For a fibrous filter where multiple capture mechanisms act simultaneously, the overall single-pass collection efficiency η_total is best expressed as which combination of the individual efficiencies by impaction (ηimp), interception (ηint), and diffusion (ηdif)?

Introduction:
When impaction, interception, and diffusion act in parallel on a filter element, the correct way to combine their contributions is crucial for predicting overall filter performance. Each mechanism removes a fraction of the particles that survive the others, leading to a multiplicative survival relationship rather than a simple sum of efficiencies.

Given Data / Assumptions:

• ηimp, ηint, ηdif are single-pass removal fractions for each independent mechanism.
• Mechanisms act concurrently and independently on the same flow element.
• Re-entrainment is neglected for this expression.

Concept / Approach:
Let survival fractions be S_imp = (1 - ηimp), S_int = (1 - ηint), and S_dif = (1 - ηdif). If mechanisms are independent, the total survival is S_total = S_imp * S_int * S_dif. Therefore, overall efficiency is η_total = 1 - S_total = 1 - (1 - ηimp)(1 - ηint)(1 - ηdif). This avoids double counting and stays bounded between 0 and 1 even when individual efficiencies are high.

Step-by-Step Solution:

Verification / Alternative check:
Monte Carlo particle tracking and experimental filtration data support multiplicative survival over additive efficiencies when mechanisms are independent, especially at moderate efficiencies.

Why Other Options Are Wrong:

• 1 - ηimpηintηdif: Overestimates efficiency and mishandles independence.
• 1 - (ηimpηint/ηdif): Dimensionally inconsistent and not physically meaningful.
• None of the above: Incorrect because the standard relationship is listed.

Common Pitfalls:
Summing efficiencies directly can produce η_total > 1, which is impossible. Always combine via survival fractions when mechanisms act in parallel.

Final Answer:
1 - (1 - ηimp)(1 - ηint)( 1 - ηdif)