Ind. Eng. Chem. Res. 1995,34, 3399-3404
3399
A Fluid Mechanics Model for an Axial Cyclone Separator Frans T. M. Nieuwstadt* and Maarten Dirkzwager Laboratory for Aero and Hydrodynamics, Delft University of Technology, Rotterdamseweg 145, 2628 AL Delft, The Netherlands
The design of an axial cyclone for the removal of liquid droplets from a liquid stream is considered. The design constraints are that the swirl in the separation section is to be maximized and that a n adverse pressure gradient in the flow is to be avoided everywhere in order to prevent droplet breakup. First, a simplified expression is derived for the separation efficiency, defined as the maximum length needed to remove droplets or particles from the flow. The result shows that it is advantageous to position the vanes that generate the swirl at a radius larger than the section where actual separation takes place. This leads to a design in which the swirl generator is mounted on a central body positioned in a wider section of the pipe than the separation section. To calculate the flow and the pressure field in this cyclone geometry, we develop a stream function method which uses a superposition of axisymmetric sources and sinks. We show by means of a n example that we can indeed design a geometry in which the undesirable adverse pressure gradients is avoided.
Introduction
I
The cyclone separator is a well-known device to remove small particles or droplets from a gas or liquid stream. Its operation principle is based on the fact that the flow is made to rotate strongly. The resulting centrifugal acceleration then forces particles with a higher density than the fluid t o migrate toward the outside wall where they are subsequently removed from the flow. One may distinguish two types of cyclones which differ in the way the flow enters in the apparatus. The first is the so-called tangential or reversed flow cyclone. The flow in this case enters tangentially into a cylinder or pipe section, and this produces a strong rotation. Within the cyclone the flow direction is reversed after which the flow exits axially at the top and the particles are collected at the bottom. For further details and discussion one may refer t o the literature such as Panton (1984,p 619)and Dirgo and Leith (1986). The advantage of the tangential cyclone is its very good separation efficiency. On the negative side stands the fact that the flow in this type of cyclone is usually highly turbulent and nonstationary. These points together with the flow reversal result in a high pressure drop across the cyclone. The second type is the axial cyclone. In principle, it consists of a pipe geometry and an example is shown in Figure 1 which illustrates an axial cyclone in the form of a cylindrical pipe. The flow through this pipe is put into rotation by a so-called swirl generator. This is a cylindrical body centered in the pipe on which vanes are mounted that give a tangential component to the axial flow. Downstream of the swirl generator in the so-called separation section or settling zone, the flow rotation forces the particles t o the outside wall where they are removed through slits or annular rings. For a more detailed description we may refer to Swanborn (1988). It should be mentioned at this stage that an axial cyclone does not always have a cylindrical geometry. In particular, in the geometry to be studied here, we will allow the pipe section that contains the swirl generator to have a different diameter than the separation section
* To whom correspondence should be addressed. E-mail: f.nieuwstadt@wbmt. t u d e l f h l .
' tgas
Out
I I
1 gas
&. fluid in
Figure 1. Schematic drawing of an axial cyclone. Gas with fluid droplets enters at the bottom and is put in a swirling motion by the swirl generator; the fluid droplets are deposited at the wall in the settling zone and both gas and fluid are removed at the top of the cyclone.
