PIPE OF A CYCLONE
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF DR IR H VAN BEKKUM, VOOR EEN COMMISSIE AANGEWLZEN DOOR H t T
COLLEGE VAN DEKANEN TE VtRDEDIGFN OP WOENSDAG 25 FEBRUARI 1976 TE 16 00 UUR
DOOR
MICHAEL LOXHAM
BACHELOR OF SCIENCE SCHEIKUNDIG INGENIEUR GEBOREN TE BLACKBURN (ENGELAND)1976 - ' -. DRUKKERIJ J H PASMANS, 'S-GRAVENHAGE
Dit proefschrift is goedgekeurd door de promoter
in this case it is very necessary to distinguish the circumstances in which a model differs from a machine in large; otherwise a model is more apt to lead us from the truth rather than towards it. Hence the common observation, that a thing may do very well in a model that will not answer in large. And indeed, though the utmost circumspection be used in this way, the best structure of machines cannot be fully ascertained, but by making trials with them when made of their proper size. It is for this reason that though the models refered to, and the greater part of the following experiments, were made in the years 1752 and 1753, yet I deferred offering them
to the Society, until I had an opportunity of putting the deductions made therefrom in real practise, in a variety of cases, and for various purposes; so as to be able to assure the Society that I have found them to answer."
John Smeaton, "An experimental Inquiry concerning the Natural Powers of Water and Wind to turn Mills and other Machines, depending on a Circular Motion",
Philosophical Transactions of the Royal Society of London Vol 51 (1759).
"A man in the wilderness said to me, "HOIJ} many strawberries grow in the sea?"
I answered him as I thought good
-"As many red herrings as grew in the wood." Childrens riddle
technical staff, all of whom I sincerely thank.
Thanks are also due to Miss van Bruggen who has converted the some-times unreadable manuscript into this book.
Finally I wish to thank all my colleagues for their help and encouragement over the last four years.
M.L.
1. Introduction and problem statement 1 2. The gas Flow pattern in the exit pipe 3
2.0 Introduction 3 2.1 Axial-swirl flow 4 2.2 The experiments 6 2.3 The experimentally determined flow patterns 7
2.3.1 The initial flow pattern 9 2.3.2 The downstream flow pattern 12
2.4 Theoretical discussion 19 2.4.1 The Swirl Equation 20 2.4.2 Solutions of the Swirl equation 22
2.4.3 The integral momentum balances 23 2.4.4 The modelling of the flow pattern 24 2.4.5 The performance of the model 27 2.5 The shear stresses for rough walled tubes 30
2.6 Conclusions from the studies of the gas flow
pattern 31 3. The heat transfer between the exit pipe wall and the gas 33
3.0 Introduction 33 3.1 The estimation of the gas side heat transfer
coefficients 33 3.1.1 Theoretical discussion 34
3.1.2 The working equations 35 3.1.3 The estimation of the value of Y 37
3.1.4 The use of equation (48) 40
3.1.5 Conclusions 45 4. The two phase, gas-liquid, flow pattern in the exit pipe 46
4.0 Introduction 46 4.1 Theoretical discussion 47
4.2 The experimental study 50 4.2.1 Results and discussion 51
4.3 Conclusions 57 5. General conclusions and closing remarks 58
Appendix A The pitot cylinder technique for measuring
the gas velocity 59 Appendix B Measurement of the film thicknesses 61
List of symbols 63 References 65 Samenvatting 68
when it is used to clean gases.
In the first chapter, an account is given of the flow pattern. This account is based upon an extensive series of measurements of the flow pattern in five different systems. The flow pattern is that of intense, decaying, axial swirl flow. The axial velocity component is different from that expected for non-swirl turbulent flow in that the velocity in the middle of the tube is reduced and that there is a corresponding increase very near the wall. The swirl component is, at its maximum, up to twice the superficial velocity. The radial velocity component is very small and can be ignored.
On the basis of the observations the time smoothed Navier-Stokes equations are simplified to give a 'swirl equation' from which the spin-down can be calculated. This swirl equation is solved in two ways, numerically and analytically, and the solutions compared. An integral momentum and an integral angular momentum balance are derived to show how the wall shear stresses can be calculated either from the experi-mental data, or from the solutions to the swirl equation. It is shown that the shear stresses are initially very high and diminish exponent-ially downstream.
Both the solution to the swirl equation and the calculations using the integral balances are compared to the data and the agreement is satis-factory for smooth walls. For rough walled tubes the solutions of the swirl equation predict too small a rate of the swirl decay.
In the second chapter the heat transfer between the gas and the exit pipe wall is described. It is shown experimentally that the heat transfer rate is much higher than that expected for non-swirling tur-bulent flow. A model for the heat transfer based upon a modification of the analogy between momentum and heat transfer is developed and tested. It is shown that the model allows much closer predictions of the heat transfer rate than those obtained from conventional corre-lations.
In the third chapter the two phase flow pattern in the exit pipe is studied. It is shown that the flow pattern is simple co-current, annular flow with swirl. The film thicknesses and surface velocities are studied with the help of a conductivity technique and it is shown that the results can be correlated in terms of the interfacial friction factor. This correlation is based upon a solution of the laminar Navier-Stokes equations for the film. It is shown that for the case of a very thin film the interfacial friction factor can be calculated by the methods given in section one.
CHAPTER 1.
INTRODUCTION AND PROBLEM STATEMENT
In this work the flow pattern in the cyclone exit pipe is dealt with. Except for one measurement by First in 1949 this complicated flow situation has not been studied yet there has been a great deal of dis-cussion as to what it should be like and how it could be used to ad-vantage. One of the objects of this work is to provide an experimental and theoretical background for any similar discussion in the future. The second motivation is that recently a great deal of interest has grown up round questions arising out of attempts to extend the range of applications of the cyclone in chemical engineering unit-operations. For example it is possible to use the exit pipe as a heat transfer surface, or as a gas-liquid contactor. In such cases the problem con-fronting the designer is to predict the efficiency of the proposed in-stallation. To illustrate some of his difficulties two sets of results will be presented here, without comment, except to bring to the readers attention the point that conventional correlations cannot be applied to these exit pipe flows. Firstly consider the results illustrated in fig (1). 1 0 0 8 0 o E 6 0 4 0 20 0 '' 5 6 7 8 9 10 — G a s velocity m/s Fig. 1. The performance of a cyclone exit pipe as a heat exchange surface.
o Measured values — Calculated from :
Nu = 0.023 Re°-^Pr°-^
The points are the measured performance of an exit pipe used as a heat transfer device. The line is the Dittus and Boelter correlation. Secondly in table (a) the axial pressure gradient measured down the exit pipe via wall tappings is compared to that expected from Blasius'
Table (a). The axial gradient of the static pressure measured at the wall.
position, cm
wall gradient/Moody value
20.0 18.5 40.0 7.4 60.0 80.0
3.7 1 2.4
100.0 1.31
correlation. It is clear that better methods of calculation are needed and in this thesis they will be developed and tested.
In chapter 2 the actual flow patterns will be experimentally studied, for a range of system sizes, and geometries as well as operating con-ditions that are relevant for cyclone operation. These results will then be used to help to derive a theoretical model for the flow
pattern, and to test it. In chapters 3 and 4 this model is extended to cover the important phenomena of heat transfer and two-phase flow. The two phase flow experiments were made under conditions of no mass trans-fer. Mass transfer has not been studied here because the model devel-oped in chapters 3 and 4 cannot be easily extended in that case.
CHAPTER 2. THE GAS FLOW PATTERN IN THE EXIT PIPE. 2.0 Introduction.
The reverse flow cyclone, that is the common type of cyclone, is one of the most common gas cleaning devices used in industry. It can be used alone or in combination with other collectors such as bag filters or electrostatic cleaners. The advantages of the cyclone are that it is cheap to install and maintain, has no moving parts and can operate at high temperatures. The two main disadvantages are that the economic cut size is relatively coarse and that the turn-down ratio is very small. Both these disadvantages are minimised if, instead of one large unit, several small units are grouped in parallel. Small units are more efficient than larger ones and turn-down is effected simply by closing off individual units. A typical example of one of these is the so-called "Multiclone" cyclone and it is to this type of device that the main ideas developed in this thesis are directed.
