Question

A manometer containing oil \(\left(\rho=850 \mathrm{kg} / \mathrm{m}^{3}\right)\) is attached to a tank filled with air. If the oil-level difference between the two columns is \(80 \mathrm{cm}\) and the atmospheric pressure is \(98 \mathrm{kPa}\), determine the absolute pressure of the air in the tank.

Short answer

Answer: The absolute pressure of the air in the tank is 104.68 kPa.

Step-by-step solution

Write down the given information

We are given the following information: - Oil density: \(\rho = 850 \, kg/m^3\) - Oil-level difference: \(h = 0.80 \,m\) (converted to meters from centimeters) - Atmospheric pressure: \(P_{atm} = 98 \,kPa\)

Apply the manometer equation

The manometer equation allows us to determine the difference in pressure between two points. For our case, the equation is: $$ \Delta P = \rho \cdot g \cdot h $$ where \(\Delta P\) is the difference in pressure between the tank and atmosphere, \(\rho\) is the oil density, \(g\) is gravitational acceleration, and \(h\) is the oil-level difference.

Calculate the pressure difference

Now, we can plug in the given values and the gravitational acceleration (\(g = 9.81 \,m/s^2\)) to find the pressure difference: $$ \Delta P = 850 \, kg/m^3 \times 9.81 \, m/s^2 \times 0.80 \, m = 6681.98 \, N/m^2 $$ The pressure difference is \(6681.98 \, N/m^2\), or \(6.68 \, kPa\) (since \(1 \,N/m^2 = 1 \,Pa\), and there are \(1000 \,Pa\) in \(1 \,kPa\)).

Determine the absolute pressure in the tank

To find the absolute pressure of the air in the tank, we need to add the pressure difference to the atmospheric pressure: $$ P_{absolute} = P_{atm} + \Delta P = 98 \, kPa + 6.68 \, kPa = 104.68 \, kPa $$ The absolute pressure of the air in the tank is \(104.68 \, kPa\).

Key concepts

Fluid Mechanics and the Manometer

Understanding fluid mechanics is crucial when dealing with the principles of pressure and flow within a fluid. A common device used in fluid mechanics for pressure measurement is the manometer. It's an instrument that uses a column of liquid to measure the difference in pressure between two points. In a simple U-shaped manometer, one end of the tube is often open to the atmosphere, or it can be connected to a container where the pressure is unknown, while the other end is exposed to the pressure to be measured.

For the manometer problem involving oil, which is a fluid with a known density, we use the properties of the fluid (density, height of fluid column) to calculate the pressure difference. The height of the liquid column directly correlates to the pressure exerted by the fluid at the base, with a greater height indicating higher pressure exerted by the fluid's weight due to gravity. These principles allow us to interpret manometer readings and thereby determine unknown pressures, such as the air pressure in a tank.

Pressure Measurement with a Manometer

Pressure measurement in fluids can be made using various instruments, and a manometer is a classic example. This device functions based on the hydrostatic principle which states that the pressure of a fluid at rest changes linearly with changes in height. In other words, pressure differences in a static fluid are caused solely by the weight of the fluid above the point where the pressure is being measured.

Interpreting Manometer Readings

To interpret manometer readings, one must consider the density of the manometer fluid, the gravitational force, and the height difference in the manometer columns. The formula derived from these considerations is
\[\begin{equation}\Delta P = \rho \cdot g \cdot h\end{equation}\]where \(\Delta P\) is the pressure difference, \(\rho\) is the fluid density, \(g\) is the acceleration due to gravity, and \(h\) represents the height difference of the fluid columns. The exercise demonstrates how to calculate the pressure difference using this formula and understanding the concept helps simplify the process for students.

Understanding Absolute Pressure

Absolute pressure is the total pressure exerted on a system, including the atmospheric pressure and the gauge pressure (the pressure difference read from a device like a manometer). It is important as it provides a true representation of pressure acting on a fluid, irrelevant of the surrounding atmospheric pressure.

Absolute Pressure Calculation

In our example, we first measure the pressure difference using a manometer. Then, to obtain the absolute pressure of the air in the tank, we must add the atmospheric pressure to the pressure difference we calculated. This method ensures that we are accounting for all the forces exerted on the fluid, giving us a more accurate measure of the actual pressure within the tank. Absolute pressure is vital in various applications, such as calculating fluid flows and thermodynamic processes.