for reasons that will be explained below. As a disadvantage of an axial cyclone we should mention that it is far from straightforward t o produce a strong swirl by the vanes. This makes it for instance difficult to reach a high separation efficiency. Moreover, there is much less experience with this type than with the tangential cyclone and as a result its optimal design is far from being settled. On the other hand, a clear advantage is, however, that the flow is much less disturbed than in a tangential cyclone and as a result the pressure drop across an axial cyclone is small. In this paper we face the special case of separation of liquid droplets from a liquid flow where both liquids are immiscible. The cyclones used in this case are usually
Q888-5885/95/2634-3399$09.QQIQ 0 1995 American Chemical Society
3400 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
referred to as hydrocyclones (Thew, 1986; Svarovsky, 1984). In the application that we consider here, the density of the droplets is larger than the density of the main fluid flow. An example is the case of small water droplets dispersed in oil which is a common problem at offshore installations. The water has to be removed at the platform itself because it is clearly not very economical t o pump water through the pipeline used to transport the oil to the shore, apart from the fact that one also wants t o avoid the danger of corrosion in this pipe. With the aging of the oil well, the water fraction in the oil increases. This implies that, to reach the same production, one has to remove more water and this calls for an increase in the separation capacity on the platform. A complication in the design of this so-called liquid/ liquid cyclone is the fact that the flow should not be strongly disturbed or turbulent. The reason is that the shear connected with these flow disturbances will break up the droplets into smaller ones. For instance the sizes of the water droplets in an oil dispersion usually vary between 10 and 100 pm and these sizes can be still removed by a cyclone. However, a break-up process due t o shear will produce droplets of -1 pm and these can no longer be handled by a cyclone. Avoidance of strong flow disturbances means for instance that nowhere in the cyclone should flow separation be allowed because this is almost always accompanied by strong turbulence. Flow separation occurs as a result of an adverse pressure gradient in the flow, and this is directly connected to the flow geometry. For instance, a flow that has to pass around a corner or has to change direction suddenly usually experiences an adverse pressure gradient at some position and consequently will separate. The strong changes of flow direction in the tangential cyclone determine that it is not applicable to our problem. An axial cyclone on the other hand has a minimal flow disturbance, as we have argued above. This and the additional advantage of a small pressure drop favors the use of an axial cyclone for our case. In this study we concentrate on the optimization of an axial cyclone design with respect to its flow field. For the geometry we will choose a design in which the swirl element is mounted on a central body centered in a wider section of the pipe than the section where the separation takes place. As a result, there are convergent and divergent parts which must be properly designed in order to avoid the adverse pressure gradients which cause separation. We shall develop and apply a stream function method by which we can calculate the flow field and pressure distribution for a given cyclone geometry. On the basis of the results of this method, we can then adjust the geometry toward the most optimal shape. In the next section we first discuss the geometry of our axial cyclone and we derive a simplified expression for the theoretical separation efficiency. After that, we discuss the calculation method of the flow. Subsequently, the method is illustrated by an example. Cyclone Geometry
A schematic illustration of the axial cyclone that we will study here is given in Figure 2. It shows the central body with radius Rl(x)on which the swirling vanes are mounted. Behind this central body lies the separation section with a radius Rz less than R I . The profile of the swirling vanes has been optimized with the CFD
h
A
Figure 2. Schematic illustration of the axial cyclone geometry: central body (index 1)and separation section (index 2).
package ISES (Drela and Giles, 1987). With the aid of this package we have been able to find a vane geometry which produces a tangential flow that is twice the axial velocity at the trailing edge of the vanes. In other words the flow is deflected by an angle of approximately 63" without any separation. From the geometry illustrated in Figure 2, it should be also clear that the vanes on the central body are positioned a t a larger radius than the radius of the separation section. The reason for this will become clear later. Let us estimate the separation efficiency for this geometry. The annular pipe section around the central body has a width A(x). The Rl(x) gives the central radius of this annulus. The velocities in the annulus and in the separation section are assumed to be uniform over the cross section and are denoted by U1 and UZ, respectively. Conservation of mass then implies
2nRlAUl = nR:Uz
(1)
For the tangential velocity we choose the following relationship
w = ul2nr (2) where K is independent of r and also of x . The latter condition is true, because the cross section remains constant after the swirling vanes and also because frictional dissipation of the swirl is neglected. Expression 2 denotes a so-called potential vortex along the centerline of the cyclone. The reason for this choice, which is clearly an approximation of a real velocity distribution, will become clear in the following section. Because of the result given above, that the tangential velocity downstream of the swirling vanes is twice the axial velocity, we find K
= aU,Rl
(3)
where a is equal to 4n. Next we consider the motion of a small droplet in this flow. Let us assume that the droplet is so small that the force on it is given by the Stokes law (Bird et al., 1960, p 59) where we have neglected the fluid motion in the interior of the droplet. Another assumption, which seems also reasonable in this case, is that we neglect accelerations of the droplet. This implies that the particle has the same axial and tangential velocity as the flow. The radial velocity follows from a balance between the Stokes drag and the centrifugal force, which leads to the following equation for the radial position r, of the droplet as a function of time (4)
where a is the droplet radius, eP is the droplet density, ef the fluid density, and v the kinematic viscosity. Equation 4 can be integrated. With the initial condition that r = ro at time t = 0, we find that the droplet reaches the wall of the separation section after a time
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3401
........
