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Chapter 3: Problem 196

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

Expert verified
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

01

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: \[r = \frac{d}{2}\]
02

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: \[A = 4 \pi r^2\]
03

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.
04

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, \(T_{surf}\): \(T_{surf} = \frac{qr}{k} + T_{ground}\)
05

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.

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