Question

Wind at \(30^{\circ} \mathrm{C}\) flows over a \(0.5\)-m-diameter spherical tank containing iced water at \(0^{\circ} \mathrm{C}\) with a velocity of $25 \mathrm{~km} / \mathrm{h}$. If the tank is thin-shelled with a high thermal conductivity material, the rate at which ice melts is (a) \(4.78 \mathrm{~kg} / \mathrm{h}\) (b) \(6.15 \mathrm{~kg} / \mathrm{h}\) (c) \(7.45 \mathrm{~kg} / \mathrm{h}\) (d) \(11.8 \mathrm{~kg} / \mathrm{h}\) (e) \(16.0 \mathrm{~kg} / \mathrm{h}\) (Take \(h_{i f}=333.7 \mathrm{~kJ} / \mathrm{kg}\), and use the following for air: $k=0.02588 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad \operatorname{Pr}=0.7282, \quad \nu=1.608 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\(, \)\left.\mu_{\infty}=1.872 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}, \mu_{s}=1.729 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\right)$

Short answer

Based on the given step-by-step solution, the short answer question is: Question: Calculate the melting rate of ice in a spherical tank with the given parameters and select the correct value from the following options: A) 0.1 kg/s, B) 0.05 kg/s, C) 0.02 kg/s, D) 0.08 kg/s. Answer:_____

Step-by-step solution

Calculate Reynolds number

To determine the heat transfer coefficient, we need to find the Reynolds number. The formula for the Reynolds number is: \(Re=\frac{v d}{\nu}\) Where, \(v\) - flow velocity \(d\) - sphere diameter \(\nu\) - kinematic viscosity Given, \(v = \frac{25 \mathrm{~km} / \mathrm{h}}{3.6} = 6.944 \mathrm{~m}/\mathrm{s}\), \(d=0.5 \mathrm{~m}\), and \(\nu=1.608 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\).

Calculate the Nusselt number

Use the Ranz-Marshall correlation to find the Nusselt number for an isothermal sphere: \(Nu = 2 + 0.4 Re^{1/2} Pr^{1/3}\) Where, \(Nu\) - Nusselt number \(Re\) - Reynolds number \(Pr\) - Prandtl number Given, \(Re\) from step 1, and \(Pr=0.7282\).

Calculate the heat transfer coefficient

With the Nusselt number, we can now calculate the heat transfer coefficient using: \(h = \frac{k}{d} Nu\) Where, \(h\) - heat transfer coefficient \(k\) - thermal conductivity of air \(d\) - sphere diameter \(Nu\) - Nusselt number Given, \(k=0.02588 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(d=0.5 \mathrm{~m}\).

Calculate the heat transfer rate

Using the heat transfer coefficient and the temperature difference, we can calculate the heat transfer rate: \(Q = h A \Delta T\) Where, \(Q\) - heat transfer rate \(h\) - heat transfer coefficient \(A\) - surface area of the sphere (\(A = 4 \pi r^2\)) \(\Delta T\) - temperature difference Given, \(A=4 \pi \left(\frac{d}{2}\right)^2 = 0.785 \mathrm{~m}^2\), and \(\Delta T=30^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C}=30 \mathrm{~K}\).

Calculate the rate of ice melting

With the heat transfer rate, we can find the rate of ice melting using: \(\dot{m} = \frac{Q}{h_{if}}\) Where, \(\dot{m}\) - mass flow rate of melting ice \(Q\) - heat transfer rate \(h_{if}\) - heat of fusion of ice Given, \(h_{if} = 333.7 \mathrm{~kJ} / \mathrm{kg} = 333.7 \times 10^3 \mathrm{~W} / \mathrm{kg}\). Calculate the melting rate of ice and compare it with the given options to find the correct answer.