Explanation Given: The diameter of the spherical hailstone is d = 5 mm . The initial temperature of the spherical hailstone is T i = − 30 °C . The air temperature is T ∞ = 5 °C . The temperature of the surface of the sphere is T s u r f a c e = 150 ° C . The convection heat transfer coefficient is h = 250 W / m 2 K . The radius of the sphere is R = 40 mm . Formula used: The expression for the Biot number is given as, B i = h L c K Here, B i is the Biot number, L c is the characteristic length and K is the thermal conductivity of the material. The expression for the time required is given as, T − T ∞ T i − T ∞ = e − h A ρ V c p × t Here, c p is the specific heat of the material, t is the time, ρ is the density of the material, A is the area of a sphere and V is the volume of the sphere. The expression for the Fourier number is given as, F o = α t r 2 Here, α is the thermal diffusivity and r is the radius of the sphere. Calculation: Refer to table A.3, “Properties of ice” ρ = 920 kg / m 3 K = 1.88 W / m ⋅ K c p = 2040 J / kg ⋅ K The Biot number can be calculated as, B i = h L c K = h × d 6 K = ( 250 W / m 2 ⋅ K ) × 5 mm 6 ( 1 m 1000 mm ) ( 1.88 W / m ⋅ K ) = 0.110 The inverse of Biot number can be calculated as, B i − 1 = 1 0.110 = 9.09 The time required is calculated as, T − T ∞ T i − T ∞ = e − h A ρ V c p × t t = ρ ( A × L c ) c p h A ln ( T i − T ∞ T − T ∞ ) = ( 920 kg / m 3 ) × ( 8

7th Edition

ISBN: 9780470501979

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5, Problem 5S.7P

To determine

The time taken by spherical hailstone before the outer surface begins to melt.

The temperature of the center of the stone.

The energy transferred to the stone.

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Chapter 5, Problem 5S.4P

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The time taken by spherical hailstone before the outer surface begins to melt. The temperature of the center of the stone. The energy transferred to the stone.

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The time taken by spherical hailstone before the outer surface begins to melt. The temperature of the center of the stone. The energy transferred to the stone.

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Chapter 5 Solutions

The time taken by spherical hailstone before the outer surface begins to melt.

The temperature of the center of the stone.

The energy transferred to the stone.