixing quality wili have occurred_ In such a case good mixing may be more readily obtained and handling and storage of the mixture can be avoided by using a continuous mixer placed immediately before the next part of the process. A change to continuous mixing to some extent changes the problem, as it is now necessary to ensure satisfactory controlled feeding of the components. However, since the technical advantages of continuous *Paper presented at the 5th CHISA Congress, Symposium on Mixing, Segregation and Sampling pf Particulate Solids, Prague, August 24 - 29.1975. <, \
mixing are most marked with components which tend to segregate, these will be freeflowing and the problem of controlling feed rat& is generally then more tractable than obtaining good batch mixing_ In addition a continuous mixer can be used to smooth out fluctuations in the feed rates, so that the requirements of the feeders can be more easily met.
ADVANTAGES CONTINUOUS
AND DISADVANTAGES MIXING
OF
The advantages and disadvantages of continuous mking and the conditions under which it is most likely to be the preferred method of mixing are summarised below. Aduantages
(1) It is easier to obtain good mixing of segregating materials with continuous mixing. (2) The mixer can be placed immediately before the next part of the process, reducing the risk of segregation in handling and storing the mixture. (3) Labour costs involved in filling and emptying the mixer are reduced_ (4) The cost of materials held up in the process can be reduced_ (5) The plant will generally require less space. Disadvcntages a.-zd limitations (1) The capital cost of the equipment
may
be higher, although this will be offset by the fact that the continuous mixer will be smaller than the batch mixer which it replaces and provision need not be made for storing the mixture_ (2) The most difficult problem in running a continuous mixing system is likely to be
238
the maintenance of the continuous feeders. (3) Continuous mixing may be unacceptable when a large number of components have to be mised. (4) For feeding small amounts of trace elements, commercial weigh feeders may not be avaiIabIe to give the required delivery rates. (5) A continuous mixing system is generally less flexible in accommodating changes in the materials handled.
EQUIPMENT
FOR
CONTIW_IOUS
MIXING
The simplest type of continuous mixer is one which can be used when the materials to be mixed are free-flowing and very prone to segregation_ The components are fed, one on top of the other, onto a conveyor belt which then carries a ribbon of material like a striped toothpaste_ If any short length of this ribbon couId be cut out from the stream it v:ouId contain the correct proportions of the components, subject to the fluctuations of the delivery rates of the feeders. To obtain a mixture it is therefore necessary only to cause transverse mixing of the components in the stream- This can be done as the material faIIs off the conveyor belt by arranging that it passes through a set of rotating blades The mixture then goes directly, without storage, to the next part of the process, which might be a bagging machine, tableting machine, glass melting tank or other equipment. Since there is a negligibly small residence time in the mixer, no appreciable amount of back-mixing takes place, and the quality of the mixture obtained is therefore only as good as the feeding systems will allow_ For very segregating materials this may be the best method to use, and it is used in at least one place to my knowiedge with satisfactory rest&s. In general, however. it is possible to use a continuous mixer with an appreciable residence time in which back-mixing takes pIace and this smooths out some of the fluctuations in the composition of the stream entering the mixer. There is very Little information avaiIabIe about the performance of industriaI continuous mixers. Some general conclusions can, however, be drawn from the results for batch mixers_ Mixers which rely predominantly on
the tumbling of the ingredients are known to be unsuitable for segregating materials in batch mixing, and it is reasonable to suppose that this will also apply to continuous mixing. If the materiaIs are not Iiable to segregation, generally because they are very fine or cohesive, any type of mixer shouId give satisfactory results, provided the mean residence time of the mixer (capacity divided by volume feed rate) is large enough. The best operating conditions for a given type of mixer can be found by experimenting using the components to be mixed.
