Ind. Eng. Chem. Process Des. Dev. 1903, 22, 68-73
00
Lamella and Tube Settlers. 2. Flow Stablllty Woon-Fong Leung Department of Mechenical Engineering, Massachusetts Institute of Technology, Cam-,
Massachusetts 02139
The role of flow instability in decreasing lamek settler performance is examined. Emphasis is on the supercritical operating mode in which the feed layer contracts as it moves down the settler channel. From observations of the flows in Plexiglas settlers, the unstable nature of the interface between the clarified layer and feed layer is characterized as a function of settler angle and flow rate. I t is shown that at low settler angles the destabilizing mechanism is assoclated with an inflectional point in the flow due to shear, and at high angles with a gravity destabilizing mechanism. At low and high angles, theory and experiment are used to define the dependence of the critical flow rate for the onset of turbulence at the interface on the settler angle, the channel helght, and the density difference between the clarified and feed layers. Experiments on the operating efficiency of settlers in the supercritical mode are in accord with the stability criteria. The subcrlticai mode is shown to be inherently more unstable because the velocity profiles lead to entrainment of the feed layer in the overflow.
Introduction A generalization of the model for lamella settlers originally presented by Probstein et al. (1977) was given in part 1of this paper (Leung and Probstein, 1983) together with some experimental results. The experiments confirmed the earlier findings of Probstein and Hicks (1978) that for a fixed settler throughput there are two operating modes-a supercritical mode and a subcritical mode. The experiments also showed that the operating efficiency of the supercritical mode is always higher than that of the subcritical mode for a fixed settler angle and feed slurry concentration. However, the experiments also showed that at large enough settler angles and feed solids concentrations both the supercritical and subcritical flows become unstable resulting in a substantial decrease in operating efficiency. Probstein et al. (1977) reported a nearly 50% drop in efficiency for lamella settlers operating in the subcritical mode. This was attributed to flow instability. It is also believed that in batch sedimentation in inclined channels flow instability and turbulence accounts for much of the discrepancy between the measured enhanced settling rate and the theoretical rate predicted by the well-known Nakamura and Kuroda equation (Pearce, 1962). In this paper a theoretical and experimental study is presented on the flow stability of the supercritical operating mode, with the goal of optimizing the lamella configuration in this new mode for process engineering uses. The aims of the study are to determine the most stably operating configurations as a function of settler geometry, suspension properties, and operating conditions, and to see if the instabilities can be suppressed by design changes. Some consideration is also given to subcritical operation. Observations on Flow Instabilities The experimental arrangement described in part 1employing a single channel Plexiglas lamella settler was used to observe the flow instabilities. In the stability experiments discussed below, the sludge was not withdrawn as in the experiments in part 1, but instead it was allowed to accumulate in the collecting box during the short period of observation. In the stable operation of the supercriticalmode the feed suspension layer and the clarified layer are laminar, as is the interface between them. The sedimentation is not disturbed, the overflow is clear, and there are no waves or *Gulf Research and Development Co., Pittsburgh, PA 15230. 0196-4305/83/ 1122-0068$01.50/0
disturbances present in the flow. For a fixed settler inclination angle the following behavior is observed with increasing flow rate: (i) There is the same behavior as at low flow rate except that waves appear at the interface. (ii) The interfacial waves amplify downstream toward the sludge collecting end and eventually begin to break. (iii) The wave breaking moves upstream. (iv) Moderate breaking of the interfacial waves causes slight mixing and entrainment of the solids upstream and the overflow becomes slightly contaminated (as measured by a turbidity meter). (v) Moderate entrainment causes turbidity in the clarified overflow to exceed an arbitrarily set clarity limit as measured by a turbidity meter. (This flow rate, defined as Q ,, was used in eq 51 of part 1to evaluate the settler operating efficiency). (vi) The whole interface breaks and there is large mixing of the two layers at the interface, although each layer may still be laminar. (This flow rate is defined as Qturb) (vii) At a large enough flow rate transition to turbulent flow occurs throughout the channel in both layers. Measurements of the flow rate QtWbat which the whole interface breaks are shown in Figure 1 as a function of settler inclination angle. The data are taken with settler A, without the Plexiglas insert (see part 1 and Figure 4 below for geometry) at two different concentrations of the aluminum oxide particles in the feed. The data are normalized with.respect to the value of Qturbat the settler angle 8 = loo. This angle is seen to be the approximate transition point between which the breakdown is angle independent and steeply angle dependent. That two different breakdown mechanisms are at work, one at low angles and one at high angles, is observed visually from the nature of the wave forms and their breaking characteristics. Figure 2 is a sketch of the wave forms and their breaking patterns at low and high settler angles. Some important features to be noted are that at low angles the ra_tioof the wave length X to the suspension layer thickness A is about 3 and the breakdown is essentially two-dimensional. On the other hand, at large angles the ratio of the wave length X to the suspension layer thickness is about 10, there is a distinct three-dimensional structure to the breakdown, and the transition is characterized by waves that roll up like a “jelly roll” with alternate clear and dense layers of fluid. Based on analysis and examination of the data discussed below, the instability at low angles is assumed to be associated with an inflectional point destabilizing 0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 60
INTERFACE
Figure 3. Sketch for supercritical mode of clarified layer and feed suspension layer flow profiles neglecting thin sludge layer.
