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# HW 5 - CHEN 4820 Fall 2010 Homework 5(filtration...

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CHEN 4820 Fall 2010 Homework 5 (filtration equipment) (1) Harrison prob 4.4. (scale-up of rotary vacuum filter) Note: interpret “How significant is the medium resistance” to mean how resistance from the filter medium alone compares to the resistance from the filter cake. Ans: see Harrison key. (2) Harrison prob 4.6 (feed and bleed microfiltration). Note – in this problem, the variables x f , x r , and Q f are “known”. Also, how Q p depends on x, i.e. Q p (x), is “known” (so how x depends on Q p is also known). Use material balances for the flows and for the cells to find the unknowns, x and R, in terms of these knowns. Hint: use material balances drawn around the entire system and use material balances around the point where the recycle stream and feed stream join together. Ans: First get an expression for Q p in terms of knowns, by using the whole system balance for streams and for cells. Whole system balance for streams: Q p = Q f – Q r Whole system balance for cells: x f Q f = x r Q r (there are no cells in the permeate stream) Combine them and solve for Q p : g1843 g3043 = g1843 g3033 ( 1 g3051 g3281 g3051 g3293 ) . Everything on the right side is known, so this eqn yields a value for Q p . However, Q p is a known function of x, therefore it can be inverted, i.e. we can get a value for x given a known value for Q p . With the known relationship, x = x(Q p ), we get a value for x. With x now known, develop an expression for R, the recycle ratio, in terms of other known variables, including x, from material balances applied to the point where the recycle stream and the feed stream merge: Balance for streams at merger point: Q f + RQ r = Q . Balance for cells at merger point: x f Q f + x r RQ r = xQ . Multiply the first balance by x on both sides, then combine this equation with the second balance to eliminate the xQ term: xQ f + xRQ r = x f Q f + x r RQ r . Rearrange this eqn to solve for R: g1844 = ( g3051 g2879 g3051 g3281 ) ( g3051 g3293 g2879 g3051 ) g3018 g3281 g3018 g3293 . Use the whole system balance for streams again to eliminate Q r , leaving: g1844 = ( g1876 g1876 g3033 ) ( g1876 g3045 g1876 ) g1843 g3033 ( g1843 g3033 g1843 g3043 ) Everything on the right side is known, so now R is known.

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Summary of the overall algorithm: get Q p in terms of known quantities; get x from Q p ; get R in terms of x and other known quantities. (2) How bleed concentration depends on recycle ratio for feed and bleed: For the feed and bleed scheme in Harrison Figure P4.6, the operation must arrive at the right recycle ratio, R, to achieve the intended degree of concentration in the bleed stream relative to the feed stream, x r /x f . Here, R is the ratio of the flowrate of the recycle stream to the flowrate of the bleed stream (the bleed stream flowrate is Q r in figure P4.6). If R is too low, the permeate flowrate is insufficient to achieve the desired concentration. If R is too high, the permeate flowrate overshoots and the bleed stream is too concentrated.
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HW 5 - CHEN 4820 Fall 2010 Homework 5(filtration...

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