400
is negligible. This assumption is reasonable because the pressure field in the annulus is primarily determined by flow distortion connected to the flow around the central body and frictional effects, and thus also turbulence, can be considered as a second order effect. First, we introduce the so-called Stokes stream function, q, which in the case of an axially symmetric incompressible flow field integrates the continuity equation. For the case of a stationary flow, as considered here, this function depends only on the axial ( x ) and the radial ( r )direction in cylindrical space coordinates. The axial and radial velocities, u and u , respectively, are then given by (see also Bird et al. (19601, Table 4,2-1)
05
........ ..............- ..................... 0
----
500
- * '
1000
"
"
"
1500
'
"
"
2000
"
2500
A/a Figure 3. Separation efficiency L$D, as derived in the text for
(ep- efYef 0.2 and a = 4n, as a function of the ratio d/a for several values of the Reynolds number Re. T given by
-
The maximum value for T will regult when ro 0. In that case ( 5 ) may be rewritten in terms of a separation length Ls = UZT,and by using of (1)and (3) we find
(7) from which it can be shown that the value of the stream function IJJ is constant along a streamline. Next, we search for a stream function which can describe the flow around an arbitrarily shaped swirl element inside a pipe with variable diameter. Let us introduce the following expression
(6) where D = 2 R 2 is the diameter of the separation section and the Reynolds number Re is defined as U a l v . A smaller value of the ratio L$D implies a higher separation efficiency. It then follows that the separation efficiency improves when the Reynolds number of the flow becomes larger, but more importantly, the separation efficiency also increases when the ratio of the width A to the particle radius a becomes smaller. As we have assumed a constant cross section after the swirling vanes, a small value of A implies a large value of R1 (see (1)).This underlines the advantage of having a larger diameter for the central body than for the separation section to which we already referred above. With oil and water for which (ep- ef)/ef= 0.2 and with a = 4n, (6)becomes equal to L$D = 5.625(A/~)~/ Re. We have illustrated (6) for these values in Figure 3 as function of A/a for several realistic values of Re. This figure clearly illustrates the importance of A/a and the Reynolds number for the separation efficiency.
The Stream Function Method A good design of the central body and its smooth connection to the settling zone (see Figures 1 and 2)is essential from a fluid mechanics point of view in order to avoid disturbances such as flow separation. Here, we shall develop a calculation method for the flow based on a representation of the central body by means of sources and sinks in combination with a straining flow which is assumed to be a function of x . For a general background on the flow resulting from sources and sinks, we refer t o any standard text on fluid mechanics (e.g., Batchelor, 1967). The idea is that with this method one is able to estimate the pressure field in the annulus under the assumption that influence of friction
which consist of three contributions. The first is a straining flow along the axial direction which is taken to be a function of x only. The second contribution is a continuous source/sink distribution with a strength q per unit length (divided by 4x1 along the centerline between x = 0 and x = L, and the third contribution denotes a point source with strength mp (divided by 4n) at the location x = t, (with 4 < t, < L). The L is here the length of the central body, i.e., the swirl element. The first condition that must be satisfied is that the stream function (8) describe the flow around a closed body. It can be shown that this condition requires
(9) Next we have to generate conditions for the functions U(x)so that (8)describes the flow around the central body and within the outer pipe wall. We know that these two surfaces must be streamlines so that we may apply the condition that the stream function is constant along them. As a matter of fact, it follows from (9)that the streamline which describes the central body has the value zero. Before imposing these conditions, we divide the centerline between x = 0 and L in N intervals [&-I, til with i = 1,....N. In each interval, we take the strength of the source distribution q equal t o a constant, qi. We also impose that the stagnation points on the front and back of the central body coincide with x = 0 and x = L , respectively. It then follows that a constant source strength is not suitable in the first and last interval and that a quadratic function should be used here: i.e., q(t) = t2 for 0 It tl and q(t) = ( t t ~for)~ N~- I St s t ~ The . fact that the stagnation points must lie at x = 0 and x = L results in two additional
q(t) and
3402 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
conditions which are
where the integral has to be divided over the intervals introduced above. At the stagnation points on the front and back of the central body, i.e., at x = 0, r = 0 andx = L, r = 0, the stream function (11)is singular. An asymptotic expansion can be found by first letting x 0 or x L and then by letting r 0 also. For the frontal stagnation point we find for the streamline = 0 the following expression
-
The positions xi with i = 1, ...,N - 2 are placed a t the center of the intervals Lti-1, til, xo a t t o and XN-1 at tN. The function U(x)is evaluated at the positions xi with i = 0, ...,N - 1. The stream function at a position xi can now be written as
- t)t2
1
40
(Xi
2
t12
((Xi- t)2
q(xi,r)= -U(xi>r2- - Jtl
c SJ-‘
N-2
(Xi
- t)
Q.
