Question

A spherical stainless steel tank with an inner diameter of \(3 \mathrm{~m}\) and a wall thickness of \(10 \mathrm{~mm}\) is used to contain a solution undergoing an exothermic reaction that generates \(450 \mathrm{~W} / \mathrm{m}^{3}\) of heat. The tank is located in surroundings with air at \(15^{\circ} \mathrm{C}\). To prevent thermal burns to people working near the tank, the outer surface temperature should be at \(45^{\circ} \mathrm{C}\) or lower. Determine whether the outer surface of the tank should be polished \((\varepsilon=0.2)\) or painted black \((\varepsilon=0.88)\). Evaluate the air properties at $30^{\circ} \mathrm{C}\( and \)1 \mathrm{~atm}$ pressure. Is this a good assumption?

Short answer

Answer: To answer this question, we need to compare the heat transfer rates for both polished and black-painted surface conditions, calculated in steps 3 and 4, with the required heat transfer rate through the tank wall. If either of the calculated heat transfer rates is higher than the required rate, then the corresponding surface type is suitable to maintain the desired outer surface temperature. Additionally, we need to verify whether the resulting outer surface temperature is close to the assumed \(30^{\circ}\mathrm{C}\) to justify the property evaluation assumption.

Step-by-step solution

Calculate the heat generation inside the tank.

First, we need to determine the heat generated inside the tank. We have the heat generation rate, and we need to find the volume of the tank to calculate the total heat generated. The volume of a sphere can be given by the formula: $$ V = \frac{4}{3} \pi r^3 $$ where \(r\) is the radius of the sphere. The given inner diameter of the tank is \(3 \mathrm{~m}\), so the radius of the inner surface is \(1.5 \mathrm{~m}\). Using the formula, we can find the volume of the inner surface. $$ V = \frac{4}{3} \pi (1.5\,\mathrm{m})^3 \approx 14.14 \,\mathrm{m}^3 $$ Now we can determine the heat generated inside the tank as: $$ Q_\mathrm{gen} = V \times q_\mathrm{gen} $$ where \(q_\mathrm{gen}\) is the given heat generation rate of \(450 \,\mathrm{W} / \mathrm{m}^{3}\).

Calculate the required heat transfer rate through the tank wall.

Since the tank is insulated, we assume that all of the heat generated inside the tank must be transferred through the tank wall. So, the required heat transfer rate through the tank wall is equal to the calculated heat generation from the previous step. $$ Q_\mathrm{wall} = Q_\mathrm{gen} $$

Calculate the heat transfer rate for polished surface condition.

To calculate the heat transfer rate for the polished surface condition, we will use the Stefan-Boltzmann Law: $$ Q_\text{pol}=\varepsilon\sigma A\left(T_\text{surf}^4-T_\text{air}^4\right) $$ where: \(\varepsilon = 0.2\) (given) \(A\) is the outer surface area (will be found using the given wall thickness), \(\sigma = 5.67 \times 10^{-8}\,\mathrm{W}/\mathrm{m}^2\mathrm{K}^4\) (Stefan-Boltzmann constant) First, we need to find the area of the outer surface. Since the sphere has a wall thickness of \(10 \, \mathrm{mm}\) (or \(0.01 \, \mathrm{m}\)), the radius of the outer surface will be \(1.5\,\mathrm{m} + 0.01\,\mathrm{m} = 1.51\,\mathrm{m}\). The outer surface area can be calculated as: $$ A = 4 \pi r_\mathrm{outer}^2 = 4 \pi (1.51\, \mathrm{m})^2 \approx 28.69 \,\mathrm{m}^2 $$ Using the Stefan-Boltzmann Law, calculate the heat transfer rate for the polished surface: $$ Q_\text{pol} = Q_\mathrm{wall} $$

Calculate the heat transfer rate for the black-painted surface condition.

Now we will repeat the same process as in the previous step to calculate the heat transfer rate for the black-painted surface condition, with \(\varepsilon=0.88\). Using the Stefan-Boltzmann Law, calculate the heat transfer rate for the black-painted surface: $$ Q_\text{black} = Q_\mathrm{wall} $$

Compare the heat transfer rates and choose surface type.

Now we have the calculated heat transfer rates for both the polished and black-painted surface conditions. Compare the two rates with the required heat transfer rate through the tank wall. If either of the calculated heat transfer rates is higher than the required rate, then the corresponding surface type is suitable to maintain the desired outer surface temperature.

Evaluate if property evaluation at \(30^{\circ}\mathrm{C}\) and \(1\,\mathrm{atm}\) pressure is a good assumption.

Look at the assumptions regarding air property evaluation at \(30^{\circ}\mathrm{C}\) and \(1\,\mathrm{atm}\) pressure, and verify if they are appropriate based on the obtained outer surface temperature. If the resulting surface temperature is close to the assumed \(30^{\circ}\mathrm{C}\), then the assumption is justified.