The rotor shaft of an ice cream freezer consists of 3 scraper blades. The temperature difference between the ice cream mix and the refrigerant during freezing of ice cream is 30⁰C. Density and latent heat of fusion of ice are 917 kg m⁻³ and 335 kJ kg⁻¹, respectively. The overall heat transfer coefficient is 2000 kJ m⁻²h⁻¹⁰C⁻¹. If the maximum thickness of ice formed before being scraped off is 10 µm, the minimum speed of the scraper shaft in rpm is
Options:
(A) 88
(B) 109
(C) 121
(D) 149
Official Answer Key:
B
Step-by-Step Solution & Explanation
Heat flux through the forming ice layer, q = U×ΔT = 2000×30 = 60,000 kJ/(m²·h).
Rate of ice-layer mass formation per unit area = q/λ = 60,000/335 = 179.1 kg/(m²·h).
Rate of ice-layer thickness growth = (mass rate)/density = 179.1/917 = 0.1953 m/h.
Time to build up the maximum allowable thickness δ=10µm=1×10⁻⁵ m: t = δ/(growth rate) = 1×10⁻⁵/0.1953 = 5.121×10⁻⁵ h = 0.1843 s.
With 3 scraper blades sweeping the wall, the shaft must complete enough revolutions so that a blade passes any given point before that point's ice exceeds δ: time between successive blade-passes at a point = (1/N)/3 (N=rev/s, 3 blades per rev), which must equal t ⇒ N = 1/(3t) rev/s.
N = 1/(3×0.1843) = 1.809 rev/s = 108.5 rpm ≈ 109 rpm. Answer: (B).
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