58
Ind. Eng. Chem. process Des. Dev. 1983, 22, 58-87
tributions of R. C. Binning, D. C. Boyce, L. J. Breuklander, D. Cova, S. D. Koban, A. C. Pauls, and especially R. A. Murray. Nomenclature a = interfacial area for mass transfer, ft2 ft3 DE = eddy diffusion coefficient for liqui flow, ft2/s Ew = Murphree tray efficiency, vapor concentration basis, fractional E, = overall column efficiency, fractional E,, = point efficiency, overall vapor concentration basis, fractional F, = superficial vapor F factor, (ft/~)(lb/ft~)’/~ GM= vapor molar mass velocity, Ib-mol/(s-ft2) hL = liquid holdup on tray, in. KO,= overall mass transfer coefficient, vapor basis, lb-mol/ (s-ft2-atm) L = liquid molar rate, lb-mol/s m = slope of equilibrium line M, = molecular weight of vapor N , = actual plates in column No, = number of overall vapor transfer units Nt = theoretical stages P = pressure, atm Pe = dimensionless Peclet number fL = average residence time of liquid on tray, s TS = tray spacing in column, in. U, = vapor velocity through active, or bubbling, area, ft/s Us = vapor velocity based on total, or superficial area, ft/s U,, = superficial vapor velocity at flood point, ft/s V = vapor molar rate, lb-mol/s W = length of flow path, ft zf = froth height on tray, in. Y = vapor mole fraction Y* = vapor mole fraction in equilibrium with exit liquid
d
Greek Letters X = ratio of slopes of equilibrium and operating lines
p
= density, lb/ft3
Subscripts L = liquid n = tray n (or stage n) v = vapor 1 = small scale 2 = commercial scale
Literature Cited American Instltute of Ctmmlcal Englneers, “Bubble Tray Design Manual”, New Ywk. 1958. Anderson, R. H. M.S. Thesis, The Univsrslty of Texas, Aucltin, TX, 1974. Anderson, R. H.; Ga~&t,G. R.; Van Wlnkle, M. W. Ind. Eng. Chem. process Des. D e w . 1976, 15, 96. Barker. P. E.; Self, M. F. Chem. Eng. Scl. 1962, 17, 541. BkJbauer, F. A.; Oakley, H. T.; Porter, C. E.; Staib, J. H.; Stewart, J. Ind. Eng. them. 1957, 49, 1673. Chan, H.; Falr, J. R. Paper presented at AIChE Meeting, Anaheim, CA. June 1982. Cooke, 0. M. Anel. 0. 1967, 39, 286. Economldes, M.; Maloney, J. 0. A I C E Svmp. Ser. No. 783 1979. 75, 80. Falr, J. R. Pebo/chsm. Eng. 1961, =(IO), 45. Fak, J. I?.In Smith, B. D. “Dedgnof Equ#lbrlum Stage Processes”; McGrawHa: New Ywk, 1963; Chaptsr 15. Garrett, (3. R. Ph.D. MsecKtetkn, The Univeslty of Texas, AwrUn, TX, 1975. Qarrett,0. R.; Andereon, R. H.; Van W l e , M. W. Ind. Eng. Chem. Roces.5 L b . D e v . 1977, 16, 79. Mertin, H. W. Chem. Eng. Rug. 1964, 60(10), 50. Mead, R. W.; Rehm, T. R. Chem. Eng. Prog. Symp. Ser. No. 70, 1967, 63, 44. Olderehew, C. F. Ind. Eng. C b m . , A n d . Ed. 1941, 13, 265. Perry, R. H.; Chitton, C. H., Ed. ”chemicel Engineers’ Handbook”; McGrawHill: New York, 1973; sectkn 18. Sakata. M.; Yanapi, T. I . Chem. E . Symp. Ser. No. 56 1979, 3.2121. Slhrey, F. C.; Keller, 0. Chem. €ng. Rug. 1966, 62(1). 68. Slkey, F. C.; K e k , G. I . chsm.E . Symp. Ser. No. 32 1969, 418. Smuck, W. W. Chem. Eng. Rog. 1963, 59(6), 64. Veatch, F.; Callahan, J. L.; Idol, J. D.; MHberger, E. C. Chem. Eng. Prog. 1960, 56(10), 65. Yanagl, T.; Sakata, M. Paper presented at A I C M Meeting. Houston, TX, Apr 1981.
