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Mixing of highly viscous liquids: flow geometrics for streamline subdivision and redistribution
Chemical Engineering
Science, 1973. Vol. 28, pp. 109 I-1098.
Per&mm
Press.
Printed
in Great
Britain
Mixing of highly viscous liquids: flow geometries for streamline subdivision and redistribution Shell Development
Company,
C. J. SHEARERt Emeryville, California, U.S.A. U.S.A.) (Received
(now located in Houston,
Texas,
24 July 1972)
Abstract-Subdivision and redistribution of flow streamlines is an important principle in the bulk mixing of highly viscous liquids. The principle is discussed with reference to three flow geometries, viz. (a) a stacked array of helical flighted ducts, (b) an assembly of rotating blades, and (c) an assembly of planetary rollers. Streamline redistribution in these geometries has been demonstrated by experiment, and simple theories are proposed to predict the degree of subdivision. These flow geometries have potential use in the processing operations of blending, devolatilization, heat exchange and reaction. Further work is required to reduce them to practice. INTRODUCTION
THE PROBLEM of mixing two liquids to produce
a homogeneous mass is a common one. Methods of solving this problem are numerous, and raise interesting questions in science and economics. When liquids are of low viscosity, turbulence provides a randomising influence, and the solution of the problem is generally obtained from a knowledge of the bulk or turbulent diffusivities. When the liquids to be mixed are highly viscous, the situation is quite different. Here the fluids are in laminar flow, and material is transported along streamlines. For initial mixing, flow streamlines must be redistributed relative to each other, and final mixing obtained by molecular diffusion. The redistribution of flow streamlines can be achieved in numerous flow geometries. How to achieve streamline redistribution systematically is the scientific basis of this work. Equipment for handling viscous liquids such as polymer melts and pastes is bulky and expensive. Relevant processing operations include blending, devolatilization, heat exchange and reaction. The capital cost of the equipment generally makes the largest contribution to the process total; power cost is also an important item. These cost observations suggest that equiptPresent
address: Department
of Chemical Engineering,
ment for processing highly viscous liquids should generally be designed for (a) continuous operation, (b) short residence time of liquid, and (c) tolerable power consumption. However, the present uncertainties in design of equipment for processing viscous liquids force the designer in many cases to choose batch-wise over continuous operation. A rewarding field of study should therefore be the development of continuous, inline equipment which ‘will achieve the required degree of mixing, and satisfy the performance criteria of short residence time and tolerable power consumption.
THE
PRINCIPLE OF STREAMLINE DIVISION AND REDISTRIBUTION
SUB-
The first step in the solution of any problem involving the mixing of highly viscous liquid streams will generally be the specification of the initial state of the stream before mixing and the required final state of the stream after mixing. We shall assume for the initial state that two streams, A and B, have to be mixed, and that the streams are in steady flow. A transverse crosssection through the flow will therefore show the initial state before mixing to be a distribution of Pembroke Street, Cambridge, England.
1091
C. J. SHEARER
A and B comprising a single patch of A beside a single patch of B. For the final state of mixing, we shall assume that a transverse cross-section through the flow should show a distribution of A and B comprising numerous and equally small patches of A distributed evenly among patches of B. As mixing progresses, we must go from large patches of A and B to smaller and smaller patches of A and B. This reduction in patch size can be achieved by successive subdivision and redistribution of patches. The fluid flow is laminar and steady, and material is transported along streamlines; for all points on a streamline, the velocity vectors meet the streamline tangentially, and no fluid can cross a streamline. Therefore, reduction in patch size in viscous liquid systems is to be obtained by successive subdivision and redistribution of flow streamlines. Streamline redistribution can be accomplished in many types of flow geometry having stationary and/or moving walls. Three flow geometries have been chosen as being representative of the types available. These flow geometries are shown diagrammatically in Fig. 1, and are as follows: (i) Stationary ~walls only. A diametral flight in the shape of a 180” helix is located in a hollow cylinder. If a viscous liquid Ilows through the cylinder, material will flow from one side of the cylinder to the other, and the streamlines will be redistributed. (ii) Stationary and mooring walls. A wall slides perpendicular to the central axis of a stationary rectangular groove as shown in Fig. lb. If a viscous liquid flows through the groove, material will flow from one side of the groove to the other, and the streamlines will be redistributed. (iii) Moving walls only. Two parallel flat plates move at the same velocity in opposite directions, and drive two rollers located between the plates as shown in Fig. lc. If viscous liquid flows through the channel between the plates and rollers, material will flow from one side of the channel to the other, and the streamlines will be redistributed. That each of the above flow geometries can
-..
diractim (a)
FIOW dnction
Stationay
walls
‘Staticcay wall
(b)
Stationary and moving walls ~rmsversevelocity
/
--
Strwr4ims (end !yeff?ts sstimoted)
A
Transverse r.ctii of the
channel
Cc) Moving walls
Fig.
1. Flow geometries
for redistribution of streamlines.
be arranged to achieve systematic streamline subdivision and redistribution is indicated by the experimental and theoretical evidence described below. A STACKED
ARRAY
OF HELICAL DUCTS
FLIGHTED
Two viscous streams A and B are allowed to flow through a stacked array of helical flighted ducts shown diagrammatically in Fig. 2. The initial state of the streams before mixing is seen from section XX’ to be an inner core of liquid “B” of diameter D surrounded by an annulus of liquid “A” of o.d. 20. The streams pass through two rows of ducts. The ducts of diameter D/2 are in triangular array; the rows are staggered relative to each other as shown in Fig. 2. For a first approximation, we shall assume that the liquid flows through each duct as a plug, and
1092
Mixing of highly viscous liquids
of ducts with duct diameter D/2, D/6 and D/18, the original core of stream B of diameter D should be subdivided and redistributed over the pipe cross-section in the form of numerous patches or filaments having approximate diameter of D/33. If there are n pairs of duct rows, the approximate diameter of filaments leaving the last row should be D/3”.
fi-
ELEMENT
._
f-
Y
EXPERIMENT
DUCT
2.
