Cumulative undersize mass fraction from mixed particle data\nGiven particles i = 1, 2, 3, … with diameters d_i and masses m_i, what is the expression for the cumulative mass fraction of particles smaller than or equal to a size d_j?

Introduction / Context:
Particle size distributions (PSDs) are commonly represented in cumulative undersize (passing) form. When raw data consist of individual particle masses and sizes, the cumulative mass fraction at a cut size is the mass-based proportion of particles not exceeding that size. This definition underpins PSD plots, mass balancing, and screen efficiency calculations.

Given Data / Assumptions:

• A set of particles indexed by i with measured diameter d_i and mass m_i.
• Mass-based PSD (not number-based) is required.

Concept / Approach:
The cumulative undersize mass fraction at size d_j, denoted F(d_j), is the total mass of all particles with diameters less than or equal to d_j divided by the total sample mass. This directly corresponds to “% passing” if multiplied by 100. Number-based PSDs would instead count particles rather than summing masses, which is a different metric and often less useful for process calculations.

Step-by-Step Solution:

Verification / Alternative check:
Integrating a continuous mass-based size density function up to d_j yields the same cumulative passing function; this discrete form is its sample analogue.

Why Other Options Are Wrong:

• Summing diameters or number counts does not produce a mass-based fraction.
• Using ≥ d_j yields a cumulative oversize fraction, not undersize.

Common Pitfalls:
Confusing number-based and mass-based PSDs; always specify the basis when comparing distributions or setting specs.

Final Answer:
F(d_j) = (Σ m_i for d_i ≤ d_j) / (Σ m_i for all i)