A steady state filtration process is used to separate silicon dioxide (sand) from water. The stream to be treated has a flow rate of 50 kg/min and contains 0.22% sand by mass. The filter has a cross-sectional area of 9 square meters and successfully filters out 90% of the input sand by mass. As the filter is used, a cake forms, which we will assume is pure sand (SG of sand = 2.25). The filter needs to be replaced once this cake has a thickness of 0.25 meters. Required:
How long can a new filter be used before it needs to be replaced, in units of days?

3.19 days

Explanation:

Given data :

stream flow rate = 50 kg/min

stream contains ; 0.22% sand by mass

Cross sectional area of filter = 9 m^2

Filter successfully filters out 90% of the input sand by mass

SG = 2.25

thickness of cake formed = 0.25 meters

Determine how long a new filter can be used before replacement

Given that for every 1 minute  5.43*10^-5 m thickness of sand layer(cake) forms on the filter and the replacement of filter is done once the cake thickness = 0.25 meters

To determine the number of days ( X ) before replacing filter we apply the relationship below

5.43 * 10^-5 = 1 min

0.25 m = X

hence ; X = 0.25 / (5.43 * 10^-5 )  = 4604.051 minutes ≈ 3.19 days

Attached below is the beginning part of the detailed solution

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