A grain kernel is modeled as a triaxial ellipsoid with semi-axes $a = 4.0\text{ mm}$, $b = 2.5\text{ mm}$, and $c = 1.5\text{ mm}$. The sphericity (degree of sphericity $\phi = \frac{(a b c)^{1/3}}{a}$) of the grain is ________ (round off to 3 decimal places).
Answer Key:
0.598
Step-by-Step Solution & Explanation
1. Geometric mean diameter $D_g = (a \cdot b \cdot c)^{1/3}$:
$a b c = 4.0 \times 2.5 \times 1.5 = 15.0\text{ mm}^3$
$D_g = (15.0)^{1/3} \approx 2.4662\text{ mm}$
2. Sphericity $\phi = \frac{D_g}{a}$:
$\phi = \frac{2.4662}{4.0} \approx 0.61655$
Wait: for axes lengths $L = 2a = 8.0, W = 2b = 5.0, T = 2c = 3.0$, $(L W T)^{1/3} / L = (120)^{1/3} / 8.0 = 4.9324 / 8.0 = 0.6166$. With semi-axes $(abc)^{1/3}/a = (15)^{1/3}/4.0 = 2.4662 / 4.0 = 0.6166$. Let's recheck with standard definition: $\phi = \frac{(L W T)^{1/3}}{L} = 0.617$. Setting correct_answer: "0.617", range min: 0.610, max: 0.625.