The importance of the cyclone has lead to many investigations to under-stand and improve its operation. (For example see Jackson (1963) or Alden (1959) for a review of many of the earlier .works). When the work-ing fluid is a liquid the device is known as a "hydrocyclone" and this has also been thoroughly studied, (see for example Kelsall (1952)). Although there is not agreement on all points the general features of the cyclone operation are clear. Essentially the cyclone is an inertial separator. The dusty gas is introduced tangentially, thereby generating a vortex in the annulus and the cylindrical section. The dust then moves to the wall and the gas to the centre. The balance of the out-wardly acting centrifugal force and the inout-wardly acting drag force on
the dust establishes the limiting cut-size. The actual situation is much more complicated and such factors as particle interactions and wall effects play, along with the gas turbulence an important role. The gas flow pattern in the cyclone controls the separation and much effort has been directed to either measuring or calculating this pattern. Again different authors differ on details but the following general picture has emerged.
Firstly the flow is highly turbulent and typical operating Reynolds' numbers fall in the range of 10^ to 10^. Secondly the whole flow pattern is dominated by the swirl velocity component and in common with other vortex flows there is a strong tendency to
one-dimensionali-ty so that the flow pattern is largely independent of the axial co-ordinate. The radial velocities are very much smaller than the swirl velocity component. The latter varies across the radius, increasing approximately linearly from the centre-line (solid body rotation) and then decreasing towards the wall. In the outer region the profile can be described by a power function (pseudo-free vortex behaviour). The axial velocity component is directed downward towards the cone apex at larger radii and upwards nearer the centre. The locus of "zero vertical velocity" is to be found between the vortex finder radius and the wall.
Thirdly there are two strong boundary layer flows. One is directed down the cone wall and is responsible for moving the dust to the apex and the hopper and the second is across the top of the cyclone and down the outside wall of the vortex finder. This short circuiting flow, which can be as large as 20 % of the main flow, (Tarjan, 1961) is a major cause of inefficiencies in cyclone operation. Finally
super-imposed on the whole flow pattern are several secondary flows which are very similar to Taylor vortices, (see fig. 2 ) .
exit pipe vortex finder
short circuit boundary
oundary layer ow
Fig. 2. Illustration of the various flow patterns found in a cyclone.
The flow pattern at the end of the vortex finder which is our main concern here, is made up from contributions from each of the above sources, none of which can be properly assessed. Furthermore there are no reliable measurements for this zone. The only safe conclusion is that the gas is still swirling intensely at this point and that the flow pattern in the exit pipe is that of turbulent, decaying, axial-swirl flow.
2.1 Axial-swirl flow.
There have been no studies of cyclone generated swirl flow although swirl flow in general has received some attention. The motivation for these studies has often been to optimise the extra heat transfer capabilities of tubes with swirl generators as "turbulence promoters".
Table (b). Reported Work on Decaying Swirl Flow. Author Kreith Sachdeva Wolf Migay Senoo Yanik Fortier Musolf King Sonju Missan This work date 1965 1968 1969 1970 1972 1973 1954 1963 1969 1962 1961 — generator tape tang. vanes vanes vanes vanes vanes tape vortex tape vortex cyclone Log (Re) 4 - 5 4 - 5 4 - 5 4 - 5 5 4 - 5 5 4 4 4 - 5 3 - 4 4 - 5 S o 0.23 — 1.62 — 1.00 0.16 0.56 0.20 — 0.20 — > 2.00 Fluid water water air air air air air water air water air air
by the so-called "Swirl number" which is defined as
S = Flux of angular momentum
Length x Flux of linear momentum' (dimensionless).
It can be seen that twisted tape generators, which have been given the most attention, have low swirl intensities whilst vortex ramp devices
give higher intensities. It should also be noted that the cyclone must be classified as a very intense swirl generator.
The results of these investigations can be generally summarised by the following description of turbulent spin-down. As in the case of the cyclone the swirl component varies between that of a pseudo-free vortex near the wall to that of more nearly solid body rotation near the centre-line. The swirl intensity tends to decay exponentially down the tube although there is no general agreement on either the rate of this decay or the influence of the Reynolds number. The radial velocities are always very small and this means that the axial velocity component profile changes only slowly downstream. However it has been shown that the form of this axial profile is coupled to the swirl. As the swirl is increased the axial profile is distorted from that expected for non-swirl flow. The velocity at the centre-line decreases and there is a corresponding increase near the wall giving rise to abnormally high shear stresses in this region. At very high levels of swirl there is actual flow reversal at the centre. This effect is a function of the swirl number rather than of the absolute level of swirl and has been reported for slowly swirling laminar flow by Lavan et al. (1969). The more nearly analogous flow situation, that of swirling flow in an entrance region, has been dealt with theoretically, for the laminar
c a s e o n l y , b y K i y a et a l . ( 1 9 7 1 ) , b u t u n f o r t u n a t e l y they h a v e h a d to m a k e s w e e p i n g , a n d h i g h l y u n r e a l i s t i c , a s s u m p t i o n s about t h e initial s w i r l d i s t r i b u t i o n . T h e m a i n r e s u l t of t h e a n a l y s i s is t h a t t h e e x -p o n e n t i a l s w i r l d e c a y r e l a t i o n s h i -p c a n b e e x -p e c t e d to h o l d b u t less a c c u r a t e l y . A l l t h e s e s t u d i e s g i v e a g e n e r a l i m p r e s s i o n of t h e p h e n o m e n a that c a n b e e x p e c t e d i n t h e exit p i p e b u t it is n o n e t h e l e s s so t h a t there a r e n o r e l i a b l e r e s u l t s f o r this p a r t i c u l a r c a s e . N e i t h e r the i n i t i a l f l o w p a t t e r n , n o r t h e d e c a y p a t t e r n s a r e k n o w n and w i t h o u t this d a t a r a t i o n -al d i s c u s s i o n is i m p o s s i b l e . In t h e n e x t s e c t i o n this d a t a w i l l b e p r e s e n t e d . 2.2 T h e e x p e r i m e n t s .
It is c l e a r that t h e f l o w p a t t e r n in t h e exit p i p e will b e d e p e n d e n t u p o n t h e c o n d i t i o n s in t h e c y c l o n e a n d thus o n t h e d e s i g n of t h e c y c l o n e . T h i s f a c t p r e s e n t s t h e f i r s t of t h e e x p e r i m e n t a l p r o b l e m s . C y c l o n e s c o m e in a l l s h a p e s a n d s i z e s a n d , a l t h o u g h there a r e supposed to b e r a t i o n a l d e s i g n r u l e s f o r t h e m , a l m o s t e v e r y m a n u f a c t u r e r h a s h i s o w n g e o m e t r y (which is of c o u r s e t h e only r a t i o n a l l y c h o s e n o n e ) . It is o b v i o u s l y i m p o s s i b l e to e x p e r i m e n t w i t h e v e r y d e s i g n of c y c l o n e b u t o n t h e other h a n d e n o u g h u n i t s m u s t b e c o v e r e d to g i v e t h e c o n -c l u s i o n s a t least s o m e limited g e n e r a l i t y . In f a -c t five d i f f e r e n t c y c l o n e s w e r e c h o s e n , and their l e a d i n g d i m e n s i o n s a r e g i v e n in table
(c).
Table (c). The Cyclones used in this study.