j=l
J t j
((Xi- t)2
(tN - tN-1)
((Xi
dt -
+r2y2 +
(11) ((xi- tPl2 r2Y2
+
This results in N unknown values U(xi),N unknown values of qi, and an unknown point source strength mp, i.e., a total of 2N 1unknowns. To obtain a set of equations from which these unknowns can be solved, we apply the condition of I#= constant along the central body, i.e., on the inner streamline, a t the positions xi with i = 1,..,,N - 2. This leads to the following N - 2 equations
+
(12) A similar condition is applied along the pipe wall, i.e., along the outer streamline, at the positions xi with i = 0, ..., N - 1which leads to the N equations =C
(13)
where the constant C follows from the flow rate through the cyclone. The equations (12) and (13) together with the conditions (9) and (10) form a set of 2N 1linear equations for the 2N 1 unknowns which can be solved by standard matrix methods. The axial and radial velocity distribution can then be calculated with the aid of (7) and this leads to
+
+
2
dt -
dt - t>2 r2Y2 mp(xi- tP)
V(Xiaouter,i)
-
+ r2Y2
(Xi- t)(tN- t)2
QN-1
-
This means that the front of our body is not sharp but it is given by a logarithmic singularity. For the back of the central body a similar expression can be derived. Finally, we consider the tangential velocity introduced by the vanes on the central body in the annular pipe section. One may wonder whether the introduction of such tangential velocity will change the streamline pattern which follows from stream function (11).However, if we select a tangential velocity according to (2), which is known as a potential vortex, it can be shown that the streamline pattern will not change. The presence of a tangential velocity appears in that case only in the pressure field. When the tangential velocity is not given by a potential vortex, the radial pressure field is not in equilibrium with the centrifugal acceleration. In that case flow circulations will develop which subsequently will change the streamline pattern. In addition, these secondary flow circulations may also diminish the separation efficiency. Next we consider the pressure field. For this case, where the flow is stationary and, where frictional effects are neglected, we can apply Bernoulli’s equation. This equation implies that the sum of the dynamic and static pressure is constant along a streamline. In principle, however, this constant may vary for each streamline. As a simplification, we adopt the approximation that the constant is the same for all streamlines. This implies that we assume the flow to be irrotational and this can be true only when U is constant as function of x . Therefore, the pressure distribution calculated with a varying U(x) can be only an approximation and, consequently, we must limit ourselves to a not too strongly varying cross section. With these assumptions we find
E + -1 (u2+ u2 + w2>= constant ef
2
(16)
where the constant follows from applying the upstream conditions. Equation 16 leads to a pressure jump across the vanes in the annular pipe section. The pressure jump should interpreted as the power that is extracted from the flow by the vanes in order to deflect the axial velocity into the tangential direction. As such it contributes to the pressure drop across the cyclone. Now that the total flow and pressure field can be calculated for a given geometry, one is able to examine the influence of geometry variations in order to optimize the design of the cyclone. The optimization criterion that we will use is related t o the pressure gradient. We demand that the pressure gradient along walls should
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3403
i 10
0
X
20
30
40
50
.
Figure 4. Contour plot of the stream function. 1.5
->E.
sharp point. It was found by using our streamline method that such a geometry is necessary to keep the I _ _ _ _ _ _ _ _ _ _ _ _ _ . ~ . . - - . ~ . . . _ _ _ pressure gradient everywhere negative. When the tail is shortened, the pressure gradient will become positive at some position. This is undesirable for the reasons mentioned before. Furthermore, this clearly illustrates the advantages of basing a design on flow calculations. Finally, we may mention that at the position of the nose of the central body we have applied a slight decrease of the cross section in order that separation 0.2 0.4 0.6 on the walls is avoided. After the vanes, the cross x (m) section has been kept constant.
- radial velacity _-.axial velocity
1
22
\
0.5
I
'..,'
3
0
-0.5
0
Figure 6. Axial and radial velocity components on the outer pipe wall.
Conclusions 0 t
P 3
'
'
% -52 . a . -10
-
- central body .-___.
-.- pipe wall 0
0.2
0.4
0.6
x (m)
Figure 6, Pressure at the outer pipe wall (dashed line) and central body (solid line). The negative pressure values are due to the choice of the undisturbed pressure level.
not become too strongly positive, because this would imply separation of the flow and we have already argued that this needs t o be avoided.
