A spherical tank \((k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) with an inner diameter of \(3 \mathrm{~m}\) and a wall thickness of \(10 \mathrm{~mm}\) is used for storing hot liquid. The hot liquid inside the tank causes the inner surface temperature to be as high as \(100^{\circ} \mathrm{C}\). To prevent thermal burns on the skin of the people working near the vicinity of the tank, the tank is covered with a \(7-\mathrm{cm}\) thick layer of insulation \((k=0.15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and the outer surface is painted to give an emissivity of \(0.35\). The tank is located in a surrounding with air at \(16^{\circ} \mathrm{C}\). Determine whether or not the insulation layer is sufficient to keep the outer surface temperature below \(45^{\circ} \mathrm{C}\) to prevent thermal burn hazards. Discuss ways to further decrease the outer surface temperature. Evaluate the air properties at \(30^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Step by step solution

01

Define known quantities

We are given the following information: - The thermal conductivities of the tank material \((k_{tank})\) and insulation material \((k_{insul})\): \(k_{tank} = 15 \mathrm{\frac{W}{m \cdot K}}\) \(k_{insul} = 0.15 \mathrm{\frac{W}{m \cdot K}}\) - The inner diameter \((D_{in})\) and outer diameter \((D_{out})\) of the tank: \(D_{in} = 3 \mathrm{~m}\) \(D_{out} = D_{in} + 2 \cdot (10^{-3} \mathrm{~m})\) - The thickness of the insulation \((t_{insul})\): \(t_{insul} = 0.07 \mathrm{~m}\) - The emissivity \((\epsilon)\), inner surface temperature \((T_{in})\), and air temperature \((T_{air})\): \(\epsilon = 0.35\) \(T_{in} = 100^{\circ} \mathrm{C}\) \(T_{air} = 16^{\circ} \mathrm{C}\)

02

Calculate the equivalent thermal conductance

In order to find the heat transfer through the tank and insulation layers, we need to find the equivalent thermal conductance \((U)\), which is the reciprocal of the sum of the thermal resistances of the tank material and insulation: \(R_{tank} = \frac{1}{k_{tank}} \cdot \frac{D_{out} - D_{in}}{D_{in}} = \frac{1}{15} \cdot \frac{0.02}{3}\) \(R_{insul} = \frac{1}{k_{insul}} \cdot \frac{t_{insul}}{D_{out}} = \frac{1}{0.15} \cdot \frac{0.07}{3.016}\) \(U = \frac{1}{R_{tank} + R_{insul}}\)

03

Calculate the heat transfer

Use the equivalent thermal conductance (\((U)\)), to calculate the heat transfer through the tank and insulation layers: \(q = U \cdot (T_{in} - T_{air})\)

04

Calculate the outer surface temperature

With the calculated heat transfer rate, we can find the temperature of the outer surface of the insulation, which should be below 45°C. \(q = \epsilon \cdot \sigma \cdot (T_{out}^{4} - T_{air}^{4})\) Solving for \(T_{out}\): \(T_{out} = \sqrt[4]{\frac{q}{\epsilon \cdot \sigma} + T_{air}^{4}}\) Where \(\sigma\) is the Stefan-Boltzmann constant, approximately \(5.67 \times 10^{-8} \mathrm{\frac{W}{m^{2} \cdot K^{4}}}\).

05

Determine the sufficiency of the insulation layer and discuss ways to decrease the outer surface temperature

Compare the calculated outer surface temperature \((T_{out})\) to the given limit of 45°C. If \(T_{out}\) is greater than 45°C, the insulation layer is not sufficient, and we need to propose ways to further decrease the outer surface temperature. Some possible suggestions include: 1. Increasing the thickness of the insulation layer 2. Changing the insulation material to a lower thermal conductivity material 3. Adding a reflective or emissive coating on the outer surface of the insulation.

06

Evaluate the assumption for air properties

It was mentioned to evaluate air properties at 30°C and 1 atm pressure. Considering the given air temperature of 16°C, evaluating air properties at 30°C is acceptable since it's within a reasonable range. Also, assuming a pressure of 1 atm is a standard assumption for near sea level conditions.

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