3
The purpose of this experiment was to demonstrate the principle of streamline subdivision and redistribution and to confirm the above prediction of patch-size reduction. Figure 3 is a photograph of the apparatus. The helical flights were manufactured by painting several coats of polyester resin on a helical aluminium former. The resin was allowed to cure. The helical flights were then removed and glued into glass tubes. The following table summarises the dimensions of the ducts which were used in the experiment.
Y’
Table 1. Dimensions of ducts SCCIION
XX’
SECTION
YY’
Fig. 2. Streamline subdivision and redistribution with helical-flighted ducts.
a streamline entering a duct on one side will leave the duct from the corresponding diametral position on the opposite side of the duct. Hence, the position of the streams after each row of ducts can be drawn. Figure 2 shows the position and distribution of the streams at the exit from the second row of ducts, section YY’. It is seen from section YY’ that mixing of streams “A” and “B” has been obtained; the original inner core of stream “B” of diameter D has been subdivided and redistributed over the pipe crosssection in the form of irregularly shaped patches having approximate size of D/3. If the stream leaving the second row of ducts is now allowed to flow through a third and a fourth row of ducts having one third the diameter of the ducts in the first two rows, subdivision and redistribution of stream B should again be obtained with a further threefold reduction in patch size of stream B. For three pairs of rows
Row Number l-2 3,4 5,6
Diameter o.d. i.d. (cm) (cm) S-08 1.7 0.8
4.75 1.4 0.56
Length (cm)
Baffle Width (cm)
5.08 l-9 O-8
0.25 0.15 o-13
The ducts were stacked in triangular array in a 17 cm i.d. column as shown in Fig. 3. Two isoviscous streams, one transparent and the other dyed red, were fed to the column from constant-head tanks. The red stream was fed to the duct assembly through an inner concentric tube of diameter 9 cm, fixed in position flush with the inlet to the duct assembly; the transparent stream was fed to the outer annulus. Photographs of the experiment are given in Figs. 3 and 4. Figure 3 shows the successive subdivision and redistribution of streamlines through each row of ducts. Figure 4 shows the dispersed filaments of liquid, 75-O cm downstream of the duct assembly. The diameter of the filaments is of the order of O-35 cm which is in good agreement with
1093
C. J. SHEARER
the predicted 33 reduction in size of the initial 9 cm dia. core. The liquid used in this experiment was glycerol of viscosity 6 poise; the same effects were observed with corn syrup of viscosity 3000 poise. Some pressure drop measurements through the duct assembly were taken for both glycerol and corn syrup experiments. These results are summarised in Table 2. Inspection of these results show that within the accuracy of the experiment, the pressure drop is directly proportional to the product of the first power of viscosity and flow-rate which is in agreement with the relationship for Poiseuille flow in straight tubes. An accurate prediction of pressure drop in this flow geometry is complicated by the helical nature of the flow geometry.
STATOR
PICTORIAL VIEW OF SCRAPER
ENTERlNG
BLADE
ROW I DISTRIBUTION
/
/‘,,’ I
,,:,’ _r _--
ENTERlNG
ROW2 DlSTRlEUTlON
LNTIMING
ROW 4 DlSlRlBUTlON
ENTERING
,x
ROW 3 DlSTRIEUTlON
Table 2. Pressure drop through the array of helical flights of Fig. 3
Liquid
Glycerol
Corn syrup
Flow-rate, cmYmin
Temperature “C
Viscosity, poise
Pressure drop, cm H,O
1,350 2,800 5,250 7,920
21.6 23.1 20.5 22-3
5.10 4.63 5.40 4-95
2.72 4.52 10.00 13.00
27.8 61.5 156.0 187.0
17.2 17.2 19-o 17.5
3150 3150 2050 2875
34.40 69.50 133.00 195+MJ
AN ASSEMBLY
OF SCRAPER
Fig.
BLADES
Two viscous streams A and B flow through the compartments between two cylinders, a stationary, hollow cylinder and an inner concentric rotor with radial, protruding blades. The blades are set in staggered rows, scrape the outer stator, and are located parallel to the axis of rotation. The streams enter the annulus between the cylinder, and their initial state before mixing consists of two semi-annular patches of “A” and “B” as shown in Fig. 5. For a first approximation and analogous to the earlier helical-baffle case, we shall assume that as the stream flows through each compartment the material moves as a rotating plug with 180
5. Streamline
subdivision and scraper blades.
redistribution
with
displacement such that a streamline which enters the compartment on one side leaves the compartment from the symmetrical position on the opposite side. The position and number of scraper blades in each row have been selected from geometrical considerations to give the maximum reduction in patch-size per row. The number of scraper blades in successive rows is 4, 12 and 36; their circumferential position is shown in Fig. 5. It is seen that, by subdivision and redistribution, mixing of the patches takes place and a three-fold reduction in patch size is obtained per row. For three successive rows of scraper blades, the original annulus of outer circumference 27rR which consists of semi-annular patches of A and B will be systematically subdivided and redistributed in the form of numerous annular segments of size ~Rl3~. With n rows of scraper
1094
Fig. 3. Stacked array of helical-tlighted ducts: blending of two iso-viscous streams. Liquid: Glycerol, Diameter: Tube- 17 cm; Inner Core-9 cm, Viscosity: 6poise, Duct Sizes: See Table 1.
Fig. 4. Dispersed filaments of liquid downstream of apparatus shown in Fig. 3.
(Facing page 1094)
Fig. 7. Experiment to show streamline redistribution between scraper blades. Wall velocity: 2-4 cm/n&. Compartment dimensions: 7-6 cm X 1.9 cm X 9.5 cm, Liquid: Glycerol, Viscosity: 5.7 poise.
(b)
(4
(4
(4
04 Fig. 8. Experiment to show streamline redistribution between planetary rollers. Wall velocities (all four): 10.5 cmlmin, Compartment dimensions: 8.9 cm i.d., 15.3 cm o.d., 120”between rollers, 15 cm depth, Liquid: Silicone Oil, Viscosity: 600 poise.