DEFINITION
V u B r
= = = =
OF TERMS
AND
SYMBOLS
volume of material in mixer (L3) volume feed rate to mixer ( L3 T ’ ) mean residence time = V/u (T) reduced time = time/O (-)
Both the input and the output streams of a continuous mixer can be characterised by a composition variance and a correlation coefficient_ The variance (of, af) is a measure of the magnitude of the fluctuations from the mean value. The function of a continuous mixer is to smooth out fluctuations in the composition of the input stream. A measure of its ability to do this is the variance reduction ratio (V-R-R.), which is the ratio of the variances of the input and output streams (~$/a~). This ratio is used as a measure of the effectiveness of the mixer. The correlation coefficient of a stream is a description of the order in which the fIuctuations are arranged; for points separated by a time r the correlation coefficient R(r) is given by R(r)
=
cov 6%
Xt*r)
var et ) where X~ represents the fraction of one component in the stream at time t. Smce the correlation coefficient varies with the separation time, the effect of correlation over aU separaticm times is represented by a correlation index, where
1, = f 0
R(r)
emrle dr
239
THE
IDEAL
MIXER
The best that can be expected from a continuous mixer is that at any time its contents shall be randomly mixed. Such a mixer is referred to as an “ideal” mixer. The concept is useful in that it gives the highest value of the variance reduction ratio that can be expected from a mixer. In design it gives the smallest capacity of mixer to perform a given duty. Danckwerts and Sellers [l J have shown that for an ideaI mixer the variance reduction ratio is given by the equation 1 p=V.R.R_
0
V
-
os
--cr'v R(r) dr e
(1)
To inte,ate this espression it is required to know how the correlation coefficient varies with the interval r. Goldsmith [ 2 J suggested that it could be assumed that as the interval r between the points examined increases, R falls off in geometrical progression, that is,
R(r) = d
(2)
where ial & Z_ Substituting this into eqn. (2) and integrating gives V.R.R.=l--ina
(3)
0
Equation (3) can be used to calculate the variance reduction ratio for an ideal mixer or the least capacity of a miver required to a given dutyEquations (1) and (3) apply when the input and output are continuous. Beaudry [ 3 3 and Goidsmith [ 2 J have discussed the effect of various types of intermittent feeding and discharging of an ideal mixerThe work of JKramers and Alberda [4 J and Walker and Cholette [ 5 J suggests that it is sometimes possibIe to obtain better mixing by using a number of small mixers in series rather than one large mixer. These results were obtained for liquid mixing systems and have not yet been applied to solid particle mixing.
THE
NON-IDEAL
MIXER
Generally, it cannot be assumed that a continuous mixer will be ideal. To predict the
performance of a non-ideal mixer it is necessary to obtain information about the amount of back-mixing occurring in it. This can be found by determining the residence time distribution which is obtained by a stimulus response technique. The methods most convenient for studying particulate solids mixers are the introduction of a step change, which gives the internal age distribution 1, or the introduction of a pu!se, which gives the exit age distribution E_ If either distribution is obtained by experiment, the other can be obtained from the relation
ELd$ Most of the pubhshed papers on the performance of continuous systems are concerned with fluids in connection with the design of continuous reactors 16, 7 J , distillation columns 18 J , heat eschangers [9 J , etc. Levenspiel 110 J has discussed the interpretation of residence time distributions_ In the past few years, although many papers dealing with the batch mixing of particulate solids have appeared, there seems to be little movement towards the study of continuous mixing systems. Poole et al_ [ 11,12 J considered the continuous mixing of cohesive powders in a ribbon mixer. In their experiments they used very accurately controlled feeders, so that there were no appreciable fluctuations in the input. The problem was thus reduced to the mixing of the components in the direction perpendicular to that of the flow. They investigated the homogeneity of the mixture by extracting samples from random positions in the mixer and in the output. They found that the degree of homogeneity obtained was as good as in batch mixing. They also found, in mixing urania and thoria, that if the minor component was fed intermittently, maintaining the same overall rate as in continuous mixing, the homogeneity of the mixture at the outlet was the same as in the case of continuous feeding, provided the interval between feed batches was less than one-fifth of the mean residence time. They obtained residence time distributions for the mixer but did not investigate the relation between residence time distribution and mixer performance. Sugimoto and his workers 113 - 151 attempted to explain the performance of a
continuous horizontal drum mixer in terms of the mechanisms of flow and particle segregation inside the mixer. They found that segregating zones occur in the same way as had previously been reported in batch mixing. In the case of a binary mixture of particles of different size, they studied the situation where the smaller particles are f-me enough to fit into the voids between the larger particles without disturbing their packing arrangements and the proportion of fine particles present is higher than that which can be accommodated in the voids. They then found that two zones were formed, one consisting of coarse particles with fines filling the voids and the othe; consisting of fines only_ They argued that the bulk density of the former zone would be higher than that of the latter, and that this would account for fluctuations in the discharge rate from the mixer. Predictions of the