### Fluid Mechanics

# Flow Rate of Water through Ion-Exchange Column

Figure shows a water softener in which water trickles by gravity over a bed of spherical ion-exchange resin particles, each 0.05 inch in diameter. The bed has a porosity of 0.33. Calculate the volumetric flow rate of water.

Calculations:

Data:

m = 1 cp = 1 x 6.72x10^{-4} lb/(ft.sec)

e = 0.33

r = 62.3 lb/ft^{3}

D*x* = 1 ft

g = 32.2 ft/sec^{2}

Formula:

Applying Bernoulli's equation from the top surface of the fluid to the outlet of the packed bed and ignoring the kinetic-energy term and the pressure drop through the support screen, which are both small, we find

g(Dz) = h_{f}

h_{f} = Dp/r

For laminar flow, (Blake-Kozeny Equation)

Calculations:

Therefore, V_{s} = 32.2 x 1.25 x (0.05/12)^{2} x 0.33^{3} x 62.3 / ( 150 x (1 x 6.72x10^{-4}) x (1 - 0.33)^{2} x 1)

= 0.035 ft/sec = 0.011 m/sec.

Q = AV_{s} = (2/12)^{2} x (p/4) x 0.035 = 0.00075 ft3/sec = **21 cm ^{3}/sec**.

Before accepting this as the correct solution, we check the NRe_{m}.

NRe_{m }= (0.05/12) x 0.035 x 62.3 / (1 x 6.72x10^{-4} x (1 - 0.33) ) = 20.2

This is slightly above the value of 10 (up to which the Blake-Kozeny Equation can be used), for which we can safely use without appreciable error.