In depth-air filtration modeling, the characteristic thickness X10 satisfies X10 = 2.303 / K.\r On which parameters does K (the first-order removal constant per unit depth) primarily depend?

Introduction:
Depth filtration models frequently assume an exponential decrease in microbial or particle concentration with penetration depth. The constant K captures capture efficiency per unit thickness and embodies fiber properties and hydrodynamic conditions. Understanding what determines K guides media selection and operating setpoints.

Given Data / Assumptions:

• Model form: C(x) = C0 * exp(−K * x).
• X10 = 2.303 / K gives the depth for 90% reduction.
• We consider steady, uniform flow through a homogeneous fibrous medium.

Concept / Approach:
K depends on the probability of capture per unit path length, which is dictated by fiber diameter, packing density, surface chemistry (nature of the material), and by the approach velocity, which alters residence time and transport mechanisms (diffusion, interception, impaction). Therefore, both the nature of the filter and the linear velocity of the bypassing air influence K. The thickness itself is the integration variable and does not define K.

Step-by-Step Solution:
Relate K to micro-scale capture coefficients that depend on fiber/media properties.Relate K to flow rate via velocity: lower velocity → longer residence → higher effective K.Conclude that both media characteristics and air velocity govern K.

Verification / Alternative check:
Single-fiber theory expresses overall efficiency as a function of packing density and velocity-dependent dimensionless groups (Re, Pe, NSt), all embedded in K.

Why Other Options Are Wrong:

• Filter thickness: Appears in X10 but does not determine K; it is the depth variable.
• Temperature only: Secondary unless it significantly changes air properties and mechanisms.

Common Pitfalls:
Assuming K is a constant independent of operating conditions; scaling velocity or changing media often requires recalibration.

Final Answer:
both (a) and (b)