variations of composition and discharge rate agree only moderatoIy with experimental results. The same workers [ 141 argued that a continuous drum mixer may be divided into two parts, one where the segregating zones had already been formed and no further changes were taking place in the asial direction, and the other where the zones were not yet completely formed. From a residence time distribution they obtained an axial dispersion coefficient, and showed esperimentally that asial dispersion increased with increasing proportion ofthe smallerparticlesand stoppedwhenthesegregatingzones hadbeen formed; the rate of axial dispersion also increased with drum speed within the range 10 - 30% of the c&i&l speed. Sugimoto 1153 attempted to predict the residence time distribution of a continuous horizontal drum mixer by considering the paths followed by different particles_ The model he proposed was that each particle follows a path in which it alternately moves upwards in a circular path and downwards in a straight line in a direction parallel to the maximum slope of the surface of the particle bed- During each drum rotation the particle moves rorward a distance which depends on its position within the bed_ By assuming that each particle remains at the same relative position in the bed in passing through the mixer (e-g- a particle which is in the outer surface of the bed continues to move in the oY.ttersurface), the residence time distribution oL^the particles
001
J
’ 0
05
10
15
20
25
30
Fig. 1. Effect of drum speed on the performance of a drum mixer. Internal age distribution, I_ 30, 40, 50, 130, 60, 100, SO r.p.m. can be predicted_ In spite of the simplicity of the model, reasonable agreement was obtained between predicted and experimental results. The papers referred to above approach the problem by attempting to predict the performance of a mixer by using a model which describes the mechanisms of flow and mixing Such attempts are valuable in that they increase our understanding of what is going on inside the miser, but it is unlikely that they will lead to sufficiently reliable predictions of mixer performance and of the effect of operating conditions on the quality of mixing produced for use in the design or selection of mixers for particular duties. Even in the comparatively simple case of a rotating drum, only partial success can be claimed, and for mixers with more complex mechanisms (e-g_ ribbon blenders), there would be little hope of producing a satisfactory working model. In studying continuous systems involving the mixing of liquids, it has been found that the most successful approach has been to use residence time distributions to give information about the effectiveness of the mixer and to examin e the effect of operating variables_ This approach to the continuous mixing of solid particles was taken by WiIIiams and
211 TABLE
1
Effect of drum speed on mixer performance Drum speed. r.p.m.
20
40
50
60
80
100
120
Variance reduction ratio
2.0
2.9
5.5
10.6
13.6
12.1
8.8
For ideal mixing,
V.R.R.
= 11.5.
Rahman [16 - 18]. Their aim was to develop a method for predicting the performance of a mixer from the result of a stimulus response test. The work described was the first phase of a more extensive programme, being confined to the study of non-segregating systems_ The mixer used was an inclined rotating
drum
which
was
fed
10
15 seconds_
-
_______________-
_ &
The
_ -o&+&~z~~_~%~~_
_
= 2 /
I
0
ci
_ _ _
intermittently
outlet stream was examined by taking E-second samples, the whole of the stream being collected. Under these conditions the results are the same as if the feeding were continuous_ The effects of feed rate, drum inclination and drum speed were first examined to establish the optimum conditions for future tests, using a pulse input test to determine the residence time distribution. The effect of varying the drum speed is shown in Fig. 1, where the internal age distribution is plotted in the form log I against reduced time_ For an ideal mixer the internal age distribution is of the form I = e-‘, SO that a plot of log I against r will be a straight line of slope -l/2.3. This line is shown for comparison in Fig. 1, which shows that at low drum speeds the graph departs every
very much from that for-an ideal mixer. As the speed increases, the graph approaches a straight line of slope -l/2.3. At 80 r_p.m., the graph is very nearly a straight line parallel to the ideal miuing line but displaced to the right. Further increase in speed leads to lines of greater slope, indicating a fall-off in mixing performance. Two methods were developed for predicting the performance of the mixer, as characterised by the variance reduction ratio, from the residence time distribution_ In the first, the residence time distribution is plotted as in Fig. 1 (log I against reduced time) and is approximated by one or more straight lines. Using a theorem proved by Danckwerts: E(t) E(c f r) R(r) dr dt
V.R.R. can be evaluated provided the correlogram R(r) can be represented in a mathematical form. If it is assumed that the correlation coefficient falls off geometrically with time. R(r) = ,r where [al < 1. the integration can be performed and the value of the variance reduction ratio obtained. For the experimental results shown in Fig- 1 these the
calculations
have
been
carried
out,
assuming
a = 0.11,
and the values of the variance reduction ratio are shown in Table 1. This method has the disadvantage that it requires the assumption of a mathematical form for the correlogram of the input stream. A second method was developed which did not have this limitation. It provides a set of equations which give the composition of any sample in the output streams in terms of the
Meosumxf
_
123
6 t
Fig. 2. Comparison of predicted and measured output sample compositions. (Input serial correlation index = 0.1.) % Composition_ Variance: input 1155; output-predicted 1.19, measured 1.23.
