Question

A spherical tank, with an inner diameter of \(D_{1}=3 \mathrm{~m}\), is filled with a solution undergoing an exothermic reaction that heats the surface to a uniform temperature of \(120^{\circ} \mathrm{C}\). To prevent thermal burn hazards, the tank is enclosed by a concentric outer cover that provides an evacuated gap of \(5 \mathrm{~cm}\) in the enclosure. Both spherical surfaces have the same emissivity of \(0.5\), and the outer surface is exposed to natural convection with a heat transfer coefficient of $5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and radiation heat transfer with the surroundings at a temperature of \(30^{\circ} \mathrm{C}\). Determine whether or not the vacuumed gap is sufficient to keep the outer surface temperature below $45^{\circ} \mathrm{C}$ to prevent thermal burns. If not, propose a solution to keep the outer surface temperature below \(45^{\circ} \mathrm{C}\).

Short answer

If not, what are some possible solutions? Answer: No, the 5 cm vacuum gap is not sufficient to keep the outer surface temperature below 45°C. Some possible solutions include increasing the vacuum gap's size, adding insulation between the inner and outer surfaces, using reflective coating on both surfaces to reduce radiation heat transfer, or increasing the heat transfer coefficient by adding an external fan to enhance convection performance.

Step-by-step solution

Find Heat Flux Through Radiation

We will use the following formula to find the heat flux through radiation: \(q_r = \epsilon \sigma (T_{hot}^4 - T_{cold}^4)\) Where: \(q_r\) is the heat flux (\(\mathrm{W} / \mathrm{m}^{2}\)) \(\epsilon\) is the emissivity (0.5, given) \(\sigma\) is the Stefan-Boltzmann constant, which is \(5.67 \times 10^{-8} \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\) \(T_{hot}\) is the temperature of the hot surface in Kelvin (393 K, given) \(T_{cold}\) is the temperature of the surrounding in Kelvin (303 K, given) Let's calculate the heat flux through radiation: \(q_r = 0.5(5.67 \times 10^{-8})(393^4 - 303^4)= 1685.78 \mathrm{W} / \mathrm{m}^{2}\)

Find Heat Flux Through Convection

We will use the following formula to find the heat flux through convection: \(q_c = h(T_{hot} - T_{cold})\) Where: \(q_c\) is the heat flux (\(\mathrm{W} / \mathrm{m}^{2}\)) \(h\) is the heat transfer coefficient (5 W/m²·K, given) Let's calculate the heat flux through convection, assuming the outer surface temperature is equal to \(T_{hot} = 45^{\circ} \mathrm{C}\) (318 K): \(q_c = 5 (318 - 303) = 75 \mathrm{W} / \mathrm{m}^{2}\)

Determine the Total Heat Flux and Temperature on the Outer Surface

Now, we need to calculate the total heat flux at the outer surface to see if it can be achieved. The total heat flux (\(q_{total}\)) will be the sum of the heat flux through radiation (\(q_r\)) and convection (\(q_c\)). \(q_{total} = q_r + q_c = 1685.78 + 75 = 1760.78 \mathrm{W} / \mathrm{m}^{2}\) The heat flow through the vacuum gap for this configuration is too high to keep the temperature of the outer surface below 45°C, so we need to find a solution.

Propose a Solution to Keep the Outer Surface Temperature Below \(45^{\circ} \mathrm{C}\)

To keep the outer surface temperature below the threshold, we can increase the vacuum gap's size or add insulation between the inner and outer surfaces. Another option is to use reflective coating on both surfaces to reduce the radiation heat transfer, or increase the heat transfer coefficient by adding an external fan to improve the convection performance and allow more heat to be dissipated. In summary, the vacuum gap is not sufficient to keep the outer surface temperature below 45°C, and an additional solution must be implemented, such as adding insulation, reflective coating or enhancing convection.