Received for reuiew September 21, 1981 Accepted July 2, 1982
Lamella and lube Settlers. 1. Model and Operation Woon-Fong Leung. and Ronald F. Probsteln Department of Mchanlcal Engineering, Massachusetts Institute of Technology, Camb-,
Massachusetts 02139
The three-layer, stratified viscous channel flow model of Probsteii, Yung, and Hicks for lamella settlers is generalized and applied to evaluate the performance of cocurrent flow lamella settlers and countercurrent flow tube settlers. For topfeeding lamella settlers a subcritical mode in which the feed layer expands down the channel and a supercritiii mode in which it contracts are confirmed. The mode obtained depends, respectively, on whether the clarified layer thickness at the outlet is less than or greater than about half the channel height, with the exact value dependent on the ratio of the solids fraction in the feed to that in the sludge. I n the bottom-feeding countercurrent flow tube settler there is shown to exist only a subcritical mode. Data from experiments on bench-scale Plexiglas settlers show that for a given settler angle and slurry concentration the efficiency of the supercritical mode is always higher than that of the subcritical mode. The efficiency decreases for both modes as the settter angie and slurry concentration are increased. The efflclency decrease with settler angle is sharp and results principally from flow instability.
Introduction Lamella settlers (Forsell and Hedstrom, 1975) are high-rate sedimentation devices consisting of inclined parallel plates stacked to form into which a s l ~ n y is fed for gravitational separation. Typical dimensions of *Address correspondenceto this author at GulfResearch and Development Co., Pittsburgh, P A 15230. 0196-4305/83/1122-0058$01.50/0
a conventional lamella plate are 150 cm wide by 240 cm long with channel spacings of 4 cm and settler angles between 45 and 60’ (Culp et d . , 1978). In each lamella channel partidea in the influent sediment toward the lower wall Of the channel. The settled sludge slides downward and is removed as the underflow. The downward movement of the feed slurry and sludge, which is commonly referred to &8 “cocurrent flow” (Ward, 1979), induces an upward flow of the clarified effluent. For most lamella 0 1982 Amerlcan Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 59
settlers the suspension is fed in somewhat below the top or near the middle, although the entry designs for the influent and effluent for the middle-feeding settlers are somewhat more complicated than when the feed is close to the top. Tube settlers (Culp et al., 1968) are designed with a pressurized bottom feed. Typical dimensions of the tubes are 5 cm2by 60 to 120 cm long. Since the feed slurry flows up the channel in a direction opposite to the falling sludge, it is referred to as “countercurrent flow” (Ward, 1979). In countercurrent flow the shear stress exerted by the upflow suspension on the sludge opposes the gravitational force acting to pull the sludge down, 80 that a higher settler angle is required to remove the same density sludge. Conventional tube settlers operate with angles up to 60’. Lamella and tube settlers can settle out very fine suspended particles at a high rate. Moreover, the settler capacity per unit volume can be made large because the settling area is equal to the horizontal projected area of each plate or tube multiplied by the number of plates or tubes. Another important advantage of these devices is that they can be designed for operation at high pressures and temperatures. Studies on sedimentation in inclined channels may be traced to the observation of Boycott (1920), that the settling rate is much higher in a channel when it is inclined than when it is vertical. Numerous studies based on purely kinematic arguments have been made (Nakamura and Kuroda, 1937; Oliver and Jenson, 1964; Zahavi and Rubin, 1975) to characterize the phenomenon. In most cases the predicted settling rates are consistently higher than the experimentally measured values. Probstein et