GATE 2019 BT Q.53: solution and answer

GATE 2019 · BTQ.53 · 2 marks · NAT

In a cross-flow filtration process, the pressure drop ( $Δ P$ ) driving the fluid flow is 2 atm, inlet feed pressure ( $P_{i}$ ) is 3 atm and filtrate pressure ( $P_{f}$ ) is equal to atmospheric pressure. The average transmembrane pressure drop ( $Δ P_{m}$ ) is ________ atm.

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Answer and solution

ANSWER

The average transmembrane pressure drop is $Δ P_{m} = 1 atm$ .

Official answer: 1Source: Official answer key, GATE 2019 BT (IIT Madras)

In cross-flow (tangential-flow) filtration the feed flows along the membrane surface. The pressure drop $Δ P$ that drives this flow along the channel is the fall from the inlet feed pressure $P_{i}$ to the outlet (retentate) pressure $P_{o}$ : $Δ P = P_{i} - P_{o}$ .
Retentate outlet pressure: $P_{o} = P_{i} - Δ P = 3 - 2 = 1 atm$ .
The feed-side pressure falls along the membrane from $P_{i}$ to $P_{o}$ , so the average feed-side pressure is taken as the arithmetic mean of the inlet and outlet values: $\overset{ˉ}{P}_{f ee d} = \frac{P _{i} + P _{o}}{2} = \frac{3 + 1}{2} = 2 atm$ .
The filtrate (permeate) side is at atmospheric pressure. Since the feed pressures are given as absolute values, $P_{f} = 1 atm$ .
The average transmembrane pressure drop is the average feed-side pressure minus the filtrate pressure: $Δ P_{m} = \frac{P _{i} + P _{o}}{2} - P_{f} = 2 - 1 = 1 atm$ Equivalently, $Δ P_{m} = P_{i} - \frac{Δ P}{2} - P_{f} = 3 - 1 - 1 = 1 atm$ .

KEY IDEA

In cross-flow filtration, $Δ P = P_{i} - P_{o}$ is the pressure drop along the channel and $Δ P_{m} = \frac{P _{i} + P _{o}}{2} - P_{f}$ is the average pressure difference across the membrane. Put together, $Δ P_{m} = P_{i} - \frac{Δ P}{2} - P_{f}$ .

WATCH OUT FOR

Taking atmospheric pressure as $P_{f} = 0$ (a gauge value) while $P_{i} = 3 atm$ is an absolute value. That gives $2 atm$ . The other common slip is subtracting $P_{f}$ from the inlet pressure alone ( $3 - 1 = 2 atm$ ) instead of from the average feed-side pressure, which ignores the pressure drop along the membrane.

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