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Chapter 2: Problem 4

(a) A large tank contains water at \(20^{\circ} \mathrm{C},\) and the absolute pressure \(12 \mathrm{~m}\) below the surface of the water is measured as \(200 \mathrm{kPa}\). Estimate the atmospheric pressure above the tank. (b) If the water in the tank was replaced by a liquid with a specific gravity of \(0.85,\) what absolute pressure and what gauge pressure would be measured \(6 \mathrm{~m}\) below the surface of the liquid?

Short Answer

Expert verified
(a) Atmospheric pressure is 82.28 kPa. (b) Absolute pressure is 132.253 kPa, and gauge pressure is 49.973 kPa.

Step by step solution

01

Understanding the Problem

We need to find the atmospheric pressure when given the absolute pressure at a certain depth in water, and then determine the absolute and gauge pressures for a different liquid under specified conditions.
02

Calculating Atmospheric Pressure

Given that the absolute pressure at 12 m depth is 200 kPa, we use the formula for pressure in liquids: \( P = P_0 + \rho gh \), where \( \rho \) is the density of water (\( 1000 \text{ kg/m}^3 \)), \( g = 9.81 \text{ m/s}^2 \), and \( h = 12 \text{ m} \). \( P_0 \) is what we want to find. Rearranging gives \( P_0 = P - \rho gh = 200 \text{ kPa} - (1000 \text{ kg/m}^3 \times 9.81 \text{ m/s}^2 \times 12 \text{ m}) \). Calculate to find \( P_0 = 200 \text{ kPa} - 117.72 \text{ kPa} = 82.28 \text{ kPa} \).
03

Finding Absolute Pressure for New Liquid

We use the same pressure formula for the liquid with specific gravity 0.85, at 6 m depth: \( P = P_0 + \rho g h \). First, find \( \rho \) for the new liquid: \( \rho = 850 \text{ kg/m}^3 \) (since specific gravity = 0.85). Use \( P_0 = 82.28 \text{ kPa} \), \( g = 9.81 \text{ m/s}^2 \), \( h = 6 \text{ m} \). Then \( P = 82.28 \text{ kPa} + 850 \text{ kg/m}^3 \times 9.81 \text{ m/s}^2 \times 6 \text{ m} \). Calculate to find \( P = 82.28 \text{ kPa} + 49.973 \text{ kPa} = 132.253 \text{ kPa} \).
04

Calculating Gauge Pressure for New Liquid

Gauge pressure is the absolute pressure minus atmospheric pressure: \( P_{gauge} = P - P_0 \). Using the absolute pressure for the new liquid: \( P_{gauge} = 132.253 \text{ kPa} - 82.28 \text{ kPa} = 49.973 \text{ kPa} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atmospheric Pressure
Atmospheric pressure is the force exerted by the weight of the air above a surface. This pressure plays an essential role in various calculations in fluid mechanics, particularly when dealing with absolute and gauge pressures.
  • At sea level, the standard atmospheric pressure is about 101.3 kPa. changes based on altitude and weather conditions.
  • It acts as a reference point when calculating other types of pressures found in fluids.
In the exercise, we calculated it by subtracting the hydrostatic pressure (pressure due to the fluid) from the given absolute pressure. Knowing atmospheric pressure is crucial for accurate measurement of other pressures in fluid systems.
Specific Gravity
Specific gravity is a dimensionless quantity that represents the ratio of the density of a substance to the density of a reference substance, typically water for liquids and solids.
  • Water has a specific gravity of 1. If a liquid's specific gravity is less than 1, it is less dense than water.
  • In the exercise, the liquid with a specific gravity of 0.85 means it is 15% less dense than water, which affects the calculations of pressures at a given depth.
Understanding specific gravity helps in determining how a liquid will behave under different pressure conditions, especially when replaced in systems originally designed for water.
Absolute Pressure
Absolute pressure is the total pressure exerted on a system, including atmospheric pressure. It is measured relative to a perfect vacuum.
  • The formula for calculating absolute pressure is: \( P_{absolute} = P_{gauge} + P_{atmospheric} \).
  • In fluid systems, it helps in understanding the total pressure the fluid system exerts.
In the exercise, we determined absolute pressure using the atmospheric pressure and the additional pressure due to the liquid column above the measurement point. This calculation is essential in applications like submersible systems and diving.
Gauge Pressure
Gauge pressure is the pressure relative to atmospheric pressure, mostly used in practical applications to indicate pressure differences.
  • It is the pressure reading you get from instruments like pressure gauges, which do not account for atmospheric pressure, but only measure the deviation from it.
  • Formula: \( P_{gauge} = P_{absolute} - P_{atmospheric} \).
In our exercise, we found the gauge pressure for a new liquid by calculating the difference between the absolute pressure and the atmospheric pressure at a certain depth. Understanding gauge pressure is vital for ensuring safety and functionality in systems where pressure levels need to be monitored and controlled.

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Most popular questions from this chapter

A vertical circular gate of diameter \(D\) is installed such that the top of the gate is a distance \(d\) below the water surface in a reservoir. Calculate the hydrostatic force on the gate and show that the location of the center of pressure is given by $$ y_{\mathrm{cp}}=D+\frac{D(8 d+5 D)}{16 d+8 D} $$

The head of an adult male giraffe is typically \(6 \mathrm{~m}\) above the ground. The normal blood pressure of a giraffe (at heart level) is typically stated as \(280 / 180,\) where 280 the maximum arterial pressure in \(\mathrm{mm} \mathrm{Hg}\) and 180 is the minimum arterial pressure in \(\mathrm{mm} \mathrm{Hg}\). The density of a giraffe's blood is approximately \(1060 \mathrm{~kg} / \mathrm{m}^{3}\). (a) What is the change in the blood pressure in the giraffe's head, in millimeters of mercury, as it moves from grazing on a tall tree to drinking from a pond at ground level? (b) Assuming that there is a static distribution of blood pressure in the giraffe's body and that its heart is at approximately the mid-elevation of its body, estimate the maximum blood pressure in its head.

A barometer located at the entrance to the ground floor of the Burj Khalifa building in Dubai estimates a pressure of \(100.8 \mathrm{kPa}\) on an average day in August, when the temperature is \(37^{\circ} \mathrm{C}\). The height of the Burj Khalifa is reported to be \(829.8 \mathrm{~m}\). Estimate the barometric pressure at the top of the building.

A 1.6-m-diameter cylinder is filled with a liquid to a depth of \(1.1 \mathrm{~m}\) and rotated about its center axis. (a) Assuming that the cylinder is tall enough for the liquid not to spill, at what rotational speed will the liquid surface intersect the bottom of the cylinder? (b) If the cylinder is rotated at 60 rpm, what is the minimum height of the cylinder that prevents spillage?

A tank contains a layer of SAE 30 oil floating on a layer of water, where both liquids are at \(20^{\circ} \mathrm{C}\). A cube of dimensions \(0.15 \mathrm{~m} \times 0.15 \mathrm{~m} \times 0.15 \mathrm{~m}\) is placed in the tank, and it is observed that the cube locates itself such that its top and bottom faces are parallel to the interface between fluids and \(15 \%\) of the height of the cube is located in the oil layer. Estimate the density of the cube.
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