Methods

Methods

Our design process consisted of 3 major steps. First we designed the individual filter bed itself based on the relationship equations mentioned in the theory section. Second, we sized the pipe in our system so that the head loss experienced by the pipes is never greater than 10% of the head loss experienced by the sand. If the head loss in the sand is not greater than the head loss in the pipes, there will be preferential flow and not all of the pipes in the manifold will have equal flow. Third step was to design the minimum distance between the entrance pipes from the sedimentation tank to the height of the gutter to ensure proper back wash.
In our first step, we chose a conservative back wash velocity 14 mm/s that we know empirically to raise a layer of sand bed 30%. We also chose a conservative back wash flow rate of 3.15 L/s which is half of the maximum plant flow rate at Agalteca. Based on that given parameter, we solved for the area of the filtration bed as shown below.

We made the design a square with a side length of 0.474m which would also be the length of the filtration inlet and outlet tubes. In our second step, we solved for the head loss that occurs in the 20 layers of sand using the Carmen-Kozeny Equation shown below:

Where
h=head loss.
L=length of layer which in this case is 20 cm.
K=Kozeny constant of 5.

=porosity of sand=0.4 for our case.
g=acceleration of gravity.
d=diameter of sand which in this case is 0.55mm.
V=velocity of the fluid passing through the filter
ν=kinematic viscosity=10-6 m2/s
We calculated that the head loss is around 10cm. Since we want the head loss in the sand to dominate, we design all of our pipes to have large enough diameter so that their head loss never exceeds 10% of the sand's head loss. We utilized the manifold equation as shown below:

Where the variables not defined previously are as follows:

N = number of ports
M = Manifold
P = Port
D = Diameter as dictated by the subscript
F =friction factor

As the MathCAD file shows, we calculated the inlet and outlet tubes to be ½ inch and the manifold connecting them to be 3 inches in diameter. Because the entire back wash flow rate must pass through the inlet tubes during back wash, the inlet tubes of the bottom plane need to be 0.75 inch in diameter. The bottom manifold and the rest of the pipe system need to be 3 inch in diameter.
Our last step was to determine the minimum distance between the entrance pipes from the sedimentation tank to the gutter of the filtration unit. The minimum distance must be greater than the head loss that occurs through the pipes and the expanded bed during head loss as shown below in Figure 4 Distance from Entrance Tank to Gutter and in our MathCAD file. We first calculated the head loss in the pipes during back wash to be 6.7 cm. Most of the head loss occurs in the expanded bed calculated as shown below, an Okun equation, as shown on the Review of Existing Research section:

Where the previously undefined variables are
h= Head Loss
H=Height of Unexpanded Bed
SG=specific gravity

Figure 1: Minimum Distance from Gutter to Entrance Pipe

We calculate the head loss to be 2.145 m which means that there must be at least 1.1 m distance from the entrance pipe to the gutter. The effluent pipe of the filtration tank should be slightly higher than the top of the sand filter to ensure that even during the no flow the sand would stay wet. The high elevation of the effluent pipe would essentially trap the water in the filter until the flow of water pushes it over during normal filtration operations.