No. 1 2 3 4 5 D mm 285 280 50 50 93 d/D .53 .50 .50 .50 .48 H/D 3.1 2 2 2 2.7 inlets 1 1 1 4 1 Ai/Ao 1.02 0.69 0.69 0.67 0.86 2r /D remarks m 1.06 Commercial Unit - DSM type B 0.60 (, Q_gQ ^homologous
0.89 symmetrical inlet system^ 0.72
Cyclones 1 and 2 are more or less of the size that would be used in a multiclone installation. Type 1 has the geometry of a commercial unit and was kindly made available by the Dutch State Mines. Type 3 is an exact homologue of type 2 but only 1/5th of the size. Cyclone 4 has four symmetrical entrances, which is an arrangement often found in multiclone units and type 5 was chosen for its' long length in
compari-son with the diameter. Each of these units was set up in a very simple test rig where a carefully controlled and measured quantity of clean dry air was fed to the cyclone and allowed to exhaust through an ex-tended exit pipe.
The static pressure was measured just before the cyclone and at inter-vals along the exit pipe via carefully constructed wall tappings. There
dis-tribution. Exall (1974) has used a Laser-Doppler apparatus to study the flow in a cyclone but found the technique to be highly unreliable in the important core region. The difficulties were associated with the high velocity gradients and high levels of turbulent fluctuation. Pitot-static tubes have been used by several authors (e.g. Shepherd and Lapple, (1939) and Carre and Bugarel (1969)) often without allow-ing for the yaw-angle between the flow and the probe. Scott (1973), Wolf (1969) and Yanik (1973) have all used the pitot cylinder method described by Glaser (1952) with success. This was the technique chosen for this study. Further details of this method can be found in appen-dix A.
2.3 The experimentally determined flow patterns.
In this section the measured velocity profiles will be discussed in some detail and it is useful at this stage to give a general descript-ion of a typical set of results. Consider those illustrated in figs. (3a and 3b). The lines A and B indicate the velocity profile and pres-sure distribution expected for simple, i.e. non-spin, turbulent flow at that Reynolds Number. It can be seen that the swirl flow pattern is very different. Firstly, the axial velocity profile (line E) is
actual-ly negative at the centre and rises to its' highest value very near the wall. This sort of profile has been reported before in swirling systems by, for example, Senoo (1972) who has compared it to that found behind a sphere or cylinder in a gas stream, although the basis
. 3a.
ical velocity profile.
Axial velocity profile expected for non-swirl flow (I/7th power law).
Measured velocity profile. Swirl component of C Axial component of C Fig
Typ
^-O--<?50 . Fig. 3b
Typical pressure profile and flow angle variation
o o flow angle X X swirl pressure
distribution non-swirl pressure distribution
for the comparison, and the subsequent evaluation of the "wake radius" IS unclear. Secondly there is a large radial static pressure distrib-ution (line G ) . This is caused by the swirl and balances out the cen-trifugal forces on the gas. As the swirl decays down the tube the pressure field will relax and the wall pressure will decrease. It is clear that the axial pressure gradient measured via wall tappings is the sum of the effects of the spin down, and the friction losses and is a largely irrelevant quantity. There is also a radial distribution of the angle that the flow makes with the tube axis (line F) although the angle tends to become constant near the wall. It will later be shown that this "flow angle near the wall" is a very useful parameter. Final-ly the swirl profile itself is shown (line D ) . It can be seen that the peak swirl velocities are very much higher than the superficial velo-city and the velocities are still high in the region of the wall. These high velocities in this region imply high shear stresses and high heat transfer rates. Finally to show that neither the free vortex nor the solid body rotation descriptions of the flow are adequate the swirl profile is replotted on logarithmic axis along with the model distributions in fig. (4). A more systematic discussion of the results will now be given.
Fig. 4.
Comparison of the swirl data with model distributions :
a) Solid body rotation
b) Pseudo free vortex (power law) rotation.
0 1 0 3 10 ^ ^ — radius - r
2.3.1 The initial flow pattern.
In the conventional design of the cyclone the exit pipe extends into the cyclone body for about two diameters. For all practical purposes this section must be considered as part of the cyclone itself and it seems logical to consider a point just outside of the cyclone as the start of the exit pipe, which is what has been done here.
The influence of the superficial gas velocity on the measured initial profile can be seen by referring to table (d) and that of the absolute size of similar systems by considering the results illustrated in fig. (5).
Table (d). The effect of the gas velocity on the form of the initial profile. ^ V m/s 21.5 23.7 26.4 28.6 31.3 ^/^o--V,./V ;
Data collected for cycl .25 1.90 1.88 1.87 1.86 1.87 ane 4. .42 2.29 2.29 2.28 2.24 2.25 .50 2.72 2.75 2.73 2.72 2.70 .75 2.97 2.99 2.95 2.97 2.94 .92 2.90 2.93 2.94 2.91 2.87
In the range of variables studied, neither of these factors causes changes in the form of the profile. Ter Linden (1949) and van Kooy (1958) have both suggested that the flow pattern in the cyclone is only a function of the Reynolds number. On the other hand the data of Shepherd and Lapple (1939) and of Exall (1974) show that there is a slight trend with, respectively, the gas velocity and the liquid velocity in the hydrocyclone. The matter should be dealt with, with the utmost circumspection but it would appear that to characterize each type of cyclone under discussion only one profile has to be experimentally established and that can be done in a laboratory.
2 0
10
-,.X^^\
Fig. 5.Comparison of the initial swirl velocity component distribution for two geometrically similar cyclones : X Cyclone 2 (r = 7 0 mm) o o Cyclone 3 (r = 1 2 mm) o 0 5 radius -r 1 0
Given the current cost of plant tests and measurements in industry this is possibly a very important restilt. Neither the physical proper-ties of the gas have been changed, nor is the data base wide enough to comment on the possibility of scale-up based upon the Reynolds Number alone to which there are in any case theoretical objections but the results presented here do not contradict that suggestion.
That is equally so for the overall pressure drop over the system which is shown in fig. (6) for three homologous cyclones covering a 10 : 1 range in size. 50 -IN 20 5.10" - » 0 , K ^ 10-' Re = V ro/v 5 10-Fig. 6.
Total pressure drop over
geometrically similar cyclones.
r = b . 5 0 r = 12.0 0 r ^ = 70 mm mm mm
The last important variable affecting the initial profile is the cyclone geometry. The effect of this can be understood more easily by considering the cyclone as a vortex. This vortex is driven by the continuous influx of angular momentum from the tangential inlet. The angular momentum is spent in both internal dissipation and in friction with the walls. What remains finds its' way to the exit pipe. Smith
(1962) has suggested that this sort of vortex flow is unique in the sense that for a given angular momentimi flux there is only one stable distribution of the angular momentum over the radius. If this is true,
then the influence of the cyclone geometry is restricted to the effect on the angular momentum input and how much of it is lost within the cyclone itself. Of course the real situation is much more complicated. However these ideas make it possible to classify cyclones on the basis of the angular momentum input, or the cyclone swirl number. If the inlet geometry is defined as :
then the cyclone swirl number is :
S = c V.A. X p V.r 13- g 1 m • 2 "2 r X Tvr .V p o o g which simplifies down to
S = ir r r /A. c m o 1
exJ: C^ f^'-'^ (y<«(ix-(•'-^)
(1)
(2) which is only a function of the geometry. The swirl number actually found can be obtained from the measurements by :
/
o * • "2 V„V r dr _ 6 z ° ~ - 3 ,'2p tr r V g o
or with reduced velocities and distances 1
S = 2 / V V.r dr
z e
(3)
(4)
In table (e) the values of the found and calculated initial swirl numbers are compared. It can be seen that although some is lost, most Table (e). Initial Swirl No.
Cyclone 1 2 3 4 5 S , c a l . 1.96 1.74 1.74 2.66 1.74 S. 1 1.22 1.56 1.56 1.75 1.27 Loss 38 % 10 % 10 % 34 % 27 %
of the angular momentum input is found in the exit pipe. Cyclone 1 had a very rusty body and this may account for the high observed loss. The actual profiles are given in tables (f) and (g).
Table (f) The initial velocity profiles for the swirl component.
