IV. DIAGNÓSTICO
4.25 DISTRITO FISCAL DE PASCO
5.1 Summary and Conclusions
A simple mechanistic model is developed for the LHC. The model is capable of predicting the hydrodynamic flow field of the continuous phase within the LHC. The separation efficiency is determined based on droplet trajectories, and the inlet-underflow pressure drop is predicted using an energy balance analysis. A user friendly computer code is developed based on the proposed model. The code provides easy access to the input data and very fast output, and can be used for the design of LHC by the industry.
The prediction of the proposed model are compared with elaborated published experimental data sets. Good agreement is obtained between the model predictions and the experimental data with respect to both separation efficiency and pressure drop.
A summary of the tasks performed during this study and the most important conclusions are described as follows.
§ A set of correlations are developed to predict the hydrodynamic flow behavior of the continuous phase in the LHC. The swirl intensity, which is the ratio of tangential momentum flux to the average axial momentum flux, can be predicted using a modified form of Mantilla (1998) model, incorporating the semi-angle of a conical section and adjusting the inlet factor for LHC geometry. Good agreement with experimental data is
observed for a small angle range, from 0º to 0.75º. These are the range of values used in the Colman and Thew (1988) Design.
§ It is confirmed that the swirl intensity defines the velocity field within the LHC. The tangential velocity exhibited a forced vortex near the axis and free-like vortex in the outer region. This behavior and the order of magnitude are well predicted by the model, utilizing some of the parameters of the Rankine Vortex Equation used by Mantilla (1998) and Algifri et al (1988).
The axial velocity, which shows a reverse flow in the core region, is predicted by a third order polynomial equation, as suggested by Mantilla (1998). A modification of the relationship between the reverse flow radius and the swirl intensity is proposed. The prediction of the downward flow by the model is excellent as opposed to the reverse upward flow. No attempt to correct this is done in this study mainly because the critical parameter considered by the model is only the downward flow, where the separation is achieved.
The radial velocity is predicted using the continuity equation and wall conditions suggested by Wolbert et al. (1995). There are no data available for this velocity component.
§ A droplet trajectory analysis is developed assuming local momentum equilibrium. The only forces acting on the droplet are the centripetal and drag forces in the radial direction. For simplification it is assumed that the droplet moves at the fluid velocity in the axial and tangential directions.
§ Based on the droplet trajectory the separation efficiency of the LHC is determined using a similar procedure proposed by Wolbert et al. (1995). The underflow purity can be computed for a given feed droplet size distribution.
Through comparison with 17 cases, where the characteristic diameter of the hydrocyclone, Dc, varies from 20 to 58 mm and the flowrate ranges from 32 to 250 lpm, the model predicts the underflow purity with an average relative absolute error of 4%. One of these cases is the study by Wolbert et al. (1995), where their model predicted 90% of underflow efficiency, while the experimental results reported 81%. However, the proposed LHC model predicted 76% underflow efficiency for this same case. This may suggest that in general the present model predicts a more realistic velocity field within the LHC.
§ Based on the velocity field of the continuous phase and using an energy balance equation, the pressure drop is predicted by the model. Comparison with 20 data points reveals an average relative absolute error of 11.1% and an average relative error equal to –7.9%. The pressure drop is compared not only with the Colman and Thew’s Design but also with the Young’s (1993) Design, and good results are also achieved for the latter. It is important to mention that Young’s design has a conical section of a 3º semi-angle, what goes beyond the range for which the velocity correlation was developed.
After a critical analysis of several experimental data available for the LHC, it is possible to conclude that the LHC mechanistic model predicts with a good confidence level the performance of liquid hydrocyclones with geometrical proportions similar to the Colman and Thew’s (1988) Design.
5.2 Recommendations
The developed mechanistic model has proven to be a good tool to predict the performance of various LHC sizes, for Colman and Thew’s Design. Unfortunately, most of the experimental data published to date comes from this LHC Design. In order to use the current model as a design tool, further comparisons with experimental data from different designs are needed.
