Discussion
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 30 °C. 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-2 h-1 °C-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
Answer: B
The minimum shaft speed is about 109 rpm.
Heat removed through the wall freezes a layer of ice; the time to build up a 10 μm layer follows from a heat balance per unit area:
\[ \begin{aligned} t &= \frac{\rho L x}{U\,\Delta T} = \frac{917 \times 335 \times 10^{-5}}{2000 \times 30} \\ &= 5.12 \times 10^{-5}\ \text{h} = 0.184\ \text{s} \end{aligned} \]
Every point of the wall must be scraped at least once every 0.184 s. With three blades, the shaft need make only one revolution in three such intervals:
\[ \begin{aligned} N &= \frac{60}{3 \times 0.184} \\ &= 108.6 \approx 109\ \text{rpm} \end{aligned} \]
Ignoring the three blades gives about 326 rpm, which is not offered; 88, 121 and 149 rpm do not match the ice-layer time.
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