Question

A glass \((k=1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) spherical tank is filled with chemicals undergoing exothermic reaction. The reaction keeps the inner surface temperature of the tank at \(80^{\circ} \mathrm{C}\). The tank has an inner radius of \(0.5 \mathrm{~m}\), and its wall thickness is $10 \mathrm{~mm}$. Situated in surroundings with an ambient temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of $70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, the tank's outer surface is being cooled by air flowing across it at \(5 \mathrm{~m} / \mathrm{s}\). In order to prevent thermal burn on individuals working around the container, it is necessary to keep the tank's outer surface temperature below $50^{\circ} \mathrm{C}$. Determine whether or not the tank's outer surface temperature is safe from thermal burn hazards.

Short answer

Answer: Yes, the outer surface temperature of the tank is approximately \(40.9^{\circ} \mathrm{C}\), which is lower than the safety limit of \(50^{\circ} \mathrm{C}\), making the tank safe from thermal burn hazards.

Step-by-step solution

Recall Fourier's Law for heat conduction

Recall that the rate of heat conduction (Q_cond) through a spherical wall is given by Fourier's Law: \[Q_{cond} = k \cdot A \cdot \frac{T_{i} - T_{o}}{r_{o}-r_{i}}\] Where, \(k\) = Thermal conductivity of the material (\(1.1\mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\)) \(A\) = Surface area between the inner surface and the outer surface \(T_i\) = Inner surface temperature (\(80^{\circ} \mathrm{C}\)) \(T_o\) = Outer surface temperature \(r_i\) = Inner radius of the sphere (\(0.5\mathrm{~m}\)) \(r_o\) = Outer radius of the sphere (\(r_i + 0.01\mathrm{~m}\))

Recall Newton's Law of cooling for heat convection

Recall the equation of Newton's Law of cooling for heat convection (Q_conv) between the outer surface of an object and the surrounding fluid: \[Q_{conv} = h \cdot A \cdot (T_o - T_\infty)\] Where, \(h\) = Convection heat transfer coefficient (\(70\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)) \(A\) = Surface area of the outer surface of the sphere \(T_o\) = Outer surface temperature \(T_\infty\) = Ambient temperature (\(15^{\circ} \mathrm{C}\))

Equate the heat transfers

The heat transferred by conduction through the spherical wall must be equal to the heat transferred by convection to the surroundings: \[Q_{cond} = Q_{conv}\] Substitute the expressions for \(Q_{cond}\) and \(Q_{conv}\) from steps 1 and 2 and rearrange the equation: \[\frac{k \cdot 4\pi (r_i r_o)}{(r_o - r_i)} \cdot (T_i - T_o) = h \cdot 4\pi r_o^2 \cdot (T_o - T_\infty)\]

Calculate the outer surface temperature\(T_o\)

Plug in the known values from the problem statement and solve the equation for \(T_o\): \[T_o \approx 40.9^{\circ} \mathrm{C}\]

Compare outer surface temperature with the safety limit

Now, check if the calculated outer surface temperature is below the limit of \(50^{\circ} \mathrm{C}\): \[40.9^{\circ} \mathrm{C} < 50^{\circ} \mathrm{C}\] Since the calculated outer surface temperature is lower than the safety limit, the tank is considered safe from thermal burn hazards.