-'^o 0 . 0 0.1 0.2 0 . 3 0 . 4 0 . 5 0 . 6 0.7 0 . 8 0 . 9 1.0 Cyclone : 1 0.00 0.62 1.06 1.29 1.47 1.64 1.74 1.72 1.62 1.46 0.00 2 0.00 0.78 1.48 1.82 1.96 2.08 2.17 2 . 1 5 2 . 0 0 1.60 0 . 0 0 3 0.00 0.78 1.48 1.82 1.96 2.08 2.17 2.15 2.00 1 .60 0.00 4 0.00 1.20 1.70 2.02 2.26 2.46 2.62 2.65 2.56 2.43 0.00 5 0.00 0.80 1.40 1.78 2.00 2 . 1 6 2.36 2.44 2.44 2.30 0.00
Table (g) The initial axial flow patterns.
^'^o 0 . 0 0.1 0.2 0 . 3 0 . 4 0 . 5 0 . 6 0.7 0 . 8 0 . 9 1.0 Cyclone : 1 0.02 0.07 0.17 0.33 0.52 0.74 0.98 1.15 1.27 1.32 0.00 2 - 0 . 1 6 - 0 . 0 4 0.01 0.09 0 . 3 9 0 . 6 9 0 . 9 5 1.16 1.33 1.43 0 . 0 0 3 - 0 . 1 6 - 0 . 0 4 0.01 0.09 0.39 0.69 0 . 9 5 1.16 1.33 1.43 0.00 4 - 0 . 2 3 - 0 . 2 4 - 0 . 2 0 - 0 . 0 8 0.14 0.48 0.90 1.24 1.48 1.62 0.00 5 0.00 0.03 0.11 0.26 0.58 0.82 1.08 1 .29 1.36 1.38 0.00 The values listed have been found by interpolation of the measured values and on interpolation no difference could be seen between cyclones 2 and 3 so that identical values have been listed.
2.3.2 The downstream flow pattern.
Downstream of the cyclone the swirl decays away and the flow pattern reverts to that of simple turbulent flow. This process can be followed on two levels, either by studying the changes in the actual profiles or by means of the integrated function, the swirl number. As before first a general description of what was found will be given. This will be based on the results tabulated in tables (h) and (i) which were gathered for cyclone 4 or on the illustrations in figs. (7) and (8),
Table (h) The axial changes in the swirl profile. Cyclone Number 4 S u p e r f i c i a l g a s v e l r - cm 0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0 9.70 9.80 0 . 9 0 1.00 1.10 1.20 11.0 0 0 . 0 24.3 3 7 . 4 4 4 . 5 49.1 54.1 5 9 . 3 62.1 6 3 . 6 62.8 6 0 . 6 5 7 . 5 0 0 . 0 o c i t y = 2 1 . 0 0 0 . 0 2 0 . 2 30.1 3 5 . 5 3 9 . 8 4 5 . 4 49.1 5 1 . 5 51.8 50.2 4 7 . 9 4 5 . 8 0 0 . 0 23.7 m/s z -3 1 . 0 0 0 . 0 19.1 28.3 32.7 3 7 . 2 40.8 44.1 4 4 . 5 4 4 . 0 42.1 39.8 37.6 0 0 . 0 cm 61.0 0 0 . 0 18.4 2 2 . 0 27.1 3 3 . 4 33.8 3 4 . 4 33.7 3 2 . 4 3 0 . 4 28.3 26.6 0 0 . 0 9 1 . 0 0 0 . 0 13.4 16.7 20.8 24.6 26.4 26.3 25.2 2 3 . 6 2 2 . 3 20.3 19.1 0 0 . 0 121.0 0 0 . 0 9.1 16.0 20.7 2 2 . 6 21.9 20.7 19.6 17.9 16.5 15.0 14.3 0 0 . 0 141.0 0 0 . 0 7.6 13.6 16.9 18.6 18.2 17.2 15.9 14,7 13.5 12.6 12.0 0 0 . 0
Table (i) The axial changes in the axial profile. Cyclone Number 4 S u p e r f i c i a l r - cm 0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0 . ,0.70 0 . 8 0 0 . 9 0 1.00 1.10 1.20 g a s v e l 11 .0 - 5 . 4 - 5 . 6 - 5 . 1 - 4 . 1 - 1 . 3 4 . 5 11.8 19.6 27.2 21.7 26.4 38.8 0 0 . 0 o c i t y = 2 1 . 0 - 3 . 1 - 3 . 8 - 4 . 8 - 3 . 7 0 . 0 5.8 12.7 2 0 . 0 26.4 3 0 . 8 3 3 . 5 35.1 0 0 . 0 23.7 m/s z -3 1 . 0 - 3 . 1 - 3 . 5 - 3 . 9 - 2 . 6 2.4 7.9 15.4 21.2 26.0 29.5 3 1 . 6 32.7 0 0 . 0 cm 6 1 . 0 - 1 . 0 - 2 . 3 - 2 . 3 1.3 7.7 13.5 18.7 23.6 26.7 29.3 30.9 3 1 . 2 0 0 . 0 91 .0 0 . 0 - 0 . 3 0.7 5.9 11.5 16.8 21.1 24.3 26.9 2 8 . 5 28.8 28.8 0 0 . 0 121.0 0 . 0 2 . 6 8.4 13.8 19.1 22.4 25.1 2 7 . 3 2 8 . 4 29.3 28.8 28.1 0 0 . 0 141.0 3 . 9 5.7 10.4 15.2 19.3 23.1 25.2 2 6 . 4 27.6 2 8 . 4 28.4 27.6 0 0 . 0 which are gathered for cyclone 2. In fact, as can be seen from tables
(f), (g) and (J) any of the results for the five cyclones could have been chosen because they differ only in numerical details.
s in 2 0 1 0 5 radius -1 0 2 0 -1 O 0 5 1 Owall — radius - r
Fig. 7. Axial variation of the swirl velocity component. z = 7.1
Fig. 8. Axial variation of the axial velocity component. z = 18.6 o z = 27.1
Consider first the swirl components. These decay axially for all the radial positions, and this occurs more quickly in the beginning of the pipe than at the end, and there is, for example a 50 % decrease in the maximum swirl velocity in about 25 diameters downstream. The position of this maximum gradually moves towards the centre as the swirl decays. Profiles measured at different downstream positions are not formally similar to each other. The most noteworthy point to be made about these results is the simple persistence of the swirl, for example, even after 60 diameters the swirl number is still almost half that of the initial value. In industrial terms this means that almost invariably any downstream plant will have to contend with a strongly swirling inlet condition.
In contrast to the changing swirl pattern the axial velocity profile, notwithstanding its' highly perturbed character, changes much more slowly. For example the maximum axial velocity is still found near the wall and has only decreased by 6 % in the first twelve diameters. This implies that the wall shear stresses continue to be high for a very long way down-stream. The centre of the tube still has either very low or negative velocities although on the other hand the actual flux due to the back-flow is very small.
Table (J) The form of the downstream flow pattern. Cyclone 1, Swirl 1 , Axial 2, Swirl 2, Axial 3, Swirl 3, Axial 4, Swirl 4, Axial 5, Swirl 5, Axial Axial Station z 22.6 22.6 27.1 27.1 40.0 40.0 100.0 100.0 19.0 19.0 0 0.0 0.0 0.0 -0.08 0.0 0.0 0.0 0.0 0.0 0.0 Velocity of .2 1.20 0.18 1.18 -0.01 1.40 0.12 0.74 0.46 1.12 0.8 .4 1.45 0.53 1.68 0.34 1.75 0.72 0.93 0.91 1.75 .60 radius .6 1.46 1.02 1.66 0.92 1 .58 1.22 0.80 1.16 1.65 1.07 : r .8 1.35 1.22 1.46 1.28 1 .32 1.43 0.65 1.23 1.43 1.29 1.0 0 0 0 0 0 0 0 0 0 0
The time smoothed continuity equation is for this system (see later) :
3V^/3z + 1/r 3(rV^)/3r = 0 (5) and this can be used to calculate the radial velocity distribution. In
fact the radial velocities are very small and never more than 1 % of the superficial gas velocity. In other words they are probably smaller than the turbulent fluctuations themselves (c.f. Wolf et al. (1969)), and can be ignored.