Some recommendations that may improve the performance of the model and help to understand the limitations of its application are as follows.
§ Acquire local velocity measurements for the axial and tangential velocity distribution at different tapered section angles, from 0º to 10º semi-angle section. These data can be used to improve the set of correlation that defines the LHC flow field.
§ The axial velocity profile needs to be further investigated, since under high values of swirl intensity double reversal may occur, for which the equation that the model uses will no longer be valid.
§ The model assumes a stable core. However, vortex instability may occur under certain conditions, as confirmed by Weispfenning and Petty (1991). They found that this phenomena is strongly dependent on the swirl
intensity and a characteristic Reynolds Number. Knowledge of the swirl intensity values where these undesirable conditions occur will provide a realistic range of applicability of the model.
§ This proposed model does not consider recirculation zones or short circuits at the inlet. These two phenomena cause either the return to the main flow of some of the fluid that goes with the oil core to the overflow outlet, or cause the feed to go directly to the reject orifice. These conditions may affect to some degree the separation efficiency, and they have to be included in order to have a more robust model.
§ The model does not consider the overflow to underflow split ratio. This parameter is crucial for a desirable operation of the LHC but does not affect considerably the LHC flow field. At this point the model assumes that the split ratio is sufficient to accommodate the volume of oil that is separated and that the efficiency does not change with the split ratio, as many researchers have reported. This assumption may be true in the typical range of operation of the LHC, namely, 1 to 10 %, but outside this range a change of the velocity field may occur, and that must be accounted for.
§ There is a relationship between the swirl intensity and the reverse flow radius. As shown by the experimental data and followed by the model’s prediction, the reverse flow radius is reduced as the swirl decays. But there is a point where there is no longer reverse flow and still some swirling motion can occur. Under this condition the model will still consider a
reverse flow. A proper improvement will be to know for which small swirl intensity values the flow will not exhibit reverse flow and incorporate this aspect in the model.
Finally, to use this model as a design tool, a good prediction of the swirl intensity with the axial position for different taper sections is crucial. It is believed that in the small angle tapered section the swirl intensity decreases at a slower rate as compared to a cylindrical section. Nevertheless, a point can be reached at the conical section where lower values of swirl intensity are generated, as illustrated in the next hypothetical diagram.
Figure 5.1 Hypothetical Swirl Intensity Decay
At this point, not sufficient information is available to confirm this notion, but it is important to note that the swirl intensity is crucial for the LHC performance and also for design purposes. If a model is able to predict accurately the swirl intensity, this can be used as a design parameter, where an optimum design will be the one with the highest possible swirl intensity.
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A = cross sectional area
B = factor that determines the peak tangential velocity radius
CD = drag coefficient d = droplet diameter D = diameter Dc = LHC characteristic diameter f = friction factor h = losses I = inlet factor k = concentration L = length
m& = mass flow rate
Mt = momentum flux at the inlet slot
MT = axial momentum flux at the characteristic diameter position
n = centrifugal force correction factor
P = pressure
Q = volumetric flow rate
r = radial position
Re = Reynolds Number
t = time
Tm = maximum momentum of the tangential velocity at the section
u = continuous phase local axial velocity
U = bulk axial velocity
v = continuous phase local radial velocity
V = volumetric fraction / velocity
Vr = droplet radial velocity
Vsr = droplet slip velocity in the radial direction
Vz = droplet axial velocity
w = continuous phase local tangential velocity
W = mean tangential velocity
Greek Letters:
= swirl intensity
= taper section semi-angle
= efficiency / purity / pipe roughness
= axis – horizontal angle
= viscosity = density
Subscripts:
av = average
c = characteristic diameter location / continuous phase
cf = centrifugal
crit = critical
d = dispersed phase / droplet
f = frictional g = gravity acceleration i = inlet is = inlet section o = overflow r = resultant
rev = reverse flow
u = underflow
z = axial position
Abbreviations:
CFD = Computational Fluid Dynamics
ESP = Electric Submergible Pump
LDA = Laser Doppler Anemometry
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