OutXow Predicted 405
Measured
,o
4 32
6
Fig_ 3. Comparison of predicted and measured output sample‘compositions. (Input serial correlation OS6.) ‘Z Composition. variance: input 11.56; outflow - predicted -I.O5. measured 1.33. TABLE
3
Effect of input serial correlation reduction ratio
index on variance
Input serial correlation index
i-1
-to_66
+O.lO
V_R.R_
1.3
2.9
9.9
Predicted Measured
l-3
2.7
9.6
-
0.19
-O.-X4
5@0
16.3 15.6
1200
(In the experiment at correlation index = - 0.44, the amplitude of composition fluctuations in the output is so small that more accurate results cannot be expected_)
composition variations in the input stream and the residence time distribution_ Physically, the method assumes that the input can be divided into a number of short time-elements and that for each such element the fraction of one of the components acts as a puke whose distribution in the discharge is known from a pulse input test. Experimental measurements were made to determine the form of composition fluctuations leaving the mixer for various types of input, the results being compared with the predicted values. ‘fwo typical results are shown in Figs. 2 and 3, which demonstrate the close agreement_ It was shown that the V_R.R. was independent of the magnitude of the input fluctuations but was heavily dependent on the serial correlation index, which can vary over the range -t- to -O-5The results are shown in Table 2. The method therefore permits the prediction of the composition of each sample in the output provided the residence time distribution and the form of the input are known In more recent work by WiRiams and Richardson (not yet published), the effect of segregation on the performance of a
index =
continuous mixer has been investigated and the above method of predicting the form of the output has been extended. For segregating materials the inclined rotating drum previously used was found to give very poor mixing, even under the most favourable conditions. Experiments were therefore carried out on a continuously operated fluidised bed mixer. It was shown that the composition fluctuations in the discharge from the mixer were the results of two effects: (i) input fluctuations and (ii) segregation within the mixer. The fiit effect was evaluated, as with non-segregating materials, by predicting the composition of each sample in the output and thus predicting the output variance. The second effect was evaluated by feeding to the mixer a stream of constant composition, measuring the composition of samples in the output stream and calculating the variance u& . The two variances were then added to find the total variance of the output stream. Experiments with a large number of inputs of different serial correlation index confirmed that the predicted output variance agreed closely with the measured value. Typical results are shown in Fig. 4. The dotted line shows the variance predicted for a non-segregating system and the full line gives values of the predicted total output variance, which agrees well with measured variances-
FUTURE
WORK
There is a need for considerably more research into the continuous mixing of solid particles for both cohesive and free-flowing materials_ Preferably, the work should be directed towards the problems of selecting xuitable mixers for different types of
243
0 -0.
-
PREDICTED(WITHOuT
x
EXPEFUMENTAL
-
PREDICTED~WITH x
o;;*
REFERENCES
1
O-&f*)
6.
INPUT
VARIANCE
Fig. 4. Predicted mixer performance materials)_
(segregating
industrial materials, the two approaches to the subject, mechanistic and stimulus response, being complementary_ Following the mechanistic approach, it would be valuable to have a better understanding of the patterns of flow and mixing in different types of mixer. The stimulus response method should be used to examine the performance of industrial type mixers, giving information which would lead to optimisation of their design and operation. The effect of scale of scrutiny on the quality of the mixing produced is being investigated at Bradford; such experiments are easier to perform with a continuous mixer than with a batch miuer.
P. V. Danckwerts and S. IM. Sellers, Ind_ Chem., (1951) 395. T. L. Goldsmith, Statistician, 16 (1966) 1. J. P. Beaudry, Chem. Met. Eng., (July 1918) 112. H. Kramers and G. Aiberda. Chem. Eng. Sci.. 2 (1953) 173. 0. S_ Waiker and A_ Cbolette. Pulp Pap. Mae. Can., (March 1958) 113. _ _ P, V. Danckwerts, Chem. Eng. Sci., 2 (1953) 1. M. Sharif, MI. SC. Thesis, Univ. of Bradford, 1966P. H. Hammond and D. L. A. Barber, Trans. Sot. Instrum. Technol., 1’7 (1965) 59. J. D. Cummins, Convention on Advance in Automatic Control, Inst. Mech. Eng., Nottingham, 5 - 9 April, 1965. 10 0. Levenspiel, Chemical Reaction Engineering, Wiley, New York, 1962. 11 K. R. Poole, R. F. Taylor and G_ P. Wall, Trans. Inst. Chem. Eng.. 13 (1965) T262. 12 K. R. Poole, R_ F. Taylor and G. P. Wail, Tram_ Inst. Chem. Eng.. 42 (1964) T305. 13 M_ Sugimoto. K Endoh and K. Tanaka, Kagaku Kogaku, 1(1966) 316. 14 M_ Sugimoto, K. Endoh and K. Tanaka, Kagaku Kogaku. 31 (1967) 145. 15 hl. Sugimoto, Kagaku Kogaku, 32 (1968) 291. 16 J. C. Williams and BI. A_ Rahman, J. Sot. Cosmet. Chem., 21(1970) 3_ 1’7 J. C. Williams and M. A Rahman, Powder Technol., 5 (1971f72) 87. 18 J. C. Williams and M. A. Rahman, Powder Technol., 5 (1971/72) 30'7.