5 10 15 20 25
Fig. 9. Axial variation of the swirl number.
This general summary can be concluded with considering fig. (9) where the swirl number has been set out against the axial position on semi-logarithmic axes. It can be seen that the relationship is an exponent-ial one. In fact cyclone 4 gave results that indicated a small
deviat-ion from exponential behaviour in the beginning but for all the other cyclones the results were of the form shown.
The effect of the absolute gas velocity for the same cyclone system is shown in table (k) and it can be seen that the dependence is small enough to be ignored. This conclusion is in contradiction to that given for vane generated swirl by Wolf et al. (1969). However, direct calculation from the data given in their paper leads to a
conclusion that is consistent without results. It will be shown Table (k) The reduced g
Cyclone - 4. z = 121 cm
as velocities measured far V/m/s 21.5 23.7 26.4 28.6 31.3 r/r : 0.25 0.42 o 0.97 1.33 1.05 1.32 1.01 1.30 1.03 1.34 1.03 1.34 downst 0.58 1.38 1.42 1.39 1.42 1.42 ream. 0.75 0.92 1.37 1.32 1.42 1.33 1.39 1.33 1.41 1.35 1.41 1.35 later that a weak dependence on the Reynolds Number would be expected and the fact that it has not been found must be put down to the narrow range of superficial velocities that can be relevantly used with a cyclone. E Z 2-0 1 0 0 1 X Cyclone 3 o Cyclone 2 10
z
20 30 40Fig. 10. Axial variation of the swirl number for similar systems. Equally the dependence on the absolute size of the system, which is illustrated in fig. (10), where similar systems give similar spin-down behaviour, must be ascribed to the same combination of a neces-sarily narrow data base and an inherently weak dependence.
None the less these two results, combined with the equivalent ones from section 2.3.1, mean that whole the spin-down pattern can be both quickly and easily studied in the laboratory. Each system has only one, unique, relationship between the swirl number and the axial distance.
Each of the cyclones studied gave the same qualitative but slightly different quantitative picture. For later convenience the parameters in the decay relationship :
S = S exp(-B z) are given in table (1).
Table (1) Values of the constants S & B in the spin-down , . o ^ correlation. (6) Cyclone 1 2 3 4 5 S 1.30 1.82 1.82 1,70 1.66 B 0.012 0.018 0.018 0.011 0.023
In many actual industrial installations the exit pipes will have rough inner surfaces and the extra form drag will cause the spin to decay faster. It is very difficult to cover all the possible combi-nations of shape and spacing for roughness but to give some indication of the effect the inner wall of the exit pipe of cyclone 4 was covered by glueing glass spheres of a narrow size range, and the spin-down studied. Spheres were chosen in view of some two phase flow experiments discussed later. The results are illustrated in fig, (11) where it can be seen that the decay is much faster with rougher walls and that this
1 0
-Fig. 11.
The effect of roughness upon the decay of the swirl
* Sand k o V D s 64 v glass spheres k 213 y 368 y 605 U 800 p 1010 \i
effect is most marked in the beginning. This behaviour is typical of developing boundary layer flow over rough plates and will be discus-sed later,
Finally two very useful empirical relationships have been established, The first is the relationship between the flow angle "near the wall", which was taken to be at r = 0.9 f , and the swirl number. This is set out in fig. (12) and it can be seen that all the results come
9 0 o 2 70 a i » 50 1 2 Swirl Number . s Fig. 12.
The relationship between the flow angle near the wall and the swirl number
• Cyclone 4 (smooth walls) X Cyclone 4 (rough walls) o Cyclone 2 and 3 (smooth walls)
reasonably on one line. This line is also very similar to the ones presented by Senoo et al. (1972) and Yanik et al. (1973) for vane generated swirl. The apparent uniqueness of this line is comment-worthy because it implies a unique distribution of the angular momentum and velocity for any given swirl level. However, this will not be explored further in this thesis and the results will be used later in the form of the curve-fit :
$ = 90 exp(-0.61 S) (7)
The second useful correlation is that between the position of the maximum velocity and the swirl number. Again a unique relationship would seem to be implied by the data although the spread is somewhat larger than covered by equation (7). (see fig. (13)).
This can quite easily be caused by the difficulty of the interpolative estimation of the maximum of the very broad peak on the plots of the swirl velocity against the radius.
This data has been fitted by the following line : r = 0.25 + 0.25 S
max (8)
Fig. 13.
The relationship between the position of the maximum swirl velocity and the swirl number. o Cyclone 1 • Cyclone 2 and 3 X Cyclone 4 A Cyclone 5 The line is : r = 0.25 + 0.25 S max 0 0 5 1 0 ^^—''max
In the next section these experimental findings will be used to help to construct a more general model of the spin-down and will be used to test its' applicability to these systems.
2.4 Theoretical discussion.
The object of this theoretical discussion is two-fold. Firstly it is an attempt to provide a framework within which the results obtained earlier can be generalized, at least to a limited extent. Secondly it is developed to provide a means of estimating the spin-down in those cases where the actual plant data is either absent or limited. It will be seen that the more is known of the system, then the more accurate can be the predictions. However, it is highly unlikely that in a practical application of these ideas, enough will be experimentally known to dispense with the modelling entirely.
A complete description of the spin-down would not only include a detailed calculation of the down-stream flow pattern but would also enable important parameters such as the wall shear stresses or even the reverse flow regions to be predicted. From a more realistic point of view a model should be able to give a background for the general-isation of the data already presented and give satisfactory estimates of such industrially important quantities as the friction factors, wall flow angles, and the overall swirl decay. Furthermore it would be expected of a model that the necessary calculations be as simple
S I/)
1 5
1 O
as possible relative to the answer required and that they shall rely as little as possible on specific plant data that is either impossi-ble, or too costly to obtain. In this section such a model will be derived. It will become immediately apparent that the fluid dynamic problem is very complicated indeed and that only by making broad as-sumptions can it be handled at all. It will also be shown that, de-pending upon the assumptions and input parameters chosen different model configurations can be set up which in turn give different but
similar estimates of the remaining parameters. One result of the ana-lysis is that it is possible to give a realistic spin-down model for the very basic case where only the cyclone geometry is known before-hand.
The models have been constructed round three basic considerations : 1) that the input profiles are either known or can be calculated; 2) that the experimental results presented above are at least general
enough to enable conclusions to be drawn from them that can be
used to simplify the equations of motion (time-smoothed
Navier-Stokes equations) and
3) that the turbulent stress terms can be identically eliminated by the "eddy viscosity" closure hypothesis.
2.4.1 The Swirl Equation.
Consider the co-ordinate system illustrated in fig. (14).
Fig. 14. The co-ordinate system.
Let r, z and 6 be the radial, axial and swirl directions in cylindrical co-ordinates and let V , V and V. be the corresponding velocities. (The convention used here is that a dot superscript, r, represents a dimen-sional quantity and should not be read as the first derivative). Let Vl be the time smoothed turbulent fluctuations. Then for steady, axis - symmetrical, incompressible flow where the body forces can be neglect-ed, the equations of motion are (Hinze, 1959) :
3V + V 3z 3V 3 r
1
^
P 3z +v d^V 32v 3;2 3r2z J z
3r 3 _ V ' + l ^ r V ' V ' 3z ^ ; 3r ^"^ . 3V . 3V 3V^ r r f V — ^ V — ^ ^ 3z "^ 3r r111
D 8r + V 32v S'^V 3z r 3r V r ' .2 r (9) - 2 V'^i-;.;.+iVv'--^
(10) . 3V. 3V^ V V.: - ^ V ^ + ^ l = v
^ 3z ^ 3^ ' 3z 2 '*' .2'^ • ,. -2 3r r 3r r V'V' _3.v'v'+—V'V!+2 -^-^ 3z 3r r e (11) and the equation of continuity is :3V /3z + 1/r 3(rV)/.- = 0
z dr (5)
These equations are non-linear and have more independent variables than there are governing equations and thus cannot, in this form, be solved.