Mixing of highly viscous liquids
blades, the size of the annular segments leaving the last row will be rR/3”. Since redistribution of material in this flow geometry is achieved by transposing circumferentially adjacent patches, and each patch is an annular segment, the maximum reduction in patch size per row will be three-fold, as shown in Fig. 5. We may also note that if the numbers of scraper blades in successive rows were 3, 6, 12.. .) only a two-fold reduction in patch size would be obtained per row, thus producing a patch size in the stream leaving the nth row of 1rR/2”. EXPERIMENT
The object of this experiment was limited to streamline redistribution, and was to check the validity of the above assumption that liquid in the compartment between scraper blades is redistributed from one side to the other. Figure 6 is a sketch of the apparatus. The grooved bottom section and the slider were made of Plexiglass. An O-15 cm dia. hole was drilled in the slider to allow a coloured dye trace to be inserted for flow visualisation. The groove was filled with glycerol. A steady slider-speed was obtained by mounting the unit on the bed of a milling machine; the bed was set to feed at 2.4 cm min. Figure 7’is a series of photographs showing the displacement of the dye at different times. Two observations can be made from these photographs, one qualitative and the other quantitative. PLLXIGL/\S FRAME
IMOUNTED 19cml __ .._...
ON
MlLLlNG
MACHINE , PLEXl‘LAS
/ * r-----l _A__y-----p __L___________~.____-.
I/\ GROOVE viscous
HOLE
,0>5ca, DIAPAI
FOR
SLIDW!
DIE INJECTION
WITH ~ LIQUID
f 95cm ’
r
I
-mm I-
Fig. 6. Apparatus
It is observed that the bulk of the material is displaced from one side of the groove to the other. Thus, streamline redistribution can be achieved in this flow geometry. The second observation concerns the time taken to redistribute the material from one side of the groove to the other. The photographs of Fig. 7 show that with a steady slider-speed of 2.4 cmlsec, the bulk of the material is redistributed in the 7.6 cm broad groove in an approximate time of 430 sec. This redistribution time, which would be important in the practical design of a mixer of this type, can be estimated II priori from the velocity profile at the centre of the groove where end effects are small. The velocity profile, which has been analysed in extruder theory [ 1,2] can be determined from a force balance over a differential element. The resulting differential equation for a Newtonian liquid has the form dp &=Pg
a%
with the following boundary flow geometry:
(1) conditions
for this
y=o,
v=o
(2)
y=w,
v=s
(3)
s w
ov dy = 0.
Here, the coordinate axes x, y are shown in Fig. lb; dpldx is the pressure gradient which is assumed to be constant in the centre of the groove remote from the ends of the channel; v is the liquid velocity in the x-direction; p is the liquid viscosity; w is the thickness of the groove; and S is the slider velocity. On integration of Eq. (1) for the boundary conditions given by Eq. (2)-(4), the velocity profile is obtained:
1 .
; = 3(;>p-2(c).
IrnL” B
to show the streamline between scraper blades.
(5)
redistribution
The 1095
profile
is therefore
parabolic
and when
C. J. SHEARER
v/S = 0, y/w = 213. This profile is in agreement with the initial dye displacement shown in the photographs of Fig. 7. The redistribution time in the groove, c$,, can be obtained from the time to displace the material from one side of the groove to the other. We have
(6)
where b is the breadth of the groove. From Eq. (5) and (6) we have for the redistribution time 27b
4.S==.
(7)
With respect to the experiment of Fig. 7, where S = 2.4 cmlmin and b = 7.6 cm, the redistribution time from Eq. (7) is 640 set compared to 430 set observed experimentally in Fig. 7. This discrepancy can be attributed to the fact that not all the material is redistributed from one side of the groove to the other as indicated by the photographs of Fig. 7. AN ASSEMBLY
OF PLANETARY
The projected degree of subdivision in this flow geometry again assumes that the material moves through the compartment as a plug with 180” rotation. This assumption is checked experimentally below. Another assumption is that the ratio of roller to rotor diameter for each row remains constant. If this ratio were to vary, successive rows of rollers would rotate at different angular velocities since the angular velocity of the roller increases with decreasing ratio of roller to rotor diameter. This mode of bulk mixing employing different ratios of roller to rotor diameter is the subject of a separate study, and is outside the scope of the present investigation.
ROLLERS
The assembly of planetary rollers is very similar in construction to the previously described assembly of scraper blades. Both have stationary outer cylinders, but instead of scraper blades protruding from the inner rotor, there are rollers which are driven through gears by the rotor. If liquid is allowed to flow in the annular compartment between the rollers, the orbiting action of the rollers will cause the streamlines to be redistributed. Subdivision of the streamlines can be achieved by staggering successive rows of rollers. The proposed number and position of the rollers in each row are identical with the scraper-blade case discussed earlier and shown diagrammatically in Fig. 5. Thus if two streams are being mixed in an assembly of planetary rollers, the size of annular segments leaving the nth row will be 7rR/3” as in the scraper-blade case.
EXPERIMENT
An apparatus was constructed to demonstrate the streamline redistribution between pairs of rollers. The apparatus consisted of two variablespeed-drive motors, one driving the inner cylinder and the other driving the outer cylinder. The rollers were geared off the inner cylinder. The upper part of the apparatus was made of Plexiglas. The liquid used was silicone oil having a viscosity of 600 poise. A strand of coloured dye was injected into the liquid at a centre section between the rollers. The outer and inner cylinders were set to rotate at the same peripheral velocity of 10.5 cm/min. Figure 8 is a series of photographs showing the displacement of the strand of dye at different times. As in the previous scraper-blade case, two observations can be made from a set of photographs of this type. First, the bulk of the material is displaced from one side of the compartment to the other; thus, streamline redistribution can be achieved in this flow geometry. Secondly, the redistribution time, c#+,,can be estimated from the velocity profile at the centre of the compartment. For the coordinate axes and the simplified flow geometry of Fig. lc, the boundary conditions for Eq. (1) will be as follows:
1096
y=o,
v=-s
(8)
y = w,
v=+s
(9)
Mixing of highly viscous liquids
s w
APPLICATION
o vdy=O.