al. (1977) departed from a kinematic approach and analyzed the dynamics of the fluid flow in top-feeding lamella settlers using a two-dimensionalviscous stratified flow model. The model was based on an inclined channel flow of three stratified layers: a clarified liquid layer, a feed suspension layer, and a sludge layer. An important result from their model is that for a given throughput rate to the lamella settler there are two operating modes (a ”subcritical” and a “supercritical” mode) with different flow profiles. The mode obtained depends on whether the clarified layer thickness at the settler outlet is less than or greater than about half the channel height. In the subcritical mode the feed layer expands in its direction of flow, while in the supercritical mode it contracts. The supercritical mode is more stable and served as the basis for the design of a new type of lamella settler (Probstein and Yung, 1979) with a higher throughput than present commercial settlers, all of which operate in the subcritical mode. The existence of the two modes was first verified in experiments of Probstein and Hicks (1978). Acrivos and Herbolzheimer (1979) used a two-layer model (consisting of a clarified layer and a feed suspension layer) to analyze batch settling in inclined vessels. They considered the limit in which the ratio of the sedimentation Grashof number to the sedimentation Reynolds number becomes asymptotically small. As a consequence, the clarified layer under the upper channel wall is very thin and moves up the channel with a high velocity. The buoyancy force driving this fluid layer is balanced by the frictional force on the channel wall. The flow in the suspension layer adjacent to the fluid interface is, however, inertia dominated and is similar to the flow in a moving boundary layer. More recently, Herbolzheimer and Acrivos (1981) extended their two-layer model to analyze batch and continuous sedimentation in very narrow (high aspect ratio)
@ - cocurrent feed @ - countercurrent feed
Figure 1. Three-layer model of lamella and tube settlers.
channels where the thickness of the clarified layer is comparable to the channel height. From order of magnitude estimates, the inertial terms in the equation of motion were shown to be small compared with the pressure gradient term which is then balanced by the gravity and viscous terms. This corresponds to the same intertia-free, channel flow criteria employed in the model of Probstein et al. (1977). All of the analyses of sedimentation in lamella settlers that have been noted were based on a continuum description of immiscible stratified fluids. Recently, Rubinstein (1980) employed a two-phase flow model, also neglecting the sludge layer and considering only the influent feed and clarified layers. One of the important findings of this analysis is that the boundary value problem can be reduced to one of a gravitational flow of two immiscible fluids with a macroscopic interface. This lends support to the stratified flow methodology adopted by Probstein et al. (1977). Moreover, Rubinstein examined the “entrance length” problem and clarified the question of how the channel entrance condition can be related to the subsequent establishment of a particular operating mode. In the first part of this paper, the three-layer model of Probstein et al. (1977) is generalized to evaluate the performance of both the cocurrent and countercurrent modes. We shall show how the flow transits from a top-feeding cocurrent configuration to one of a bottom-feeding countercurrent flow. A two-layer model with zero sludge layer thickness will be examined as a limiting case of the general three-layer model. In the second part of the paper, some experimental results from bench-scale lamella settlers are reported. The operating efficiencies of the subcritical and supercritical modes are measured and compared as a function of settler angle and suspension concentration. Finally, a comparison is made of the theoretical and measured clarified layer thicknesses in the subcritical mode.
