## Backwashing Head Loss In Granular Filters

In the early development of filters, units that had been clogged were renewed by scraping the topmost layers of sand. The scraped sands were then cleaned by sand washers. In the nineteenth century, studies to clean the sand in place, rather than taking out of the unit, led to the development of the rapid-sand filter. The method of cleaning is called backwashing.

In backwashing, clean water is introduced at the filter underdrains at such a velocity as to expand the bed. The expansion frees the sand from clogging materials by causing the grains to rub against each other dislodging any material that have been clinging onto their surfaces. The dislodged materials are then discharged into washwater troughs.

To expand the bed, a force must be applied at the bottom. Per unit area of the filter, this backwashing force is simply equal to the pressure at the bottom of the filter acting upward upon a column of water with the grains suspended. Atmospheric pressure will also act on this column from its top. If the pressure used for measurement at the bottom is pressure above atmospheric, the pressure acting on the top is considered zero. The backwashing force pressure is then simply equal to the specific weight of water yw times all of the heights hLb + l' + le or yw(hLb + l' + le), where le is the expanded depth of the bed having a depth of l and l' is the difference between the level at the trough and the limit of bed expansion le (see Figure 7.2). hLb is called the backwashing head loss; hLb does not include friction losses through the washwater pipe, contraction loss from the washwater tank into the wash water pipe, valves, bends, and loss at the underdrain system. Thus, in design, these losses must be accounted for and added to hLb.

The weight of the suspended grains is le(1 - ne)YP, where ne is the expanded porosity of the bed and yp is the specific weight of the grain particles. The weight of the column of water is (lene + l')yw. These two weights are acting downward against the backwashing force. During backwashing, the whole column will be at steady state; thus, applying Newton's second law will produce (hLb +1' + le)yw - le (1 - ne)YP -(lene - l')Yw = 0. Solving this equation for the backwashing head loss produces hLb = ^-^OWw) (7.48)

Calling Vb the backwashing velocity and v the settling velocity of the grains, the commonly used expression for ne is the following empirical equation

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