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:

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

ε = 0.33

ρ = 62.3 lb/ft^{3
}

ΔL = 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(ΔL) = h_{f
}

h_{f} = Δp/ρ

For laminar flow, (Blake-Kozeny Equation)

\(\displaystyle \frac{\Delta p}{\rho} = 150\frac{V_s\mu(1-\varepsilon)^2\Delta L}{D_p^2\varepsilon^3\rho} \)

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 (π/4) x 0.035 = 0.00075 ft^{3}/sec = **21 cm ^{3}/sec**

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

NRe_{pm}= (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.

Last Modified on: 04-Feb-2022

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