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.
m= 1 cp = 1 x 6.72x10-4 lb/(ft.sec)
r= 62.3 lb/ft3
Dx = 1 ft
g = 32.2 ft/sec2
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) = hf
hf = Dp/r
For laminar flow, (Blake-Kozeny Equation)
Therefore, Vs = 32.2 x 1.25 x (0.05/12)2 x 0.333 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 = AVs = (2/12)2 x (p/4) x 0.035 = 0.00075 ft3/sec = 21 cm3/sec.
Before accepting this as the correct solution, we check the NRem.
NRem = (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.