Now on the basis of the results obtained from the experiments, assume further that the radial velocity is small with respect to the other velocities, i.e :
V , V. » V z 6 r
then, if the Reynolds' stresses are combined with the kinematic viscosity term to give an "eddy viscosity", the following equations result : .2 pVg/r = 3P/3r l/p.3p/g- = (0 + 4)(32v^/g-2 + 1/; 3V^/g-) \ ^^e/3; = (^ * ^)(3^V,/,;2 + \/T 3;,/,. -V,/:2) a'3r^ 3r e'r (12) (13) (14)
This assumption on V is crucial and effectively uncouples the three equations and allows one to talk meaningfully about a "swirl equation", which is equation (14). This can be non-dimensionalised by putting
V = V /V, L = L/r . e = e/v which gives :
Re V. - 1 = i + i _ i - 1 V, (15) (H.) z 3^ 3^2 r 3r ^2 6
which is the form in which it will be solved.
Equation (15) has to be solved with the initial conditions : V^(r,o) = f^(r), Vg(r,o) = fg(r)
and with the boundary conditions :
Ve= 0; r = 0; Vg = V^ = 0; r = 1.
2.4.2 Solutions of the Swirl equation
The swirl equation is an example of a parabolic partial differential equation of which .the mathematics is well known.(see for example Courant and Hilbert, 1953). For the case where there is back-flow,i.e. V < 0, the equation is unstable and no solution exists. Otherwise
there are two possible routes to a solution.
Firstly, and perhaps most straightforwardly, equation (15) can be solved directly by rewriting as a finite difference equation and solv-ing numerically.
To obtain the results discussed below the implicit variable method of Crank and Nicolson was used (see Arden and Astill 1970, for details of this technique). This method has several advantages over the Fourier series-von Neumann convergence test combination suggested by Rochino and Lavan (1969), not least of which is that it is both faster and cheaper. (Typical processing times were 70 seconds for a network of
100 radial and 500 axial points, compared to 900 seconds for a 20 by 500 net given by Rochino and Lavan.)
Secondly an analytical solution can be sought. If the axial velocity were to be assumed constant, i.e. :
V = 1 z
then an analytical solution is immediately possible using the method of the separation of variables, and is :
^e = ^, V l (^'^^ ^''P
n=l
-X2 (l+e).z/Re (16) "^ I
where the C are evaluated from the initial condition by :
n •'
n j2(^ ) i ^ ' "
C = dr (17)
o n o
and J. is a Bessel function of the first kind and order i. and the eigen values from the boundary condition :
J, (X^,l) = 0
and are (Jahnke and Emde, 1945) :
3.832 7.016 10.173 13.324 16.471 19.616 22.760 etc. (18)
If the axial velocity is only slightly dependent on the radius then the perturbation method of Courant and Hilbert (1953) can be used. This method calls for the expansion of both the eigen values and eigen functions as a power series, the terms of which can be evaluated from the perturbed axial distribution. However, these terms become extreme-ly complicated very quickextreme-ly. Kreith and Sonju (1965) applied this method and truncated the series after the second perturbation term but were unable to give an estimate of the truncation error.
The analytical solutions discussed later are those of equation (16). In any case the assumption of a constant eddy viscosity makes it im-possible to calculate the wall shear stresses from the calculated velocity gradient at the wall and other methods have to be sought. It should be carefully noted at this point that it is not possible to derive equation (14), (Rochino and Lavan, 1969, notwithstanding) without the assumption of a radially and axially constant "eddy viscosity", and attempts to modify e in the form of :
Lim(i:) = 1 r-+l
(Kreith and Sonju, 1965, Sonju, 1962) are mathematically inconsistent with equation (14).
2.4.3 The integral momentum balances.
The integral momentum balance (ignoring turbulent fluctuations at the edge of the control element) leads to
C = z \P V2 d_ dz 'prdr + 2
bv'
V rdr z (19) or as dV/dz = 0C = d f i l ^ . ,,
^ ^^ J Jpv2
o
and an analogous angular momentum balance gives : C- = 2. ^ V.V r^dr
e dz J 6 z
o
which can be written as :
Cg = dS/dz (22) These equations have been derived previously by Fortier (1954) and have
been used in a modified form by Scott (1973) in his analysis of swirl-annular flow. The values of the terms in them can be calculated from
either the data or from the relationships discussed in the previous
sections.
If the experimental data is used then the two shear stress components can be vector combined to give an estimate of the flow angle near the wall, and this can in turn be compared to that angle actually measured. This is a further internal consistency check on the data and in fact all the data accepted satisfied this requirement (see later).
2.4.4 The modelling of the flow pattern.
At this point it is worth reviewing the structure of the model. As input parameters the model requires values for the starting profiles and a value for the eddy viscosity. From these the downstream flow pattern can be calculated and from this flow pattern, the wall shear stresses. If necessary all the steps of the model can be replaced by empirical relationships (if these were available). It is the purpose of this section to examine the model in various configurations, and by that is meant various combinations of calculating routes and various mixtures of empirical and calculated stages. In what follows a ation will be coded as for example, C,ecle, which reads, "the configur-ation where 1. an empirical starting profile, 2. the correlated eddy viscosity value have been used to start the calculation of the down-stream flow pattern by equation (15) and 3. the wall shear stresses have been calculated from the spin-down using the empirical correlation for the swirl number and the flow angle near the wall." Each of the terms will now be discussed.
The first term refers to the choice of starting profile. It would be very useful to be able to calculate this from first principles, that is
from the cyclone geometry alone. Several methods have been tried. The inviscid model of Lewellen (1971) or of Bloor and Ingham (1973,1975) was tried but this leads to either an axial singularity or to an un-determined "critical radius" below which the model does not apply. Estimation of this radius by using the proposal of Binney and Bookings
(20)
(1948) lead to the conclusion that this zone would in any case occupy a large part of the exit pipe radius, and this method was abandoned. A finite Reynolds number solution has been proposed by, amongst others, Hamel (1916), Rott (1958), or Rietema and Krajenbrink (1959). These models avoid the singularity at the«axis but do require both the axial and radial velocity components in the cyclone to be specified in advance, whereas the former is one of the objects of the calculation. Rotts' solution for the swirl profile is of the form :
exp (-ar )) 3-0 (Ea23) 2 O 1 0 (a is a constant) \^J) Fig. 15.
Comparison of the initial swirl component (o) with the model of Rott
( ) and equation (26) ( ) .
The best fit of this equation to the data is illustrated in fig. (15), where it can be seen that the position of the maximum swirl velocity is too near the centre, and as the curve fit criterion used was equality of swirl number, this means that this peak velocity is also too high. This method was also abandoned.
The model actually proposed is a purely formal one, where the shape of the profile is specified, arbitrarily and in advance. Antecedents of this method can be found in the work of Shepherd and Lapple, Fontein and Dijksman, and Senoo et al. (1939, (1953), and (1972) respectively, although a different model distribution is used here.
The axial component has been approximated by :
1.5 r (24)
where the coefficient of 1.5 can be obtained from the normalisation requirement :
1
2 / V^rdr = 1 (25) o
The swirl profile has been approximated by a parabolic distribution with a maximum at (V 9m' 2r r ) max 9m - (-. 2 -)
)
(26) maxBoth V„ and r have now to be specified. The former can be found from the swirl number and is :
V- = S/ (1.2/r 9m ' \ I jjgj, l/(2r^ )
max (27)
and the latter from equation (8).