problem the velocity gradient can be expressed in terms of the flow rate Q by writing 0
I
1
I
-
iarl RNGLE
e 8 10'
u:
a -02
1
I
I,
WAVE BREAKS AT CREST
The thickness of the feed layer & can be expressed in terms of QCfitif we know the theoretical velocity profiles. With zero sludge flux, the velocity profiles for the clarified and feed layers can be obtained from part 1and with Y measured downward from the upper plate of the channel (see Figure 3) are written clarified layer (0 I Y I H - 6 )
- 0 . 5 cn
feed layer (H - & I Y I H) 1 (P2 - P1) u, e -g sin 8 @(1 - A)2
2
u2
111
(
$)2
'
Figure 2. Sketch of interface wave forms and their breaking patterns at low and high settler angles.
mechanism and at high angles with a gravity destabilizing mechanism. Instability at Low Settler Angles In discussing the instability mechanism at low angles reference is made to the sketch of the flow profiles shown in Figure 3 for the clarified layer and feed suspension layer. The thin sludge layer is neglected here and the sketch is for supercritical operation. If, as assumed, the presence of the inflectional point in the flow is the destabilizing mechanism, then there should be a critical Reynolds number based on the velocity gradient at the interface at which the flow will become unstable and break down. It may be defined by
where v is the fluid kinematic viscosity. For stably stratified miscible fluids in inclined channels, Macagno and Rouse (1962)showed this critical gradient Reynolds number to be the governing parameter and to be approximately constant for small inclination angles. For the present
[
(1
+ 2A) X
- 2(1 + A)(
i)+ - h/H) 11 (4)
Here, the dimensionless feed layer thickness A (= is less than 'I2 for the supercritical mode (with A O(1)). The variation in thickness of the feed layer along a channel of length L is of the order of HIL,which is very small for high aspect ratio channels. Since the wavelength of the disturbance is small compared with the channel length, we can assume the thickness of the velocity profiles for purposes of the stability analysis to be locally constant. The flow rate Q (= Q1 = Q2) is found by integrating eq 3 and 4, from which 1 (Pz - P1) g sin 8 b3(1 -
Q=3-
111
i)3
(5)
It followsAfromeq 2 and 5 that we may write to the first order in A/H (Re,)crit = ( 6 ~ 3 l / ~ )
Qcrit4I3
113
(6)
Hv213( 2 g sin 8)
where Ap = p2 - p1 and Y = pl/pl. Figure 4 is a plot of the flow data for different settler inclination angles expressed as the gradient Reynolds number Re, as a function of the flow Reynolds number. The results are striking in that they show for low angles a constant critical interface Reynolds number of about 50. On the other hand, for the more steeply inclined settlers the critical Reynolds number drops sharply, showing
70
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 B
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Gradient Reynolds number as a function of flow Reynolds number for different settler geometries and settler angles, showing constant critical interface Reynolds number for low settler angles. Figure 4.
