MATHCAD Design of Filter and Clear Well

[MATHCAD File. | ^5.04.xmcd]

[PDF version of our MATHCAD File. | ^Mathcad - 5.20.pdf]

Mathcad Code Explanation:

Target Input:
This is where the user inputs:
* The plant flow rate (Q)
* Number of granular Filters (N)
* Backwash Time  between 5-10 min (Tbw)
* Percent the Sand Bed expands (Texpansion)

Condition Input:
This is where the parameters of the construction materials are entered:
* Diameter of particles, depth of bed, porosity, specific gravity, and d60 of sand and gravel.
* Water Treatment Filtration Rate
* Diameter of backwash orifice, clear well, and piping (which should be moved to fluid functions)
* Roughness and length of piping
* Density, and dynamic and kinematic viscosity of water

 
Calculations:
First we determine the size of the filter and head loss in normal filtration. Then we use two approachs to find the area of the clear well . The first is based upon conventional knowledge that the back wash velocity needs to be 10 times the filter velocity. From there we know  that the space between the top of the filter bed and the bottom of the clear well is linked directly to the amount of headloss between those two points (headloss from the pipe, elbows, expansion and contraction, expansion of the bed). To do this, we used information from Okun  and the "Minor Losses in Pipes" Manual.

The second approach is to use the empirical formula, called the Weber Equation.  The velocity for filtration is based on the sand's d60 (only works with sand) and the specific weight of both water and sand. Then the velocity for backwash is based on the filtration velocity, the sand's porosity and the ratio of height of the expanded and unexpanded filter bed.

min fluid

Once the back wash velocity is determined, solving the clear well and filter dimensions are the same process.
Velocity Required to Fluidize the Bed

Design Output for Agalteca Plant, Q=6.3 L/s:  

Mathcad Conclusions:

1) Our design based on simple hydraulics will work. However, it is a very large filter and will not be sustainable economically. The material cost for construction will be too high.

2) The design based on the empirical Weber equation (instead of the conservative approach) is much smaller and less expensive. However, the validity of the empirical equations is not yet certain. Therefore testing needs to be done in bench and pilot scale models.

3) If the empirical equations are valid, then we can change parts of the design, by changing the sand parameters. For example, lower BW velocity by lowering the d60 and specific weight of the media.