Question

Hailstones are formed in high-altitude clouds at \(253 \mathrm{~K}\). Consider a hailstone with diameter of \(20 \mathrm{~mm}\) that is falling through air at \(15^{\circ} \mathrm{C}\) with convection heat transfer coefficient of $163 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming the hailstone can be modeled as a sphere and has properties of ice at \(253 \mathrm{~K}\), determine how long it takes to reach melting point at the surface of the falling hailstone. Solve this problem using the analytical one-term approximation method.

Short answer

Answer: It takes approximately 183.60 seconds for the hailstone to reach its melting point.

Step-by-step solution

Calculate surface area and volume of the hailstone

Given diameter of the hailstone is 20 mm. We will first convert it to meters and then calculate the surface area (A) and volume (V) of the hailstone sphere: Diameter (D) = 20 mm = 0.020 m Radius (r) = D / 2 = 0.010 m Surface area (A) = 4 * pi * r^2 Volume (V) = (4/3) * pi * r^3 Calculating A and V: A = 4 * pi * (0.010)^2 = 0.001256 m^2 V = (4/3) * pi * (0.010)^3 ≈ 4.19 × 10^{-6} m^3

Determine the Biot number

To determine the Biot number (Bi), first we need to calculating thermal conductivity (k), heat capacity (c), and density (ρ) of ice at 253 K given data. We can consider the properties of ice at 253 K as: k = 2.180 W/m·K c = 2110 J/kg·K ρ = 920 kg/m³ Now we can calculate the Biot number: Bi = (h * r) / k where h = 163 W/m²·K (convection heat transfer coefficient) Bi = (163 * 0.010) / 2.180 ≈ 0.746

Calculate Fourier number using one-term approximation

We'll now calculate Fourier number (Fo) - which depends on time (t) - using the one-term approximation method, for half of the Biot number (Bi/2 ≈ 0.373): Fo = 0.373^2 / 3.289 = 0.042

Calculate time to reach melting point

We can now calculate the required time (t) to reach the melting point using Fourier number (Fo): t = Fo * (V/A) * c * ρ / k t = 0.042 * (4.19 × 10^{-6}/0.001256) * 2110 * 920 / 2.180 t ≈ 183.60 s So, it takes approximately 183.60 seconds for the hailstone to reach the melting point at the surface while falling through the air at 15°C.