PI
Figure 6. Measurements of flow rate for onset of interface turbulence as a function of density difference between clarified layer and feed layer at low and high settler angles and for different settler geometries.
flow, is the chief cause of the instability, while the transverse component (Ap/pl)g cos 8 is stabilizing. It suggests immediately that the critical flow rate Qcrit cot 0. A detailed linear stability analysis has been carried out (Leung, 1981) for the two-layered lamella system destabilized by gravity waves. Orr-Sommerfeld equations were obtained for the feed and clarified layers and the resulting eigenvalue problem was defined by applying the appropriate interface and wall boundary conditions. The problem was solved for long wave disturbances, that is wave w h y lengths X are large compared to the feed layer thickne_ssA, by expanding to first order in the wave number 27rA/X. The analysis is rather lengthy and only one result is discussed here, in particular the result that the critjcal flow rate for instability to the leading order term in A / H is given by
-
I
r
I O
5
L
-
2
'
1
'
6
1
8
4 IC
Recrit=
CHANNEL HEIGHT, H(cm)
Figure 5. Measurementa of flow rate for onset of interface turbulence aa a function of channel height at a low settler angle (5') and high settler angles (15-55').
clearly that another destabilizingmechanism is responsible for the instability. According to the result of eq 6, the critical flow rate Qdt should vary as the 3/4-power of the channel height H. Figure 5 is a plot of the variation of QtWb with H at a 5' settler angle, and it clearly shows the 3/4-power behavior a t small channel angles. Another sensitive verification of eq 6 is the 1/4-power dependence of the critical flow rate on the density difference between the clear and suspension layers. Figure 6 is a plot of QtWb as a function of Ap/pl at a 5' settler angle and the excellent agreement with a 1/4-power dependence is striking. According to eq 6 the flow rate for unstable operation should vary as (sin This is a very weak effect of angle since (sin 8)1/4 ranges between 0.36 and 0.65 for 8 between 1 and loo. The observation that Qmit is in fact fairly constant with 8 at low angles was already shown in Figure 1. Experiments to check the variation of Qcritwith the viscosity Y have not as yet been carried out. Instability at High Settler Angles At high settler angles it is clear that the difference in specific densities between the clarified and feed layers Apjpl leads to a gravitational instability. The longitudinal component of the densimetric gravitational acceleration (Ap/pl)gsin 8, which is the driving force for the buoyancy
Qcrit = 140 cot 8( u 57
k)
(7)
The critical Reynolds number given by eq 7 may be compared with an analogous result for the stability of a thin liquid layer flowing down an inclined plane (Benjamin, 1957, Yih, 1963), for which 5 Recrit= - cot 8 (8) 6 In both eq 7 and 8, Recrit cot 8, indicating that the component of gravity along the inclined channel destabilizes the fluid layer interface at which there is a finite shear stress or the free surface of the thin liquid layer. Using the theoretical relation between flow rate and feed layer thickness given by eq 5, we can rewrite the critical flow rate eq 7 in the form
-
Qcrit Y
-
cot 8(
zyt p-g
sin 8
)"'
(9)
P1
It should be noted that the velocity profiles (eq 2 and 3) used above and in the low angle instability analysis are derived on inertial free considerations, but the stability arguments given imply the importance of inertia. If the Reynolds number is identically zero, the flow will of course always be stable. On the other hand, for small but finite inertia this need not be so. The stability of the type of flows considered here, with small inertia, can be analyzed by carrying out a regular perturbation in the Reynolds number on a zero Reynolds number base flow. Such an analysis has been made for a thin liquid layer flowing down
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 71 4 6
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I
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i
I
SETTLER
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L : 65cm W i 25cm Chonnel S p a c i n g
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Figure 7. High angle measurements of flow rate for onset of interface turbulence as a function of settler angle for different channel heights.
an inclined plane (Yih,1963). It was shown that the effect of inertia on the instability can be represented as a nonlinear, order one function of the Reynolds number and wavelength. For high inclined plane angles and long wavelengths this function reduces to the Reynolds number criterion given by eq 8. Similar conclusions can be drawn for the stratified flows studied here. It is likely that the main inertial effects are confined to the disturbances at the interface and are of only secondary importance to the stability of the base flow. Referring to Figure 1,it can be seen that at high angles the drop in flow rate at which the interface becomes turbulent closely follows a cot 8 behavior. Further support for the cot 8 behavior is given by the experimental results at high settler angles shown in Figure 7 where Qturb is plotted against e for various channel spacings of settler B. (See geometry shown in Figure 4). The agreement of Qbb with a cot B behavior at high angles is excellent. This is @ accord with the remarks made above and with eq 7, were A / H independent of 8. However, as seen from eq 9, this is not the w e . In this regard, it should be noted that the stability result of eq 7 is based on a calculation uFing velocity profiles derived under the assumption that A O(H). o n the other hand, in the flow instability experimenta, A
