In Fig. 3.4, the vertical black line marks the approximate transition (“knee”) between the pressure-dependent and pressure-independent filtration regimes. According to Gottschalk’s rule of thumb, microfiltration should be operated:

Question

BlackTom AI · Accepted Answer

Gottschalk’s rule of thumb recommends operating microfiltration near the transition (“knee”) between the pressure‑dependent and pressure‑independent regimes.

1) Relevant concepts
- Flux (J) vs transmembrane pressure (TMP) has two regimes: a pressure‑dependent region (flux rises with TMP) and a pressure‑independent region (flux plateaus due to concentration polarization and fouling/cake formation).
- Cross‑flow velocity mitigates polarization and raises the plateau flux, but it cannot eliminate the limiting mechanisms that cause the plateau.
- The “knee” is the TMP value where additional pressure gives rapidly diminishing increases in flux.

2) Step‑by‑step reasoning
- Starting at low TMP, raising TMP increases driving force and flux roughly linearly (pressure‑dependent). This is beneficial up to the point where polarization and cake build-up begin to limit flux.
- Past the knee, further TMP increases produce little extra permeate (pressure‑independent) but accelerate fouling and cake compaction, increasing cleaning frequency and operating cost.
- Therefore the ideal operating point balances maximizing flux against minimizing fouling risk and energy/cleaning penalties.

3) (No numeric calculations required) Intermediate conceptual steps: identify knee on the plot → choose TMP just at or slightly below that knee → ensures near‑maximum flux without pushing into the plateau where fouling dominates.

4) Why the correct answer is correct
Operating near the knee captures most of the achievable flux while avoiding the large increases in fouling rate and diminishing returns that occur at higher TMP.

5) Why the other choices are incorrect
- “Exactly at zero TMP” is wrong: flux would be essentially zero (no driving force).
- “Far right (high TMP)” is wrong: yields little extra flux but much more fouling and higher costs.
- “Far left (low TMP)” is wrong: avoids polarization but gives unnecessarily low flux and poor productivity.

6) Conclusion
This reasoning confirms the correct choice: Near the line (at the transition) to maximize flux while limiting fouling risk.

In Fig. 3.4, the vertical black line marks the approximate transition (“knee”) between the pressure-dependent and pressure-independent filtration regimes. According to Gottschalk’s rule of thumb, microfiltration should be operated: 单项选择题

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