TO MIXER
DESIGN
The points which crystallise out of the earlier sections are two-fold: (i) flow streamlines can be systematically subdivided and redistributed in flow geometries having stationary and/or moving walls and (ii) the degree of subdivision in the three ;=2;-1. (11) flow geometries studied can be predicted fairly well though further experimentation is required. The redistribution time in the compartment, Although these points should prove useful to the 4 P, can again be obtained from the time to designer of mixing equipment, they fall far short displace the material from one side to the other. of providing a comprehensive guide to the We have design of mixing equipment. Some of the broader aspects of mixer design are discussed &+. w l . (12) below. First, the equipment options of the designer w/2 v dy will include not only array of helical flights, From Eq. (11) and (12) the redistribution time is scraper blades and planetary rollers discussed obtained, i.e. above but also other flow geometries described in trade journals and the patent literature. The f#Bp= p. (13) flow geometries known to the author, together with their estimated degree of subdivision, For the experiment of Fig. 8, where S = 10.5 are summarised in Table 3; references to the cmlmin and b approximates to 9.5 cm, the information sources are also given. The table redistribution time is 110 set compared to the shows that the basic mixing elements of the approximate time of 85 set observed experihelical flight, scraper blade and planetary roller mentally in Fig. 8. As in the earlier scraper-blade are already being used by equipment manucase, this discrepancy can be attributed to the facturers. It is thought, however, that their fact that material is held up at the ends of the designs could be improved by application of compartment; thus not all the material is re- some of the afore-described concepts of systedistributed as seen in the photographs of Fig. 8. matic subdivision and redistribution of flow
On integration of Eq. (1) for the above three boundary conditions, the velocity profile at the centre of the compartment is obtained
I
Table
3. Flow geometries
Flow element Stationary types 1. Arrav of helical flights 2. Singie helical flighi 3. Angled plates 4. Angled holes Moving types 5. Array of scraper blades 6. Array? of planetary rollers
for subdivision streamlines
and redistribution
Reference
of flow
Degree of subdivision after n flow elements
Present work [3] Kenics [4] Grace [5] Dow ISG [6,7] Ingles [8]
3” 2” 4” $
Present work Dulmadge [2] Present work[9, lo] Hotrock]
3” $ 3” $
t Assumes constant ratio of roller to rotor diameter. $ Not readily estimated.
1097
C. J. SHEARER
streamlines. Table 3 also includes other flow geometries which have not been discussed so far; these use stationary angled plates and angled holes for streamline redistribution. Secondly, the equipment-designer will generally have to concern himself with the final steps of the mixing process in addition to the initial bulk mixing discussed above. The viscous liquids processed in practice, such as polymeric melts, generally have low mass and thermal diffusivities. Therefore, the final stages in the mixing process will have to allow for these molecularscale processes to take place. These processes when the viscous liquid is also being sheared have been investigated by Mohr and his coworkers [ 11. Lastly, the designer will also have to consider the residence time, power consumption and ease of construction and operation of the mixer. The first two items are amenable to analytical treatment if the fluid mechanics in the mixer is understood. A particular advantage here of the planetary roller technique is that material on the wall, as shown by the flow pattern of Fig. 8, will be swept away from the wall, and hence dead zones will be avoided. The third item - ease of construction and operation of the mixer- will depend largely on mechanical engineering considerations. Much development work, therefore remains to be done to reduce to practice the flow geometries proposed herein. CONCLUSIONS
The principle of subdivision and redistribution of flow streamlines is fundamental to the bulk mixing of viscous liquids. The principle can be rationally applied to flow geometries having
stationary and/or moving walls. The resulting techniques are as follows: (a) a stacked array of helical flighted ducts, (b) an assembly of scraper blades, and (c) an assembly of planetary rollers. The streamline redistribution in these three flow geometries has been demonstrated by experiment; simple theories are proposed to predict the degree of subdivision. These techniques show promise, and they may find commercial application in the viscousliquid operations of blending, devolatilisation, heat exchange and reaction. Further development work is required to reduce them to practice. Acknowledgements-The
author is grateful to Shell Development Company for permission to publish the material presented in this paper. He also appreciated the guidance and encouragement from his colleagues of the Chemical Engineering Department, notably Messrs. J. C. Dygert, G. D. Towel1 andJ R. Street.
NOTATION
breadth of compartment diameter of duct diameter of tube length of duct pressure of liquid ii radius of tube s velocity of wall velocity of fluid in x direction V width of compartment W coordinate axes shown on Fig. 1 x9 Y viscosity of liquid redistribution time of liquid ; adjacent scraper blades redistribution time of liquid adjacent planetary rollers. b d D 1
between between
REFERENCES
HI MOHR W. D. Processing ofThermoplasticMoterids, (Ed. E. C. BERNHARDT), PI SCHENKEL G. Phtics Extrusion Technology and Theory pp. 32,37, Ilitfe 1966. SHEARER
p. 138. Reinhold, New York 1960.
C. J. U.S. Patent 3434437 1969 (Shell Development Company U.S.A.).
t:; ANON. Chern. Proc. Engng, June 1970 119. PI GRACE C. D. Chem. Proc. Engng, July 1971 57. WI HARDER R. E. U.S. Patent 3 195865 1965 (Dow Chemical Company). BRUNEMANN
H. and JOHN G. Chem. LangeTech. 197143 348.
;; INGLES 0. G., Australian Provisional Patent 21339 1962. R. H. and SHEARER C. J., U.S. Patent 3443796 1969 (Shell Development 191 OVERCASHIER HOI SHEARER C. J., U.S. Patent 3443798 1969 (Shell Development Company, U.S.A.).
