Gtheta Computations using UDF
Methodology
A User Defined FLUENT script (UDF) was created to extract the post processing result from individual cell of the model.
There are three methods in calculation the Gtheta values.
The first method is to extract the individual G and theta values from the cell and sum up all the cells to obtain the total Gtheta. The total Gtheta is then divided by the total cells to get the average Gtheta.
∑(Gcell*Өcell)/N (1)
The second method was to extract individual G value and take the average of the G value. The theta is calculated by volume divided by inlet flow rate.
Өfloc = Vol./(Vol. flow rate)
∑(Gcell)/N* Өfloc (2)
The third method is multiple the sum of the G-value and the sum of the theta value and divide by the number of cells.
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G_theta value from the flocculator. The UDF loops through all of the individual cells of the converged solution, and extracts the energy dissipation^1/2 times the cell area. The sum of this quantity for all cells divided by flow and square of viscoisty results in the G_theta value of the entire flocculator as shown in equation 1.
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Gθ_baffle=1/Q*vis^(-1/2)*∑(ε^(1/2)*cell_area) Eq. 1
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The above formulation has been determined (with Monroe's help) by weighting the value of Gθ by volume.
The Gθ_baffle values weighted by the area or number of cells were also examined, but these formulations do not produce sensible results.
Gθ Computation
Below is the Gθ values for flocculation tanks of different geometries:
Case Geometry (h-height,b-width,N-baffles) | Gθ_Flocculator |
|---|---|
b=.1, h=1, N=1, Clearance height=.15 | 430 |
b=.1, h=.3, N=5, sym bc | 1330 |
b=.1, h=.2, N=5 | 1060 |
Normalizing the Gθ parameter
The Gθ value increases with increasing flocculation tank volume. Thus, when comparing different designs, each case should be normalized by the volume (or area for 2D geometries). This results in the following normalized values.
Case Geometry (h-height,b-width,N-baffles) | Gθ_Flocculator/m^2 | |
|---|---|---|
b=.1, h=1, N=1, Clearance height=.15 | 4,300 | |
b=.1, h=.3, N=5, sym bc | 8,870 | |
b=.1, h=.2, N=5 | 10,600 |
Comparison to Flocculation Tanks
Literature rates flocculators based on the Gθ value, as well as the G, epsilon value, and θ-value. For flocculators without recirculating solids, the recommended Gθ is 20,000-150,000 (Schulz, C. R. and D. A. Okun (1984). Surface Water Treatment for Communities in Developing Countries, John Wiley & Sons). This would correspond to 2-15 m^2 of flocculator area based on the weighted Gθ-values calculated. This seems to be the area of flocculators currently used in practice in Honduras. Thus, the calculated Gθ-values seems sensible.
Another check of the accuracy of the results can be seen by comparing the dissipation rate of the flocculation tanks to values recommended by Schulz and Okun of .4-10 mW/kg. The energy dissipation plotted in this region for the fh=3 case results in the plot shown below:
Figure 1: Energy Dissipation Rate in the range of .4-10 mW/kg for fh/w=3, Re=10,000
The regions in white are outside of this region, indicating that energy dissipation values above and below this recommended region exist in the flocculator
Method | Gtheta Value for k-e realizable 1_5 W one baffle |
|---|---|
Ave of Cell's Product (1) | .6698 |
Ave G* Floc residence time (2) | 45 |
Ave G*Ave N (3) | .617 |
Gtheta was also plotted using the custom function in FLUENT.
| Wiki Markup |
|---|
\\ !aGtheta plot.png!
GӨ = Const. \[ε/υ\]^1/2^ \* \[Cell Volume/Vol. flow rate\] (4)
\\ |
With Const.=1, the Gtheta values range from 0 to 430 with the highest values at the boundary and in the separation regime. The plot ranges from 0 to .1 for contrast.
Conclusion
The Gtheta that compares best to mechanically mixed floculators is the Gtheta calculated from method 2, where the G value is averaged, and this averaged value is multiplied by the residence time of a particle. As suggested in discussions with Monroe, this simplistic definition fails to incorporate the weighted flow average.
The values for the other methods represent this weighted flow average, however they are much smaller. This reflects how most of the cells are in extremely low flow stagnant conditions. This is not physically the case and represents how the converged solution represents the time-averaged value. The fluctuations which would decrease resonance time and increase shear in all regions are not present in the flow in FLUENT.