Model The present model, described by Probstein et al. (1977), departs from the a priori kinematic and geometric reasoning manifested in earlier studies. Instead, the model is based on the assumption that the flow in any channel may be treated as a steady, two-dimensional, stratified viscous channel flow under the action of gravity (see Figure 1). This assumption is approximately valid so long as one is not too close to the overflow or underflow ends of the
60
Id.Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
channel where inertial effects are important (the “entry length’! problem). With reference to Figure 1, it is clear that a cocurrent flow is established with the influent slurry fed in at the top of the channel, whereas a countercurrent flow results when the slurry is introduced at the bottom of the channel. Three stratified layers of differing density are considered: a clarified clear layer, a feed layer containing the suspended solids, and a sludge layer. In any layer both the density and viscosity are taken to be uniform, though not necessarily the same as in any other layer. This assumption, regarding the uniformity of viscosity and density, is not a necessary one and is made only to simplify the algebra in the problem. The fluids in all the layers are assumed to behave like Newtonian fluids of constant viscosity. This is not the case in the sludge layer where it is known that such sludge slurries are generally thixotropic. However, the use of a mean viscosity for the sludge layer is shown not to alter the results significantly. The approximation (Probstein et al., 1977) of no net volumetric flux across any plane normal to the channel walls in cocurrent flow is not made here. Instead, the exact values for the net flux in cocurrent and countercurrent flows are introduced in the subsequent derivations. The conditions of no fluid slip at the wall and continuity of shear and velocity across the two stratified layer interfaces provide sufficient boundary conditions to define the flows in the three layers as functions of the layer thicknesses along the channel. Thew layer thicknesses are uniquely determined from the equation of conservation of solids and the condition defining the rate of particle settling. In the present analysis the particles are assumed to be of uniform size and to settle at a known constant hindered rate, V,. This hindered settling rate can be expressed as a product of the Stokes’ settling velocity for a single particle and a hindered settling factor G which depends only on the solids concentration of the suspension. Accordingly, we write
the thicker layer is inertial dominated (Acrivos and Herbolzheimer, 1979). However, the model and results derived in the following sections are still applicable to the thin clarified layer or thin feed layer (depending on the operating mode). With the assumption of a steady-state channel flow under the action of gravity, the equations of motion in dimensionless form may be written
0 = --aPi
aY
Pi +cot 8
where the subscripts i = 1 , 2 , 3 refer to the clarified, feed, and sludge layers, respectively, and where the coordinates are as shown in Figure 1. It is clear that in eq 3 the pressure gradient along the channel is balanced by both the gravitational and frictional forces, and in eq 4 the pressure drop across the channel is simply hydrostatic. Moreover, eq 4 implies that dpi/dx is only a function of x in each layer. By virtue of the fact that the normal stress must be continuous across the two stratified layer interfaces, we conclude that (5) The effective density pi is taken constant in each stratified layer with the values differing between the layers as sketched in Figure 1. The same is true for the viscosity, though the viscosity is taken to be the same in the clear layer and feed layer. With these assumptions the momentum eq 3 is integrated to give =
u1
Y2 -aly + c,y + c2
Y2 u:, = -a:,2
pu3
where a is the particle radius, t is the particle concentration by volume of solids, ps and pp are the solid and liquid densities, respectively, and h1 is the liquid viscosity. At any position along the channel the sludge flow rate is simply assumed to be proportional to the component of the settling rate in the direction normal to the channel. Formulation In the analysis that follows, the length and thickness variables are rendered dimensionless with respect to the channel height H, the pressure gradient with respect to p g sin 8,and the velocities with respect to the velocity U, defined by I 11-
U, = - - p z g sin 9 12 P l The dimensional and dimensionless variables are denoted by upper and lower case letters, respectively, except for the dimensional layer thicknesses which are denoted by a caret overbar. The viscous channel flow model used in th_e following analysis implies that the layer thicknesses, 6,A, and 6 are of order of the channel spacing and that the flow is inertial free except at the entrance region. In the limit, when the clarified layer for the subcritical mode or the feed layer for the supercritical mode becomes very small, the flow in
(4)
Pz
Y2
= -a3-
2
+c o+
(6)
(7)
c‘j
+ CO + c6
(8)
where aiand p are defined by
p = -P3 P1
and CI-Cg are the six constants of integration. Six boundary conditions express the conditions of no fluid slip at the channel walls and the conditions of continuous velocity and shear across the stratified layer interfaces. These conditions are written y = o
y=q
u1=0
(10)
u1=u2
(11)
UIf
y=h y = l
~
(12)
= u2f 2
=
~
3
(13)
u; = pu3’
(14)
u3=0
(15)
where the primes denote a derivative with respect to y, and h=v+A.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 61
The six constants of integration are evaluated from these six boundary conditions, and it can be shown that c1
= c3
+ (a1- az)?