1098
Company, U.S.A.).
Science, 1973. Vol. 28, pp. 109 I-1098.
Per&mm
Press.
Printed
in Great
Britain
Mixing of highly viscous liquids: flow geometries for streamline subdivision and redistribution Shell Development
Company,
C. J. SHEARERt Emeryville, California, U.S.A. U.S.A.) (Received
(now located in Houston,
Texas,
24 July 1972)
Abstract-Subdivision and redistribution of flow streamlines is an important principle in the bulk mixing of highly viscous liquids. The principle is discussed with reference to three flow geometries, viz. (a) a stacked array of helical flighted ducts, (b) an assembly of rotating blades, and (c) an assembly of planetary rollers. Streamline redistribution in these geometries has been demonstrated by experiment, and simple theories are proposed to predict the degree of subdivision. These flow geometries have potential use in the processing operations of blending, devolatilization, heat exchange and reaction. Further work is required to reduce them to practice. INTRODUCTION
THE PROBLEM of mixing two liquids to produce
a homogeneous mass is a common one. Methods of solving this problem are numerous, and raise interesting questions in science and economics. When liquids are of low viscosity, turbulence provides a randomising influence, and the solution of the problem is generally obtained from a knowledge of the bulk or turbulent diffusivities. When the liquids to be mixed are highly viscous, the situation is quite different. Here the fluids are in laminar flow, and material is transported along streamlines. For initial mixing, flow streamlines must be redistributed relative to each other, and final mixing obtained by molecular diffusion. The redistribution of flow streamlines can be achieved in numerous flow geometries. How to achieve streamline redistribution systematically is the scientific basis of this work. Equipment for handling viscous liquids such as polymer melts and pastes is bulky and expensive. Relevant processing operations include blending, devolatilization, heat exchange and reaction. The capital cost of the equipment generally makes the largest contribution to the process total; power cost is also an important item. These cost observations suggest that equiptPresent
address: Department
of Chemical Engineering,
ment for processing highly viscous liquids should generally be designed for (a) continuous operation, (b) short residence time of liquid, and (c) tolerable power consumption. However, the present uncertainties in design of equipment for processing viscous liquids force the designer in many cases to choose batch-wise over continuous operation. A rewarding field of study should therefore be the development of continuous, inline equipment which ‘will achieve the required degree of mixing, and satisfy the performance criteria of short residence time and tolerable power consumption.
THE
PRINCIPLE OF STREAMLINE DIVISION AND REDISTRIBUTION
SUB-
The first step in the solution of any problem involving the mixing of highly viscous liquid streams will generally be the specification of the initial state of the stream before mixing and the required final state of the stream after mixing. We shall assume for the initial state that two streams, A and B, have to be mixed, and that the streams are in steady flow. A transverse crosssection through the flow will therefore show the initial state before mixing to be a distribution of Pembroke Street, Cambridge, England.
1091
C. J. SHEARER
A and B comprising a single patch of A beside a single patch of B. For the final state of mixing, we shall assume that a transverse cross-section through the flow should show a distribution of A and B comprising numerous and equally small patches of A distributed evenly among patches of B. As mixing progresses, we must go from large patches of A and B to smaller and smaller patches of A and B. This reduction in patch size can be achieved by successive subdivision and redistribution of patches. The fluid flow is laminar and steady, and material is transported along streamlines; for all points on a streamline, the velocity vectors meet the streamline tangentially, and no fluid can cross a streamline. Therefore, reduction in patch size in viscous liquid systems is to be obtained by successive subdivision and redistribution of flow streamlines. Streamline redistribution can be accomplished in many types of flow geometry having stationary and/or moving walls. Three flow geometries have been chosen as being representative of the types available. These flow geometries are shown diagrammatically in Fig. 1, and are as follows: (i) Stationary ~walls only. A diametral flight in the shape of a 180” helix is located in a hollow cylinder. If a viscous liquid Ilows through the cylinder, material will flow from one side of the cylinder to the other, and the streamlines will be redistributed. (ii) Stationary and mooring walls. A wall slides perpendicular to the central axis of a stationary rectangular groove as shown in Fig. lb. If a viscous liquid flows through the groove, material will flow from one side of the groove to the other, and the streamlines will be redistributed. (iii) Moving walls only. Two parallel flat plates move at the same velocity in opposite directions, and drive two rollers located between the plates as shown in Fig. lc. If viscous liquid flows through the channel between the plates and rollers, material will flow from one side of the channel to the other, and the streamlines will be redistributed. That each of the above flow geometries can
-..
diractim (a)
FIOW dnction
Stationay
walls
‘Staticcay wall
(b)
Stationary and moving walls ~rmsversevelocity
/
--
Strwr4ims (end !yeff?ts sstimoted)
A
Transverse r.ctii of the
channel
Cc) Moving walls
Fig.
1. Flow geometries
for redistribution of streamlines.
be arranged to achieve systematic streamline subdivision and redistribution is indicated by the experimental and theoretical evidence described below. A STACKED
ARRAY
OF HELICAL DUCTS
FLIGHTED
Two viscous streams A and B are allowed to flow through a stacked array of helical flighted ducts shown diagrammatically in Fig. 2. The initial state of the streams before mixing is seen from section XX’ to be an inner core of liquid “B” of diameter D surrounded by an annulus of liquid “A” of o.d. 20. The streams pass through two rows of ducts. The ducts of diameter D/2 are in triangular array; the rows are staggered relative to each other as shown in Fig. 2. For a first approximation, we shall assume that the liquid flows through each duct as a plug, and
1092
Mixing of highly viscous liquids
of ducts with duct diameter D/2, D/6 and D/18, the original core of stream B of diameter D should be subdivided and redistributed over the pipe cross-section in the form of numerous patches or filaments having approximate diameter of D/33. If there are n pairs of duct rows, the approximate diameter of filaments leaving the last row should be D/3”.
fi-
ELEMENT
._
f-
Y
EXPERIMENT
DUCT
2.