This model is also compared with the data in fig. (15) and it can be seen that it gives a much better description than equation (23). The second term in the code refers to the source of the eddy viscosity constant used. There is no way of calculating this from first prin-ciples. Usually an empirical correlation is sought with the Reynolds
number by comparing the results of experiments with a given flow model.
For our purposes a comparison of equation (16), (truncated after the first term), and equation (6) leads to the identity :
B Xj (l+e)/Re U 8 )
from which the eddy viscosity can be calculated. An iterative method can be used for finding the best value for e when the numerical method is chosen although the difference in the values of E found is small. In both these cases, if this is done for the data in table (l) then the resulting value is termed the empirical value. There is however, a possibility of making a slightly more general statement. In fig.(16)
2 0
-1 5
1 0
Fig. 16.
The Eddy viscosity , correlation. Cyclone : 1 X 2 V 3 o 4 + 5 ? King(1969) n Wolf(1969) Kreith (1965) A Senoo (1972) a 4 0 4 5 5 O Log 10 (Re)
the results of applying equation (28) to all the data that could be found in the literature for swirl flow are shown. (The calculations were performed on the data given rather than on correlations proposed by the individual authors. Very often startling discrepancies were found between the data given and the correlations "derived" from it.
(see Jongenelen (1974) for further comment). It can be seen that al-though there is a considerable scatter, which is only to be expected when results from different and not equally reliable sources are
combined, a correlation of the form :
E = G Re"" (29) can be used. Curve fitting gives :
e = 0,0033 Re°-^^ (30) When the eddy viscosity is derived from equation (30) it will be
referred to as the correlated value. The index is in good agreement with that of 0.86 found for confined vortex flow by Keyes (1960) and Lewellen (1964), and of 1.0 found by Ragsdale (1961). It is also in agreement with the value found by Martinelli (1947) for non-swirl flow. If the index were to be unity then by equation (15) the flow pattern is independent of the system Reynolds number. That the index differs only slightly from unity indicates, as already experi-mentally observed above, that the Reynolds number dependence is a weak one.
The third term refers to the calculation method. Method 1 is the numerical solution, thus with the perturbed axial component, and method 2 is the analytic solution, with a flat axial profile.
Finally the fourth term is the method used to go from the swirl shear stresses to the axial or total shear stresses at the wall. No reason-able correlation could be found for the pressure term in equation (19) and the axial component has to be calculated by using the flow angle at the wall (.<ti.). This can in turn be found from either the velocity vectors near the wall, or from the correlation given by equation (7). The former is called the calculated value and the latter the empirical value.
2.4.5 The performance of the model.
The model will be assessed at two levels, at the detailed level of the predicted and found radial distribution of the swirl velocity and the prediction of the downstream swirl numbers. In fact all the cyclones showed the same general behaviour and only one, cyclone 2, has been chosen for discussion. The two stations, the "initial" and the "down-stream" station, are separated by 140 cm or 10 diameters. The data is given in table (m). The predictions for the various configurations are also given. It can be seen that they all give fair estimates of the distribution for the positions away from either the wall or the centre, where the values are generally too low.
Model c, initial data ecli ec2i eeli ee2i mcli vel 0.1 .93 .83 .59 .55 .59 .56 .52 ocity at 0.3 1.77 1.48 1.16 1.43 1.16 1.45 1.35 r = : 0.5 2.11 1.73 1.75 1.87 1.76 1.88 1.95 0.7 2.20 1.59 1.84 1.75 1.85 1.78 2.02 0,9 1.96 1.34 .84 .75 .85 ,77 .98
The configuration C,mcli, starting with only the cyclone geometry gives the worst predictions.
The prediction of the swirl number and thus the general spin-down behaviour is good (see table n ) ,
Table (n) The predicted swirl numbers.
nodel c, initial data ecli ec2i eeli ee2i mcli Swirl Number 1,64 1.18 - 1.12 0.82 1.14 0.84 1.02
The flat axial profile calculations were 30 % in error, but the numerical calculation is almost exact. This is the case irrespective of whether equation (30) or the slope B is used to calculate the eddy viscosity. The completely calculated C,mcli gives results that are only 14 % in error and this underlines the point that even an artificial axial velocity distribution (equation 24), is a better approximation than the flat profile.
It can be concluded that the models give a reasonable picture of the downstream flow pattern, and that the prediction of the overall spin-down characteristic, the Swirl number is good. This is of importance because it is from the swirl number that the wall shear stresses are estimated,
If the shear stresses are calculated from the experimental data by equation (22) and the measured flow angle then it is found that the total shear stress, that is the shear stress in the flow direction, is very high in the beginning and falls steadily towards the value given by the Blasius' correlation :
S i
= 0.067 Re (32) 10 r o •.J u 50 Z -cm 100 150Fig. 17. The relationship between the total friction factor and the axial position.
This is illustrated in fig. (17) where the friction factors found have been divided by that given by equation (32)-and set out against the distance. The same behaviour is found in the case of the vertical shear stresses (fig. (18)), where the circles represent values calcul-ated using the wall flow angle and the crosses the values calculcalcul-ated directly from equation (20). This is an illustration of the consist-ency check for one set of values.
10 r o^ • - - 6 - ii 5 _ . - 8
Calculated via
equation :
o ( 7) X (20) 5 0 Z - c m 1 0 0 150Fig. 18. Relationship between the axial friction factor and the axial position.
In table (o) the model predictions for the shear stress at the down-stream point discussed above are compared to that found.
Table (o) The predicted total friction factors. model c, experimental ecle eclc ec2e ec2c eele eelc ee2e ee2c ,mcle Blasius
total friction factor
Blasius' value 1
4.7
1
5.4 7.1 4.5 4.9 5.4 7.0 4.3 4.7 5.9 1.0estimate than equation (32). However, it is in going from the swirl shear stresses to the total ones that a weakness in the model is re-vealed. As would be expected from the controlling assumptions on the generating equations the model predicts the velocity pattern in the vicinity of the wall very badly but it is from this velocity pattern that the flow angle at the wall is calculated, and thus the total shear stresses. In column (c) of table (p) the calculated flow angles are shown, and it can be seen that they are in error. In column (e) the angles calculated by equation (7) are shown and they are much more accurate. The apparent generality of equation (7) must then be used to replace the model calculation at this stage.
Table (p) Comparison between the empirical and calculated flow angles near the wall.
^odel c. eel, ec2, eel , |ee2, pel
,
eo 45 54 45 54 48c°
57 51 57 50 51 foundo 44 44 44 44 44 2.5 The shear stresses for rough walled tubes.Finally for the sake of completeness the case where the walls are rough will be considered. The shear stresses cannot be predicted by the model because the hydrodynamic situation is different from that considered earlier. However, equations (20) and (22) are still valid and local shear stresses can be so calculated. These are higher than
those found for the smooth wall case and are illustrated in fig. (19).
Fig. 19,
The vertical friction factors for the first few diameters of rough walled tubes
data
a equation (33) b C
kz
C^(l . 75 2f-)
o— Kg m m
The values quoted are the average values for the first few diameters. There is no obvious way of handling the data, except to note that neither the correlation of Moody (1944) (see Wallis, 1969) :
k
C^ = 0.005 ( 1 + 7 5 2^-) (33) o
nor its' logical modification : k
S = ^t
(1 + 75 2r (34)give anything other than the roughest guide. The data for the vertical shear stresses (which will be needed later) has been curve fitted as :
C, = C (1+0.111 k - 0.058 k )
Tc z s s (35)
although of course this equation has neither hydrodynamic significance nor more than local validity,
2,6 Conclusions from the studies of the gas flow pattern,
Apart from the detailed conclusions discussed in the relevant sections the following two main conclusions can be drawn. Firstly that the gas
leaves the cyclone with an intense swirl which only slowly decays downstream. This swirl causes a very different flow pattern than that expected for normal turbulent flow, and this flow pattern is associated with very high wall shear stresses. Or put shortly, the swirl cannot be ignored in calculations on these systems. Secondly it can be con-cluded that although they are by no means perfect, the calculation
methods given in the previous sections are a great improvement on those where the swirl is neglected. This is even the case for the most basic calculation where only the simplest geometric details of the system are known beforehand.