c2 c3
= {p[a&2
(16)
=0
(17)
+ (a2 - aJ7121 + (1 - h ) 2 a 3 + 2h(l h)c*12]/2(ph+ 1 - h) (18) 1
c4 = $a1 - a 2 h 2 c5
ce = -c3
(20)
+- ( a 3 - a2)h 2
(21)
a3
The dimensionless flux in each layer is obtained by integrating the velocity profile eq 6-8, from which q1 = J'ul h
q2 =
a1
u2 dy = - 2 ( h 3- q3)
'[
~6
c1 + -q2 2
dy = - - q 3 6
c3 + -(h2 - 71 ) + C4(h - t) 2
c5
-?(l - h3)+ -(1 - h2)+ ce(l -h) 2
1
+ ~ 2 ( 3 ~+) ~ 3 ( x = ) qnet
(25)
From the continuity eq 25 and the expressions for the fluxes, eq 16-24, the pressure gradient can be evaluated as a function of the layer thicknesses, the net flow rate, and the densities and viscosities of the layers. After some lengthly algebra, the dimensionless pressure gradient can be shown to be given by
6h2(1- hI2 + 4h(l - h)(l - 2h
+ 2h2) I J
3(1 - h)2+ 3h(ph+ 2[1 - h ] )- 27(ph + P2
1-h)] J
(P3
- P2)
+
1 4h - h2 + ~ (- h)2 l
(i = 2, 3) = Paci + P P ( ~ ti)
(274 (27b)
where the subscript m denotes the effective density of the mud (sediment). With the above definitions the conservation of solids condition is expressed by P2mq2b
+ P 3 m q 3 b = 6'2mq2b) + P 3 m q 3 ( x )
(28)
where the net solids flux across any plane x = constant is set equal to that at the bottom of the channel (denoted by the subscript b). A simple measure of the solids loading to the lamella and tube settlers is the solids removal factor u, defined as the ratio of the theoretical settler capacity to the feed flow rate. The theoretical capacity per unit span of the settler is equal to the settling rate normal to the channel multiplied by the length of the channel. It follows that (29)
where I , ug, and qfed are, respectively, the dimensionless channel length, settling velocity, and settler throughput. When the solids loading is less than the settler capacity, u > 1, while when the settler is overloaded, u < 1. The ideal operating condition corresponds to u = 1. The second condition which completes the definition of the layer thicknesses makes use of the fact that the sludge flow rate is proportional to the component of the particle settling rate in the direction normal to the channel, or ~ 3 m q 3=
-~2mus(FI- x )
COS
e
(304
where F, the effective settling length factor, is defined as F(u)=l O 1 (30b) The negative sign is included in eq 30a in accordance with the coordinates of Figure 1 for which q3 is negative. To complete the formulation of the model, the net flow rate in the pressure gradient expression eq 26 has to be specified for both flow configurations. This rate is determined by the continuity eq 25 which is expressable as qnet = q 2 b + q 3 b since the clarified layer flux is zero at the bottom of the channel. The sludge flux at x = 0 is evaluated from eq 30a to give
P
P2
(p - l)h2
= &ti
(24)
In the steady state, continuity of mass requires that the net flow rate qnet be constant along the channel, i.e. ql(x)
pim
(19)
+ ( a 3 - a2)h
= c3
fluxes in the three layers. One condition used to define the local thickness of the sludge layer and feed layer is the statement of conservation of mass expressing the fact that the rate at which suspended solids are fed in must equal the mass flow rate of solids in the feed and sludge layers. The effective mass density of the solids in the two layers and the density of the layers are defined through the relations
1
[:
+ 3 - 2h + (1 - h)2
=(1 - h)2
+
The pressure gradient co (= -dpi/dx) in eq 26 is composed of a pressure drop due to the viscous flow frictional loss, the hydrostatic pressure gradient in the feed layer, a pressure drop due to buoyancy between the feed layer and sludge layer, and a positive pressure gradient due to buoyancy between the clarified layer and feed layer. The layer thicknesses in eq 26 have to be specified and this is done through the mass flow constraints defining the
Here, the ratio of mass densities has been replaced by the ratio of the solid volume fractions. The feed layer flux at x = 0 in countercurrent flow is simply the feed rate given by eq 29. For a fixed settler capacity it is a function of the solids removal efficiency. In cocurrent flow, only the case of a 100% solids removal factor is considered from which it follows that q2b = 0. We may therefore write the feed layer flux at the channel bottom in the form (32)
62
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
where j = 0 for cocurrent flow and j = 1 for countercurrent flow. Combining eq 31 and 32, the net flux is therefore
The flux in each layer is given by eq 22-24 as a function of the unknown layer thicknesses. The problem is closed, however, by the fact that the fluxes are also specified by the mass flow constraints. With eq 25,28,30a, 31,32, and 33, it can readily be shown that the layer fluxes are ql(x) = ( 1 - : ) x u .