3
The purpose of this experiment was to demonstrate the principle of streamline subdivision and redistribution and to confirm the above prediction of patch-size reduction. Figure 3 is a photograph of the apparatus. The helical flights were manufactured by painting several coats of polyester resin on a helical aluminium former. The resin was allowed to cure. The helical flights were then removed and glued into glass tubes. The following table summarises the dimensions of the ducts which were used in the experiment.
Y’
Table 1. Dimensions of ducts SCCIION
XX’
SECTION
YY’
Fig. 2. Streamline subdivision and redistribution with helical-flighted ducts.
a streamline entering a duct on one side will leave the duct from the corresponding diametral position on the opposite side of the duct. Hence, the position of the streams after each row of ducts can be drawn. Figure 2 shows the position and distribution of the streams at the exit from the second row of ducts, section YY’. It is seen from section YY’ that mixing of streams “A” and “B” has been obtained; the original inner core of stream “B” of diameter D has been subdivided and redistributed over the pipe crosssection in the form of irregularly shaped patches having approximate size of D/3. If the stream leaving the second row of ducts is now allowed to flow through a third and a fourth row of ducts having one third the diameter of the ducts in the first two rows, subdivision and redistribution of stream B should again be obtained with a further threefold reduction in patch size of stream B. For three pairs of rows
Row Number l-2 3,4 5,6
Diameter o.d. i.d. (cm) (cm) S-08 1.7 0.8
4.75 1.4 0.56
Length (cm)
Baffle Width (cm)
5.08 l-9 O-8
0.25 0.15 o-13
The ducts were stacked in triangular array in a 17 cm i.d. column as shown in Fig. 3. Two isoviscous streams, one transparent and the other dyed red, were fed to the column from constant-head tanks. The red stream was fed to the duct assembly through an inner concentric tube of diameter 9 cm, fixed in position flush with the inlet to the duct assembly; the transparent stream was fed to the outer annulus. Photographs of the experiment are given in Figs. 3 and 4. Figure 3 shows the successive subdivision and redistribution of streamlines through each row of ducts. Figure 4 shows the dispersed filaments of liquid, 75-O cm downstream of the duct assembly. The diameter of the filaments is of the order of O-35 cm which is in good agreement with
1093
C. J. SHEARER
the predicted 33 reduction in size of the initial 9 cm dia. core. The liquid used in this experiment was glycerol of viscosity 6 poise; the same effects were observed with corn syrup of viscosity 3000 poise. Some pressure drop measurements through the duct assembly were taken for both glycerol and corn syrup experiments. These results are summarised in Table 2. Inspection of these results show that within the accuracy of the experiment, the pressure drop is directly proportional to the product of the first power of viscosity and flow-rate which is in agreement with the relationship for Poiseuille flow in straight tubes. An accurate prediction of pressure drop in this flow geometry is complicated by the helical nature of the flow geometry.
STATOR
PICTORIAL VIEW OF SCRAPER
ENTERlNG
BLADE
ROW I DISTRIBUTION
/
/‘,,’ I
,,:,’ _r _--
ENTERlNG
ROW2 DlSTRlEUTlON
LNTIMING
ROW 4 DlSlRlBUTlON
ENTERING
,x
ROW 3 DlSTRIEUTlON
Table 2. Pressure drop through the array of helical flights of Fig. 3
Liquid
Glycerol
Corn syrup
Flow-rate, cmYmin
Temperature “C
Viscosity, poise
Pressure drop, cm H,O
1,350 2,800 5,250 7,920
21.6 23.1 20.5 22-3
5.10 4.63 5.40 4-95
2.72 4.52 10.00 13.00
27.8 61.5 156.0 187.0
17.2 17.2 19-o 17.5
3150 3150 2050 2875
34.40 69.50 133.00 195+MJ
AN ASSEMBLY
OF SCRAPER
Fig.
BLADES
Two viscous streams A and B flow through the compartments between two cylinders, a stationary, hollow cylinder and an inner concentric rotor with radial, protruding blades. The blades are set in staggered rows, scrape the outer stator, and are located parallel to the axis of rotation. The streams enter the annulus between the cylinder, and their initial state before mixing consists of two semi-annular patches of “A” and “B” as shown in Fig. 5. For a first approximation and analogous to the earlier helical-baffle case, we shall assume that as the stream flows through each compartment the material moves as a rotating plug with 180
5. Streamline
subdivision and scraper blades.
redistribution
with
displacement such that a streamline which enters the compartment on one side leaves the compartment from the symmetrical position on the opposite side. The position and number of scraper blades in each row have been selected from geometrical considerations to give the maximum reduction in patch-size per row. The number of scraper blades in successive rows is 4, 12 and 36; their circumferential position is shown in Fig. 5. It is seen that, by subdivision and redistribution, mixing of the patches takes place and a three-fold reduction in patch size is obtained per row. For three successive rows of scraper blades, the original annulus of outer circumference 27rR which consists of semi-annular patches of A and B will be systematically subdivided and redistributed in the form of numerous annular segments of size ~Rl3~. With n rows of scraper
1094
Fig. 3. Stacked array of helical-tlighted ducts: blending of two iso-viscous streams. Liquid: Glycerol, Diameter: Tube- 17 cm; Inner Core-9 cm, Viscosity: 6poise, Duct Sizes: See Table 1.
Fig. 4. Dispersed filaments of liquid downstream of apparatus shown in Fig. 3.
(Facing page 1094)
Fig. 7. Experiment to show streamline redistribution between scraper blades. Wall velocity: 2-4 cm/n&. Compartment dimensions: 7-6 cm X 1.9 cm X 9.5 cm, Liquid: Glycerol, Viscosity: 5.7 poise.
(b)
(4
(4
(4
04 Fig. 8. Experiment to show streamline redistribution between planetary rollers. Wall velocities (all four): 10.5 cmlmin, Compartment dimensions: 8.9 cm i.d., 15.3 cm o.d., 120”between rollers, 15 cm depth, Liquid: Silicone Oil, Viscosity: 600 poise.