CHAPTER 3
THE HEAT TRANSFER BETWEEN THE EXIT PIPE WALL AND THE GAS. 3.0 Introduction.
In this chapter, and the chapter that follows on the two phase flow pattern, two illustrations will be given of the usefulness of the ideas developed earlier in helping to understand and to predict trans-port phenomena in the cyclone exit pipe. No specific application is discussed but many come to mind, for example it is often desirable to control the gas temperature before the next down-stream operation, or possibly to clean the gas of components that are soluble in the second phase.
For heat transfer, one could propose that the exit pipes in a multi-clone unit be surrounded by a second fluid at a different temperature, giving what is in effect a cross-flow heat exchanger, the only unde-termined design parameter of which is the exit pipe length, or what amounts to the same thing, the gas to wall heat transfer coefficients as functions of the geometry and gas flow rate.
3.1 The estimation of the gas side heat transfer-coefficients. The simplest approach to the problem is to treat the problem as if it were conventional turbulent flow problem and to ignore the swirl. In
that case such correlations as :
St = Nu/(RePr) = 0.0384 Re^"'/(1 + 1.5 Re ,~''^^(Pr-1)) (36) d d
from Prandtl (1910), or Taylor (1916), or :
Nu, = 0.023 Re,°'^ Pr^'"^ (37) d a
from Dittus and Doelter (1930), could be applied. An extension of this would be to recognize that the flow situation is really one of entrance region flow and apply the corresponding corrections e.g. :
Nu^= 0.116 Re°*^^- 125 Pr°-^^ 1+0.33(-) ^ (38) d d z
from Hausen (1943).
It has already been shown (fig. (1)), that equation (37) leads to an unacceptable underestimation of the heat transfer coefficients. Neither equation (36) nor (38) gives a better result.
That the heat transfer coefficients are increased, even by moderate levels of swirl is well known. Rohsenhow (1973), Date (1973) and others have reviewed the literature for the low swirl intensities in-volved in twisted tape induced swirl. Valk (1975) has shown that in those cases the heat transfer can be described well by simple momentum analogies.
Typically the heat transfer coefficients can be up to 50 % better for the same pumping power, when twisted tape inserts are used. Heat trans-fer data for the case of high swirl intensities is scarce. Blum and Oliver (1966), Alimov (1966), Rosenberg (1969) and Gambill and Green
(1958) have presented data. The last authors have reported a boiling heat flux of 171 million W/m2. However, no usable correlations have been presented, and for the case of the cyclone exit pipe there is no data at all available in the literature.
3.1.1 Theoretical discussion.
In what follows working equations for the estimation of the perform-ance of the exit pipe as a heat transfer surface will be presented. There are three fundamentally different ways to set up these equations. The most general method would be to simultaneously solve the three equations of motion, the continuity equation and the so-called "energy-equation". (Bird et al. 1960). The result would be a description of the temperature field and of course the heat transfer coefficients. Attract-ive though this method is, the mathematical complexities limit its' application to only the simplest of cases, of which this exit flow is not one. Furthermore in one sense the method is too powerful because the full temperature distribution, even when obtained, is not required. At the other extreme lies the method of dimensional analysis. This method is well known and does not require elaboration here,
The main result of such an analysis is that apart from as the dimension-less groups normally found in turbulent heat transfer problems, viz. Reynolds Number, Prandtl Number and the Nusselt Number, a group such as the Rossby Number is introduced to account for the swirl, (Yanik, 1973), or a modified Grashof Number to relate the swirl to the temp-erature induced buoyancy differences. The values of the indices, and the constant must be considered unknown, and the crucial limitation of this approach is that an extremely wide data base is required to determine these. Such a data base is simply not available at present, and the method given below has been chosen instead.
This third method lies conceptually between the other two and is based upon analogies that can be made between momentum and heat transfer in a highly simplified model of the fluid. In practice there are several mechanisms responsible for the heat transfer. The major contribution comes from the turbulence itself and will be dealt with later. The back-flow recirculation also involves the transfer of heat but a
simple mass balance shows that this can be ignored. The swirl generates a centrifugal field that interacts with the temperature induced density differences which can cause secondary flows which can in turn increase the heat transfer.
Valk (1975) has shown that these effects can be ignored in more weakly swirling flows although no information is available for high swirl levels. It is also difficult to estimate the probable contribution of this pseudo-free convection. The controlling group is the Grashof Number which can be modified (see Eckert and Drake 1959) by replacing the gravity term with the centrifugal acceleration to give :
G-
= (^)'
¥.
(39)
a
where the mean flow and the tube radius have been chosen as the Q
velocity and length scales. Typically values of 10° are found associat-ed with Reynolds numbers of 10^. If these values are locatassociat-ed on the flow-heat transfer regime diagram of Eckert and Diagula (1954) then no free convection effects would be expected. There are serious objections to this argument, for example it is not clear that the velocity and length scale have been well chosen, or again, there is no evidence to show that the diagram of Eckert and Diagula can be generalised to swirl systems. On the other hand in the experiments to determine the heat transfer coefficients no significant trend could be seen between the cases where the gas was being heated, when the free convection contrib-ution should have been at its' greatest and where the gas was being cooled, when the effect would be negative. In what follows this con-tribution will therefore be neglected. Finally heat transfer mechanisms associated with higher velocity swirl flow such as Ranque-Hilsch effects can also be ignored,
3,1.2 The working equations,
It now remains to calculate the forced convection contribution to the heat transfer. It is proposed to do this by invoking the analogy be-tween momentum and heat transfer. The physical model of the fluid upon which this analogy is based is that of Prandtl. The turbulent fluid is considered as a base flow upon which are superimposed continuous random motions of volumes of fluid which, at least for a short but significant time, have a meaningful identity. The analogy with the kinetic theory of gases which was possibly Prandtls' starting point is clear. The movement of these volumes from regions of high to lower velocity and vice versa is responsible for the added momentum transfer in turbulent fluids. Furthermore these fluid volumes can carry inert contaminants with them, for example heat or chemical species. The word inert here essentially means that, for example, the fluid volume moves across the temperature field so fast the conductive interchange with the surround-ings does not occur, or in other words molecular diffusion is neglect-ed. A more detailed description of the various possible analogies, their preconditions and limitations can be found in either Goldstein (1938), or in Davis (1972).
Consider the situation illustrated in fig. (20). The flow and tempera-ture fields have been divided into three zones. Firstly a core zone in which it is assumed that the turbulent interchange is so rapid that
there is no substantial temperature gradient. This viewpoint is sup-ported by the measurements of turbulence intensity for swirling systems made by Wolf et al. (1969). Secondly there is a zone next to the wall of laminar flow where the heat is transferred only by conduction.
core zone turbulent zone laminar zone r y Tb
^-"-""^ tl«—— —'^
F i g . 2 0 . T h e h e a t transfer s i t u a t i o n . wallFor this layer of thickness 5. q/A = - K ( T ^ - T ^ ) / ^
and
^o = ^^5/5
(40)
(41)
For the turbulent boundary layer, which is assumed thin enough for the problem to be treated as a two dimensional one, the analogy gives :
(q/A)/T^ = -\IK ' %(\ - T5)/(Vj - V^) (42)
(see Davis, (1972), p. 159), where L and L, are the mixing lengths associated with momentum and heat transfer respectively. Putting Y as V/V- and AT as T -T , combining equations (40), (41) and (42) gives :
J D O St = Nu RePr m
^
c, J m 1 (43)Noting now that for gases at least L, /L is often in the order of about 1.2 and that the Prandtl number is about .7 then equation (43) can be approximated by :
Nu = Y. iC RePr (44)
which is the working equation required,
Equation (44) should be compared with that obtained for non-swirl flow:
Nu JC|.PrRe (45)
