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The basic premise of the stacked filtration system is that the flow of filtration is equal to the flow of backwash so that we can use normal plant flow to backwash the filter. A conservative estimate of backwash velocity requires that it be 10 times the normal filtration velocity. We achieve this requirement by stacking layers of filtration on top of each other. Each layer, or plane, consists of a set of inlet tubes that introduce water to a layer of 20 cm of sand. The water once filtered is then collected by a set of outlet tubes. Each layer is essentially its own filtration system. When you stack the layer of filtration on top of each other, area for backwash stays the same and you can technically backwash all of them with the same backwash water. Figure 1 Basic Concept of Stacked Filtration Operation and the mathematical derivations demonstrate this relationship.

{latex}${{Q_{Plant} } \over 2} = V_{BW} A_{BW} ${latex}

{latex}${{Q_{Plant} } \over {N_{Filter} }} = V_{Filter} A_{Filter} ${

{latex}${\rm{V}}_{{\rm{BW }}}  = 10{\rm{V}}_{{\rm{Filtration}}} ${latex}

Where

BW=Back Wash
Q=Flow rate of filtration, backwash, or entire plant depending on the subscript.
V=Velocity of either filtration or backwash depending on the subscript.
A=Area of either filtration or backwash depending on the subscript.
N=Number of any system, pipe, and etc which in this case is the number of filtration unit that receives the plant flow rate.

For our design, we chose a conservative available backwash flow rate of only half of the plant design flow rate.
We want the backwash flow rate to equal the filtration flow rate. So we arrange the first two equations for the plant flow rate and set them equal to each other.

{latex}$V_{Filter} A_{Filter} N_{Filter}  = Q_{Plant}  = 2V_{BW} A_{BW} ${latex}

When we substitute the 10 to 1 relationship between backwash and filtration velocity into the above equation, we derive the following relationship with regards to area.

{latex}$V_{Filter} A_{Filter} N_{Filter}  = 2{\rm{x10}}V_{Filter} A_{BW} ${

{latex}$A_{Filter}  = {{20} \over {N_{Filter} }}V_{Filter} A_{BW} ${latex}

With the velocity of filtration cancelling each other, we learn that in order to use the same flow rate to backwash and filter we need the area of the filtration to be 10x the time area of backwash. Consequently, if we were to have two filters, then we would need 10 filters stacked on top of each other. If we were to have four filters, like our design, then we would have 5 layers for each filtration system.

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