Questions: You are responsible for designing two free-standing water tanks. Both are to be 10 m high. The smaller tank has a 10 m x 10 m footprint, and the larger tank has a 100 m x 100 m footprint. (a) How does the water pressure on the tank walls vary between the top and the bottom of each tank? Justify your answer. (b) Sketch a design for the height-dependent wall thickness of each tank that will safely contain the water while minimizing the cost of building materials. (c) Discuss and justify any differences in your designs for the two tanks.

You are responsible for designing two free-standing water tanks. Both are to be 10 m high. The smaller tank has a 10 m x 10 m footprint, and the larger tank has a 100 m x 100 m footprint. (a) How does the water pressure on the tank walls vary between the top and the bottom of each tank? Justify your answer. (b) Sketch a design for the height-dependent wall thickness of each tank that will safely contain the water while minimizing the cost of building materials. (c) Discuss and justify any differences in your designs for the two tanks.

Transcript text: 2. You are responsible for designing two free-standing water tanks. Both are to be 10 m high. The smaller tank has a $10 \mathrm{~m} \times 10 \mathrm{~m}$ footprint, and the larger tank has a $100 \mathrm{~m} \times 100 \mathrm{~m}$ footprint. (a) How does the water pressure on the tank walls vary between the top and the bottom of each tank? Justify your answer. (b) Sketch a design for the height-dependent wall thickness of each tank that will safely contain the water while minimizing the cost of building materials. (c) Discuss and justify any differences in your designs for the two tanks.

Solution

Solution Steps

Step 1: Understanding Water Pressure Variation

The water pressure at any depth in a tank is given by the hydrostatic pressure formula:

\[ P = \rho g h \]

where $ P $ is the pressure, $ \rho $ is the density of water (approximately $ 1000 \, \text{kg/m}^3 $), $ g $ is the acceleration due to gravity (approximately $ 9.81 \, \text{m/s}^2 $), and $ h $ is the depth of the water.

For both tanks, since they are 10 m high, the pressure at the bottom of each tank will be:

\[ P_{\text{bottom}} = \rho g \times 10 = 1000 \times 9.81 \times 10 = 98100 \, \text{Pa} \]

At the top of each tank, the pressure is zero (ignoring atmospheric pressure, which is constant and acts on both the top and bottom).

Step 2: Designing Height-Dependent Wall Thickness

The pressure on the tank walls increases linearly from the top to the bottom. Therefore, the wall thickness should also increase from the top to the bottom to withstand the increasing pressure. A common approach is to design the wall thickness $ t(h) $ as a function of height $ h $:

\[ t(h) = t_0 + k \cdot (10 - h) \]

where $ t_0 $ is the minimum thickness at the top, and $ k $ is a constant that determines how much the thickness increases per meter of depth.

Step 3: Differences in Design for the Two Tanks

The larger tank has a much greater footprint, which means it holds significantly more water and thus exerts a greater total force on the walls. This requires a thicker wall at the bottom compared to the smaller tank. However, the pressure at any given depth is the same for both tanks because it depends only on the height of the water column, not the volume.

For the larger tank, the value of $ k $ in the thickness function $ t(h) $ should be larger to account for the increased total force due to the larger volume of water. Additionally, structural reinforcements may be necessary to handle the increased lateral forces.