A system has been designed to remove solid materials from a

A system has been designed to remove solid materials from a thick sludge (a mixture of solids and water). The sludge (A), which is 55% solids (by mass), is fed into a mixer at a rate of 4.5 ft3/h. The sludge has a specific gravity of 1.50. The outflow from the mixer (B) is fed into a settling tank at a rate of 500. lbm/h. The material that settles in the tank (C) contains 95% solids (by mass). The remaining material in the tank is well mixed, containing 2% solids (by mass), and is removed from the tank in two streams (with the same composition). One stream is the final product (D), and the other stream (E) is recycled back to the mixer to dilute the thick sludge.

A) Determine the Mass flow rate of the sludge in lbm/hr

B) Use the systematic solution approach to solve this mass balance problem for the mass flow (in lbm/h) of the settled material (C), the final product (D), and the recycled stream (E), and the composition (in %) of mixer outflow (B).

Solution

The slug is fed into the mixure at the rate = 675 lbm/h

It contains  371lbm/h of solid and 304lbm/h of water.

500lbm of this material contains (371X500)/675 lbm/h of solid=274.81 lbm/h of solid

and 225.19 lbm/h of water. The rate of volume flow in to tank C is 4.08 ft3/h

Let its x ft3/h volume settles in the tank C. Then the mass settled in C is xX1.5 /h and mass of water settled is (4.08-x)/h.

the percentage of settled mass= (xX1.5)/(1.5x+4.08-x)=0.95

which gives 1.025x=3.867 or x= 3.78 ft3/h is the rate of setteling of solid in tank C, the rate of water remaing in tank C is (4.08-3.78) ft3/h=0.3 ft3/h. The mass of solid settles in tank c is 567lbm/h and that of water is 30lbm/h. Remaining material contains[ 675- (567+30)]= 78 mass units of solid. Let the total matrial removed from the tank per hour be y ft3/h , Then ( y-78/1.5) is the rate of volume of water coming out of the tank C. Then 78/ (y-78/1.5) =.02 which gives y=3.52ft3/h. This stream of final product D contains 1.76ft3/h volume flow containing 37lbm/h rate of solid flow,