A sealed industrial holding tank contains water with a pocket of compressed air trapped ab… — Armed Services Vocational Aptitude Battery (ASVAB)

A sealed industrial holding tank contains water with a pocket of compressed air trapped above the water surface. The total depth of the water in the tank is $8.0\text{ m}$ . A pressure gauge located at the absolute bottom of the tank reads a gauge pressure of $130\text{ kPa}$ . A secondary discharge valve is located on the side of the tank, exactly $5.0\text{ m}$ above the tank's bottom. Assuming the density of water is $1,000\text{ kg/m}^3$ and the acceleration due to gravity is $g = 10\text{ m/s}^2$ , what is the gauge pressure, in kilopascals (kPa), at the secondary discharge valve?

Answer: 80 kPa

Answer

The pressure at any point in a static fluid is equal to the pressure at a lower point minus the hydrostatic pressure of the fluid column between them. Since the valve is

5.0\text{ m}

above the bottom, the pressure there is less than the bottom pressure by

\rho g \Delta h = 1,000 \times 10 \times 5.0 = 50\text{ kPa}

. Subtracting this from the bottom pressure (

130\text{ kPa}

) gives the correct pressure of

80\text{ kPa}

. Alternatively, calculating the trapped air pressure (

130 - 80 = 50\text{ kPa}

) and adding the

3.0\text{ m}

water column above the valve (

30\text{ kPa}

) yields the same correct result.

Step-by-Step Solution

Determine the vertical distance between the pressure gauge at the bottom and the secondary valve.

\Delta h = 5.0\text{ m}

Hydrostatic pressure changes linearly with vertical height within a continuous fluid.

Calculate the hydrostatic pressure difference exerted by this

5.0\text{ m}

column of water.

\Delta P = \rho g \Delta h = 1,000 \times 10 \times 5.0 = 50,000\text{ Pa} = 50\text{ kPa}

This represents the amount of pressure lost as you move upward from the bottom to the valve.

Subtract the pressure difference from the gauge pressure at the bottom of the tank.

130\text{ kPa} - 50\text{ kPa} = 80\text{ kPa}

The pressure at the valve must be lower than the pressure at the bottom by exactly the weight of the water column between them.

Key Concept

Hydrostatic pressure varies directly with depth. The pressure difference between any two points in a continuous static fluid depends only on the vertical distance between them, regardless of surface air pressure.

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