cos e
= -?( €3 F - :)ha
COS
(34)
e
where 0 I x IF1. It is clear that the flux rates are proportional to the longitudinal distance along the channel, a consequence of the linearity of the mass flow constraints. The material balance constrainta, eq 28 and 30%and the flow rate expressions, eq 16-24,26, and 33-36, uniquely define the cocurrent and countercurrent sedimenting channel flow model. Solution Behavior At the channel top (denoted by the subscript t), the thickness of the sludge layer is zero &e., 6, = 0 or ht = 1). In this limit, the pressure gradient expression eq 26 and layer flux eq 22-24 together with eq 16-19 reduce to the simplified forms corresponding to h = 1
(39) 43 = 0
(40)
In eq 37-39 the qnetterms represent an equivalent single-phase viscous channel flow, while the density difference terms represent the buoyancy flow induced by the inclined stratified fluid layers. When the layer flux expressions eq 34 and 38 (or eq 35 and 39) are set equal at x = Fl with qnstgiven by eq 33, we arrive at an important relationship between the settler throughput (or capacity) and the clear layer thickness 7, at the outlet of the channel, namely
CLEAR LAYER THICKNESS AT OUTLET, 7 ,
Figure 2. Feed flow rate as a function of clarified layer thickness at overflow end for subcritical and supercritical cocurrent flow (lamella settler) solutions with solids loading equal to settler capacity (a = 1) and for different ratios of feed layer to sludge layer volume fractions (e2/t3).
or greater than half the channel height. This result is illustrated in Figure 2, where the throughput divided by the specific density difference between the layers is plotted as a function of the dimensionless clarified layer thickness for different values of c2/e3. It can be seen that when the solids concentration in the sludge layer is much larger than that in the feed layer ( ~ 2 / ~ 3 0) the throughput curve is symmetrical with respect to the half plane 7, = 1/2. However, for finite values of eZ/e3 the curve becomes asymmetric and the subcritical and supercritical regions are somewhat displaced with respect to the half plane as shown in Figure 2. In countercurrent flow, where the feed slurry is introduced at the bottom of the channel, qfd = q2b. It would appear that eq 41 for countercurrent flow 0' = 1) also has two solutions corresponding to a fixed qfeed as for the cocurrent case. However, detailed analysis shows that the supercritical solution does not exist for x C F1 because the mass constraint equations cannot be satisfied, whereas the subcritical solution does exist for x 5 F1. This countercurrent subcritical mode is illustrated in Figure 3, where the throughput, normalized with respect to the specific density difference and the solid volume fractions of the layers, is plotted as a function of the dimensionlessclarified layer thickness for u = 1. In Figure 3 the throughput increases with 7, to a local maximum of 0.172 at 7, = 0.7. A higher feed flow rate can be accommodated with the clarified layer occupying the full width of the channel at the outflow, i.e., qt = 1.
-
Simplified Two-Layer Model Solution In cocurrent flow where the feed slurry is introduced at the top of the channel, (Ifead = 4% For cocurrent flow j = 0 and with u = 1, eq 41 shows that for a fixed throughput there are two possible solutions for qt corresponding to two operating modes. For ~ 2 / ~=30 the cocurrent flow solution was characterized as either subcritical or supercritical by Probtein et al. (1977) depending on whether vt is less than
When the feed slurry concentration is dilute (e2
