Question Bank • QB_DFE_HMT_006 NAT • 2 Marks

Section 7: Dairy and Food Engineering

Heat and Mass Transfer

Subtopic: Transient heat transfer in simple geometry

Question Statement:

A spherical meat ball of diameter $D = 30.0\text{ mm}$ ($r_0 = 0.015\text{ m}$) has thermal conductivity $k = 0.45\text{ W/(m}\cdot\text{K)}$ and surface heat transfer coefficient $h = 25.0\text{ W/(m}^2\cdot\text{K)}$ during convective chilling in cold air. The Biot number for the sphere ($Bi = \frac{h r_0}{3 k}$) using volume-to-surface characteristic length $L_c = \frac{r_0}{3}$ is ________ (round off to three decimal places).

Answer Key:

0.278

Step-by-Step Solution & Explanation

Multi-Approach Solution: **Method 1: Characteristic Length Substitution** For a solid sphere: $L_c = \frac{V}{A} = \frac{\frac{4}{3}\pi r_0^3}{4 \pi r_0^2} = \frac{r_0}{3} = \frac{0.015\text{ m}}{3} = 0.005\text{ m}$ $Bi = \frac{h L_c}{k} = \frac{25.0\text{ W/(m}^2\cdot\text{K)} \times 0.005\text{ m}}{0.45\text{ W/(m}\cdot\text{K)}} = \frac{0.125}{0.45} = 0.27778 \approx 0.278$ **Method 2: Direct Ratio** $Bi = \frac{h r_0}{3 k} = \frac{25.0 \times 0.015}{3 \times 0.45} = \frac{0.375}{1.35} = 0.2778 \approx 0.278$
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