Mixing of highly viscous liquids
blades, the size of the annular segments leaving the last row will be rR/3”. Since redistribution of material in this flow geometry is achieved by transposing circumferentially adjacent patches, and each patch is an annular segment, the maximum reduction in patch size per row will be three-fold, as shown in Fig. 5. We may also note that if the numbers of scraper blades in successive rows were 3, 6, 12.. .) only a two-fold reduction in patch size would be obtained per row, thus producing a patch size in the stream leaving the nth row of 1rR/2”. EXPERIMENT
The object of this experiment was limited to streamline redistribution, and was to check the validity of the above assumption that liquid in the compartment between scraper blades is redistributed from one side to the other. Figure 6 is a sketch of the apparatus. The grooved bottom section and the slider were made of Plexiglass. An O-15 cm dia. hole was drilled in the slider to allow a coloured dye trace to be inserted for flow visualisation. The groove was filled with glycerol. A steady slider-speed was obtained by mounting the unit on the bed of a milling machine; the bed was set to feed at 2.4 cm min. Figure 7’is a series of photographs showing the displacement of the dye at different times. Two observations can be made from these photographs, one qualitative and the other quantitative. PLLXIGL/\S FRAME
IMOUNTED 19cml __ .._...
ON
MlLLlNG
MACHINE , PLEXl‘LAS
/ * r-----l _A__y-----p __L___________~.____-.
I/\ GROOVE viscous
HOLE
,0>5ca, DIAPAI
FOR
SLIDW!
DIE INJECTION
WITH ~ LIQUID
f 95cm ’
r
I
-mm I-
Fig. 6. Apparatus
It is observed that the bulk of the material is displaced from one side of the groove to the other. Thus, streamline redistribution can be achieved in this flow geometry. The second observation concerns the time taken to redistribute the material from one side of the groove to the other. The photographs of Fig. 7 show that with a steady slider-speed of 2.4 cmlsec, the bulk of the material is redistributed in the 7.6 cm broad groove in an approximate time of 430 sec. This redistribution time, which would be important in the practical design of a mixer of this type, can be estimated II priori from the velocity profile at the centre of the groove where end effects are small. The velocity profile, which has been analysed in extruder theory [ 1,2] can be determined from a force balance over a differential element. The resulting differential equation for a Newtonian liquid has the form dp &=Pg
a%
with the following boundary flow geometry:
(1) conditions
for this
y=o,
v=o
(2)
y=w,
v=s
(3)
s w
ov dy = 0.
Here, the coordinate axes x, y are shown in Fig. lb; dpldx is the pressure gradient which is assumed to be constant in the centre of the groove remote from the ends of the channel; v is the liquid velocity in the x-direction; p is the liquid viscosity; w is the thickness of the groove; and S is the slider velocity. On integration of Eq. (1) for the boundary conditions given by Eq. (2)-(4), the velocity profile is obtained:
1 .
; = 3(;>p-2(c).
IrnL” B
to show the streamline between scraper blades.
(5)
redistribution
The 1095
profile
is therefore
parabolic
and when
C. J. SHEARER
v/S = 0, y/w = 213. This profile is in agreement with the initial dye displacement shown in the photographs of Fig. 7. The redistribution time in the groove, c$,, can be obtained from the time to displace the material from one side of the groove to the other. We have
(6)
where b is the breadth of the groove. From Eq. (5) and (6) we have for the redistribution time 27b
4.S==.
(7)
With respect to the experiment of Fig. 7, where S = 2.4 cmlmin and b = 7.6 cm, the redistribution time from Eq. (7) is 640 set compared to 430 set observed experimentally in Fig. 7. This discrepancy can be attributed to the fact that not all the material is redistributed from one side of the groove to the other as indicated by the photographs of Fig. 7. AN ASSEMBLY
OF PLANETARY
The projected degree of subdivision in this flow geometry again assumes that the material moves through the compartment as a plug with 180” rotation. This assumption is checked experimentally below. Another assumption is that the ratio of roller to rotor diameter for each row remains constant. If this ratio were to vary, successive rows of rollers would rotate at different angular velocities since the angular velocity of the roller increases with decreasing ratio of roller to rotor diameter. This mode of bulk mixing employing different ratios of roller to rotor diameter is the subject of a separate study, and is outside the scope of the present investigation.
ROLLERS
The assembly of planetary rollers is very similar in construction to the previously described assembly of scraper blades. Both have stationary outer cylinders, but instead of scraper blades protruding from the inner rotor, there are rollers which are driven through gears by the rotor. If liquid is allowed to flow in the annular compartment between the rollers, the orbiting action of the rollers will cause the streamlines to be redistributed. Subdivision of the streamlines can be achieved by staggering successive rows of rollers. The proposed number and position of the rollers in each row are identical with the scraper-blade case discussed earlier and shown diagrammatically in Fig. 5. Thus if two streams are being mixed in an assembly of planetary rollers, the size of annular segments leaving the nth row will be 7rR/3” as in the scraper-blade case.
EXPERIMENT
An apparatus was constructed to demonstrate the streamline redistribution between pairs of rollers. The apparatus consisted of two variablespeed-drive motors, one driving the inner cylinder and the other driving the outer cylinder. The rollers were geared off the inner cylinder. The upper part of the apparatus was made of Plexiglas. The liquid used was silicone oil having a viscosity of 600 poise. A strand of coloured dye was injected into the liquid at a centre section between the rollers. The outer and inner cylinders were set to rotate at the same peripheral velocity of 10.5 cm/min. Figure 8 is a series of photographs showing the displacement of the strand of dye at different times. As in the previous scraper-blade case, two observations can be made from a set of photographs of this type. First, the bulk of the material is displaced from one side of the compartment to the other; thus, streamline redistribution can be achieved in this flow geometry. Secondly, the redistribution time, c#+,,can be estimated from the velocity profile at the centre of the compartment. For the coordinate axes and the simplified flow geometry of Fig. lc, the boundary conditions for Eq. (1) will be as follows:
1096
y=o,
v=-s
(8)
y = w,
v=+s
(9)
Mixing of highly viscous liquids
s w
APPLICATION
o vdy=O.
