Question

A thin-walled spherical tank in buried in the ground at a depth of \(3 \mathrm{~m}\). The tank has a diameter of \(1.5 \mathrm{~m}\), and it contains chemicals undergoing exothermic reaction that provides a uniform heat flux of \(1 \mathrm{~kW} / \mathrm{m}^{2}\) to the tank's inner surface. From soil analysis, the ground has a thermal conductivity of \(1.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and a temperature of \(10^{\circ} \mathrm{C}\). Determine the surface temperature of the tank. Discuss the effect of the ground depth on the surface temperature of the tank.

Short answer

Answer: With an increase in ground depth, the surface temperature of the tank will also increase due to a larger temperature gradient in the heat transfer through the spherical shell.

Step-by-step solution

Determine the radius of the tank

Firstly, we need to find the radius (r) of the tank since its diameter is given. The radius is equal to half of the diameter: \[r = \frac{d}{2}\]

Calculate the heat generation rate

Now, we calculate the heat generation rate (Q) in the tank. We can do this by multiplying the given heat flux (q) by the surface area of the tank: \[Q = q \cdot A\] The surface area (A) of a sphere is given by: \[A = 4 \pi r^2\]

Apply the heat conduction formula

Use Fourier's Law of Heat Conduction for steady-state heat transfer through a spherical shell: \(q = \frac{k(T_{surf} - T_{ground})}{r}\), where k is the thermal conductivity, \(T_{surf}\) is the surface temperature we need to find, and \(T_{ground}\) is the ground temperature.

Solve for the surface temperature

As we have found the values of q, k, and T_ground, we can now solve for the surface temperature, \(T_{surf}\): \(T_{surf} = \frac{qr}{k} + T_{ground}\)

Discuss the effect of ground depth

In the above equation for the surface temperature, the term \(\frac{qr}{k}\) represents the temperature gradient due to the heat transfer through the spherical shell. If the depth of the tank increases, the temperature gradient becomes larger, resulting in a higher surface temperature of the tank, given the same heat flux and thermal conductivity of the ground. So, the surface temperature will increase with the increase in ground depth.

Key concepts

Thermal Conductivity

Thermal conductivity is an important property of materials that defines their ability to conduct heat. It is denoted by the symbol \(k\). A higher thermal conductivity means that a material can transfer heat more easily. In our exercise, the ground surrounding the spherical tank has a thermal conductivity of \(1.3 \, \text{W/m} \cdot \text{K}\). This value tells us how efficiently the heat produced by the exothermic reaction inside the tank can dissipate into the ground.
Understanding thermal conductivity helps us predict how the temperature will change based on different conditions. For the buried tank, it's crucial in determining the surface temperature since it's part of the heat conduction formula used in our calculations. Lower thermal conductivity would mean more heat would remain near the tank, raising the surface temperature.

Exothermic Reaction

An exothermic reaction is a chemical reaction that releases heat. In the context of our problem, the chemicals inside the spherical tank are undergoing an exothermic reaction that supplies a steady heat flux of \(1 \, \text{kW/m}^2\) to the inner surface of the tank. This continuous release of heat creates a temperature gradient across the tank and into the surrounding ground.
Exothermic reactions are quite common and can be found in various applications. Here, they serve as the heat source, necessitating the calculation of how this heat affects the surrounding environment, particularly the surface temperature of the tank. We determine this temperature by considering both the heat produced by the reaction and the ability of the ground to dissipate that heat.

Surface Temperature Calculation

To find the surface temperature of the spherical tank, we use Fourier's Law of Heat Conduction for spherical shells. The equation relevant to our computations is:\[T_{\text{surf}} = \frac{qr}{k} + T_{\text{ground}}\]This formula takes into account the heat flux \(q\), thermal conductivity \(k\), and the ground temperature \(T_{\text{ground}}\), allowing us to solve for the surface temperature \(T_{\text{surf}}\).
By plugging in the known values:

• Heat flux \(q = 1\, \text{kW/m}^2\)
• Radius of the tank \(r = 0.75\, \text{m}\)
• Thermal conductivity \(k = 1.3\, \text{W/m} \cdot \text{K}\)
• Ground temperature \(T_{\text{ground}} = 10^\circ \text{C}\)

We determine the surface temperature at the tank's surface, crucial for safety and engineering applications in managing thermal stresses.

Effect of Ground Depth

Ground depth can significantly impact the surface temperature of a buried object such as our spherical tank. As the depth increases, the distance through which heat must travel before dissipating into cooler areas also increases. This can cause more heat to accumulate near the tank, increasing the surface temperature.
In our exercise, the depth is relevant to our understanding of the temperature gradient, represented by the component \(\frac{qr}{k}\) of our surface temperature formula. A deeper ground implies a larger volume of soil insulating the tank, which retains more heat. Thus, deeper installations may require additional considerations for thermal management to prevent overheating, especially for materials or reactions sensitive to high temperatures.