Example Let us illustrate the method discussed in the previous section by an example. We have calculated the stream function in a geometry which consists of a central body in a pipe section with variable diameter. The calculated values of qi and U(x)lead to a contour plot of the stream function as shown in Figure 4 which can be interpreted as a streamline pattern. Results for the velocity and pressure along the pipe wall, i.e., on the outer streamline, are shown in Figures 5 and 6. From the results of the pressure it can be seen that, at x locations downstream of the swirling vanes, a negative radial pressure gradient exists. The reason for this gradient is the increase of the tangential velocity toward the centerline resulting from (2). This effect counteracts the influence of the axial and radial velocities which tend to increase the pressure. A distinct characteristic of the central body geometry shown in Figure 4 is the long tail which ends in a rather
If we restrict ourselves t o the main points that we have found in this study, the following conclusions may be drawn. 1. On the basis of a simplified theoretical treatment of the separation efficiency of an axial cyclone, we found that the following parameters play an important role: first, the ratio of the width of the annular pipe section to the droplet radius and, second, the Reynolds number. On the basis of this result, we propose an design in which the position of the vanes used t o produce swirl are positioned at a larger radius than the radius of the separation section. 2. By means of a stream function method, we are able us to calculate the velocity field and estimate the pressure along the walls in an axial cyclone. By applying this method we can search for an optimal geometry, in which, e.g., positive pressure gradients are prevented in order to avoid flow separation. 3. From working out an example with our method, we have found that the geometry of the central body should have a long tail in order to avoid pressure gradients and thus flow separation.
Acknowledgment
This investigation was sponsored by the Netherlands Technology Foundation ( S W ) under Project No. DWT002346. Furthermore, it has been our pleasure to prepare this article for the special issue marking the 35th anniversary of the textbook Transport Phenomena. Knowing one of the authors, Prof. R. B. Bird, personally, we admire his enthusiasm and originality and we hope that this article reflects some of these qualities. Nomenclature a = droplet radius C = constant value of the stream function at the outer streamline D = diameter of separation section
3404 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
L = length of the central body L, = separation length mp = point source strength divided by 4n N = number of intervals of the source distribution q(t) = source strength per unit length divided by 4n as function of the x position qi = value of the function q(t)in the intervals [ti - 1,ti] for i = 1, 2, ...,N - 2 a0 = coefficient in the quadratic function which describes q(t)in the interval [to, tll qN-l= coefficient in the quadratic function which describes q(t) in the interval LtN-1, t ~ l r = radial coordinate rp = radial position of droplet arp/at = time derivative of radial droplet position ro = initial radial position of droplet R1 = radius of the centerline of the annulus Rz = radius of the separation section Rinner = radius of the central body Router= radius of the outer wall of the cyclone Re = Reynolds number of the flow in terms of D T = time of droplet to reach the wall t = x position variable used in integrals ti = border of intervals of the line source for i = 0, ...,N t , = x position of the point source Ul = bulk velocity in the annulus Uz = bulk velocity in the separation section U(x)= straining motion u = axial velocity component v = radial velocity component w = tangential velocity component x = axial coordinate xi = discretization position of the function U(x) a = constant depending on deflection angle A = width of annulus
K
= strength of the potential vortex
v = kinematic
viscosity
ef = density of the fluid
e, = density of the droplet $J = Stokes stream function
Literature Cited Batchelor, G. K. An introduction to fluid dynamics; Cambridge University Press: Cambridge, UK,1967. Bird, R.B.;Stewart, W. E.; Lightfoot, E. N. Transportphenomena; Wiley: New York; 1960. Dirgo, J.; Leith, D. Design of Cyclone Separators. In Encyclopedia of Fluid Mechanics, Vol. 4, Solids and Gas-Solids Flows; Cheremisinoff, N. P., Eds.; Gulf Publishing Company: Houston, TX, 1986. Drela, M.; Giles, M. Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA J . 1987,25(101, 1347-1355. Panton, R. L. Incompressible Flow; Wiley: New York, 1984. Svarovsky, L. Hydrocyclones; Techrome Publishing Co. Inc.: Lancaster, PA, 1984. Swanborn, R. A. A new approach to the design of gas-liquid separators for the oil industry. Ph.D. thesis, Delft University of Technology, 1988. Thew, M. Hydrocyclone redesign for liquid-liquid separation. Chem. Eng. 1986,JulylAug, 17-23.
Received for review January 4, 1995 Revised manuscript received July 19, 1995 Accepted August 7, 1995@ IE950016B
Abstract published in Advance ACS Abstracts, September 1, 1995. @