TO MIXER
DESIGN
The points which crystallise out of the earlier sections are two-fold: (i) flow streamlines can be systematically subdivided and redistributed in flow geometries having stationary and/or moving walls and (ii) the degree of subdivision in the three ;=2;-1. (11) flow geometries studied can be predicted fairly well though further experimentation is required. The redistribution time in the compartment, Although these points should prove useful to the 4 P, can again be obtained from the time to designer of mixing equipment, they fall far short displace the material from one side to the other. of providing a comprehensive guide to the We have design of mixing equipment. Some of the broader aspects of mixer design are discussed &+. w l . (12) below. First, the equipment options of the designer w/2 v dy will include not only array of helical flights, From Eq. (11) and (12) the redistribution time is scraper blades and planetary rollers discussed obtained, i.e. above but also other flow geometries described in trade journals and the patent literature. The f#Bp= p. (13) flow geometries known to the author, together with their estimated degree of subdivision, For the experiment of Fig. 8, where S = 10.5 are summarised in Table 3; references to the cmlmin and b approximates to 9.5 cm, the information sources are also given. The table redistribution time is 110 set compared to the shows that the basic mixing elements of the approximate time of 85 set observed experihelical flight, scraper blade and planetary roller mentally in Fig. 8. As in the earlier scraper-blade are already being used by equipment manucase, this discrepancy can be attributed to the facturers. It is thought, however, that their fact that material is held up at the ends of the designs could be improved by application of compartment; thus not all the material is re- some of the afore-described concepts of systedistributed as seen in the photographs of Fig. 8. matic subdivision and redistribution of flow
On integration of Eq. (1) for the above three boundary conditions, the velocity profile at the centre of the compartment is obtained
I
Table
3. Flow geometries
Flow element Stationary types 1. Arrav of helical flights 2. Singie helical flighi 3. Angled plates 4. Angled holes Moving types 5. Array of scraper blades 6. Array? of planetary rollers
for subdivision streamlines
and redistribution
Reference
of flow
Degree of subdivision after n flow elements
Present work [3] Kenics [4] Grace [5] Dow ISG [6,7] Ingles [8]
3” 2” 4” $
Present work Dulmadge [2] Present work[9, lo] Hotrock]
3” $ 3” $
t Assumes constant ratio of roller to rotor diameter. $ Not readily estimated.
1097
C. J. SHEARER
streamlines. Table 3 also includes other flow geometries which have not been discussed so far; these use stationary angled plates and angled holes for streamline redistribution. Secondly, the equipment-designer will generally have to concern himself with the final steps of the mixing process in addition to the initial bulk mixing discussed above. The viscous liquids processed in practice, such as polymeric melts, generally have low mass and thermal diffusivities. Therefore, the final stages in the mixing process will have to allow for these molecularscale processes to take place. These processes when the viscous liquid is also being sheared have been investigated by Mohr and his coworkers [ 11. Lastly, the designer will also have to consider the residence time, power consumption and ease of construction and operation of the mixer. The first two items are amenable to analytical treatment if the fluid mechanics in the mixer is understood. A particular advantage here of the planetary roller technique is that material on the wall, as shown by the flow pattern of Fig. 8, will be swept away from the wall, and hence dead zones will be avoided. The third item - ease of construction and operation of the mixer- will depend largely on mechanical engineering considerations. Much development work, therefore remains to be done to reduce to practice the flow geometries proposed herein. CONCLUSIONS
The principle of subdivision and redistribution of flow streamlines is fundamental to the bulk mixing of viscous liquids. The principle can be rationally applied to flow geometries having
stationary and/or moving walls. The resulting techniques are as follows: (a) a stacked array of helical flighted ducts, (b) an assembly of scraper blades, and (c) an assembly of planetary rollers. The streamline redistribution in these three flow geometries has been demonstrated by experiment; simple theories are proposed to predict the degree of subdivision. These techniques show promise, and they may find commercial application in the viscousliquid operations of blending, devolatilisation, heat exchange and reaction. Further development work is required to reduce them to practice. Acknowledgements-The
author is grateful to Shell Development Company for permission to publish the material presented in this paper. He also appreciated the guidance and encouragement from his colleagues of the Chemical Engineering Department, notably Messrs. J. C. Dygert, G. D. Towel1 andJ R. Street.
NOTATION
breadth of compartment diameter of duct diameter of tube length of duct pressure of liquid ii radius of tube s velocity of wall velocity of fluid in x direction V width of compartment W coordinate axes shown on Fig. 1 x9 Y viscosity of liquid redistribution time of liquid ; adjacent scraper blades redistribution time of liquid adjacent planetary rollers. b d D 1
between between
REFERENCES
HI MOHR W. D. Processing ofThermoplasticMoterids, (Ed. E. C. BERNHARDT), PI SCHENKEL G. Phtics Extrusion Technology and Theory pp. 32,37, Ilitfe 1966. SHEARER
p. 138. Reinhold, New York 1960.
C. J. U.S. Patent 3434437 1969 (Shell Development Company U.S.A.).
t:; ANON. Chern. Proc. Engng, June 1970 119. PI GRACE C. D. Chem. Proc. Engng, July 1971 57. WI HARDER R. E. U.S. Patent 3 195865 1965 (Dow Chemical Company). BRUNEMANN
H. and JOHN G. Chem. LangeTech. 197143 348.
;; INGLES 0. G., Australian Provisional Patent 21339 1962. R. H. and SHEARER C. J., U.S. Patent 3443796 1969 (Shell Development 191 OVERCASHIER HOI SHEARER C. J., U.S. Patent 3443798 1969 (Shell Development Company, U.S.A.).
1098
Company, U.S.A.).