Question

The mass fraction retained on the \(i\)-th sieve is \(x_i\) and \(D_{pl}\) is the average opening size of \(i\)-th and \((i-1)\)-th sieves. The volume surface mean diameter (\(D_s\)) of particles retained on \(n\) number of sieves is __________.

• A. \( D_s = \frac{1}{\sum_{i=1}^{n} \left( \frac{x_i}{D_{pl}} \right)} \)
• B. \( D_s = \sum_{i=1}^{n} x_i D_{pl} \)
• C. \( D_s = \left[ \frac{1}{\sum_{i=1}^{n} \left( \frac{x_i}{D_{pl}} \right)^{1/3}} \right]^{3} \)
• D. \( D_s = \left[ \frac{1}{\sum_{i=1}^{n} \left( \frac{x_i}{D_{pl}} \right)^{2/3}} \right]^{2} \)

Hint:

In sieve analysis, the volume surface mean diameter is calculated by summing the mass fractions retained on each sieve divided by the sieve size. This provides an average particle size based on the distribution.

Answer 1

The Correct Option is A

In this problem, we are asked to find the volume surface mean diameter (\(D_s\)) of particles retained on \(n\) sieves. The formula for \(D_s\) involves the mass fraction \(x_i\) retained on the \(i\)-th sieve and the average opening size \(D_{pl}\) of the \(i\)-th and \((i-1)\)-th sieves. The volume surface mean diameter is a measure used in particle size analysis to represent the average particle size based on the sieve analysis. To calculate \(D_s\), we need to use the following formula: \[ D_s = \frac{1}{\sum_{i=1}^{n} \left( \frac{x_i}{D_{pl}} \right)} \] Where: - \(x_i\) is the mass fraction retained on the \(i\)-th sieve. - \(D_{pl}\) is the average opening size of the sieve. This formula essentially gives us the volume surface mean diameter based on the distribution of particles across multiple sieves. It accounts for both the mass fraction and the sieve opening size. Let's analyze the other options: - Option (B) involves a simple sum of mass fractions multiplied by sieve opening sizes, which does not represent the correct formula for the volume surface mean diameter. - Option (C) and Option (D) include powers of 1/3 and 2/3, which are not applicable to this formula, as the volume surface mean diameter is derived from the sum of the ratios of mass fraction to sieve size. Thus, the correct formula is option (A), \( D_s = \frac{1}{\sum_{i=1}^{n} \left( \frac{x_i}{D_{pl}} \right)} \).