Question

Lab scale tests performed on a cell broth with a viscosity of 5cP gave a specific cake resistance of 1 x 1011 cm/g and a negligible medium resistance. The cake solids (dry basis) per volume of filtrate was 15 g/liter. It is desired to operate a larger rotary vacuum filter (diameter 8 m and length 12 m) at a vacuum pressure of 80 kPA with a cake formation time of 20 s and a cycle time of 60 s. Determine the filtration rate in volumes/hr expected for the rotary vacuum filter?

Answer 1

473,400 L/h

Explanation:

For an incompressible cake and negligible medium resistance, we can find the volume of filtrate collected using the following equation:

`V_(f)^(2) = (2A^(2)t_(f)\Delta P)/(\mu \alpha \rho)` (1)

Where:

`V_(f)`: is the volume of filtrate collected

A: is the filtration area

`t_(f)`: is the cake formation time = 20 s

ΔP: is the vacuum pressure = 80 kPa

μ: is the viscosity = 5 cP

α: is the specific cake resistance = 1x10¹¹ cm/g

ρ: is the cake solid per volume of filtrate = 15 g/L

First, we need to convert the units:

`\mu = 5 cP \cdot (0.01 g/cm*s)/(1 cP) = 0.05 (g)/(cm*s)`

`\rho = 15 (g)/(L) \cdot (1 L)/(1000 cm^(3)) = 15 \cdot 10^(-3) g/cm^(3)`

`\Delta P = 80\cdot 10^(3) Pa = 80\cdot 10^(3) (N)/(m^(2)) = 80 \cdot 10^(3) (kg)/(m*s^(2)) \cdot (1000 g * 1 m)/(1 kg * 100 cm) = 8.0 \cdot 10^(5) gcm^(-1)s^(-2)`

Now, we need to find the filtration area (A), which is equal to a cylinder area:

`A = 2 \pi r^(2) + 2 \pi rh`

where r: is the radius = 8/2 m and h: is the height of the side = 12 m.

`A = 2\pi (4m)^(2) + 2\pi 4m*12m = 4.02 \cdot 10^(6) cm^(2)`

Now, we can calculate the volume of filtrate using equation (1):

`V_(f) = \sqrt {(2(4.02 \cdot 10^(6) cm^(2))^(2)*20 s*8 \cdot 10^(5) gcm^(-1)s^(-2))/(0.05 gcm^(-1)s^(-1)*1 \cdot 10^(11)cmg^(-1)*15 \cdot 10^(-3) gcm^(-3))} = 2.63 \cdot 10^(6) cm^(3)`

Finally, we can calculate the filtration rate in volumes/hr:

`rate = (V_(f))/(t_(f)) = (2.63 \cdot 10^(6) cm^(3))/(20 s) \cdot (3600 s * 1 L)/(1 h * 1000 cm^(3)) = 473,400 L/h`

I hope it helps you!