FLUID PRESSURE OVERVIEW

Introduction

Pressure in fluids refers to the force per unit area that a fluid exerts on any surface that is in contact with it. It is a scalar quantity and is measured in units of pascals (Pa) or newtons per square meter (N/m²).

The pressure in a fluid at rest is equal in all directions and is transmitted equally throughout the fluid. This is known as Pascal’s principle, which states that the pressure applied to an enclosed fluid is transmitted uniformly throughout the fluid in all directions.

The pressure in a fluid can be calculated using the formula:

P = F/A

where P is the pressure, F is the force applied on the surface, and A is the area of the surface. This formula applies to both liquids and gases.

In a liquid, the pressure at a given point depends on the density of the liquid, the height of the column of liquid above the point, and the acceleration due to gravity. This relationship is described by the hydrostatic equation:

P = ρgh

where ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the column of liquid above the point.

In a gas, the pressure depends on the temperature, volume, and number of particles in the gas. This relationship is described by the ideal gas law:

P = nRT/V

where P is the pressure, n is the number of particles, R is the gas constant, T is the temperature, and V is the volume.

Understanding pressure in fluids is important in many fields, including fluid mechanics, engineering, and atmospheric science.

 

Pressure is a physical quantity that measures the force exerted per unit area. In other words, it is the amount of force that is applied to a given area. The SI unit of pressure is the pascal (Pa), which is defined as one newton of force per square meter.

Pressure can be experienced in various forms, such as atmospheric pressure, hydraulic pressure, and gas pressure. Atmospheric pressure is the pressure exerted by the Earth’s atmosphere on its surface, and is measured using a barometer. Hydraulic pressure is the pressure exerted by a liquid, and is used in hydraulic systems to transmit force. Gas pressure is the pressure exerted by a gas, and is related to the number of gas molecules in a given volume.

In engineering and physics, pressure is an important concept because it is used to describe a variety of phenomena, such as fluid dynamics, thermodynamics, and acoustics. Pressure is also used in many practical applications, such as in the design of hydraulic systems, the measurement of blood pressure, and the operation of pneumatic tools.

 

Pascal’s principle, also known as the principle of transmission of fluid-pressure, states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of the containing vessel. This principle has important applications in many fields, including engineering, physics, and medicine.

One of the most common applications of Pascal’s principle is the hydraulic press. A hydraulic press is a machine that uses the principle of transmission of fluid-pressure to multiply force. It consists of two cylinders, one large and one small, connected by a tube filled with a hydraulic fluid. When force is applied to the small piston, the pressure in the fluid increases and is transmitted to the larger piston, which experiences a much larger force.

For example, if a small piston with an area of 1 square inch is pushed with a force of 10 pounds, the pressure in the fluid will increase by the same amount, and the force applied to the larger piston with an area of 10 square inches will be 100 pounds. This allows the hydraulic press to generate a much greater force than the input force, making it useful in a variety of applications such as manufacturing, construction, and mining.

Another application of Pascal’s principle is in car brakes. The brake system of a car uses a hydraulic fluid to transmit force from the brake pedal to the brake pads, which then press against the rotating wheels to slow down or stop the car. When the brake pedal is pressed, it applies force to a small piston in the master cylinder, which increases the pressure in the brake fluid and transmits it to the larger pistons in the brake calipers. This multiplies the force applied to the brake pads, allowing them to effectively stop the car.

Overall, Pascal’s principle plays a crucial role in many aspects of modern technology and engineering, allowing us to create machines that are much more efficient and powerful than would be possible with purely mechanical systems.

 

The pressure at a point below a liquid surface is determined by the depth of the point and the density of the liquid. The pressure increases with the depth of the point below the surface, as the weight of the liquid above the point increases.

This relationship is described by the hydrostatic pressure equation, which states that the pressure at a point in a liquid is equal to the product of the density of the liquid, the acceleration due to gravity, and the depth of the point below the surface:

P = ρgh

where P is the pressure at the point (in Pascals), ρ is the density of the liquid (in kilograms per cubic meter), g is the acceleration due to gravity (in meters per second squared), and h is the depth of the point below the surface (in meters).

From this equation, it is clear that the pressure at a point below a liquid surface increases linearly with the depth of the point, as long as the density of the liquid and the acceleration due to gravity remain constant. Therefore, the deeper the point, the higher the pressure will be.

 

Atmospheric pressure, also known as air pressure, is the force exerted by the weight of the Earth’s atmosphere on the surface below it. It is a measure of the amount of air molecules in a given volume of space.

The Earth’s atmosphere is composed of a mixture of gases, primarily nitrogen (78%), oxygen (21%), and trace amounts of other gases such as carbon dioxide, water vapor, and helium. These gases are constantly in motion and exert a force on any surface that they come into contact with.

At sea level, the standard atmospheric pressure is defined as 101,325 pascals (Pa), or 1 atmosphere (atm). This means that the weight of the air column above each square meter of Earth’s surface is approximately 101,325 newtons (N).

Atmospheric pressure decreases with increasing altitude because there are fewer air molecules in the atmosphere as you move higher up. This is why climbers on high mountains often experience difficulty breathing, as the reduced atmospheric pressure makes it more difficult for their lungs to take in oxygen.

Atmospheric pressure is measured using a barometer, which typically consists of a glass tube filled with mercury or other liquid that is open at one end and sealed at the other. The tube is turned upside down and placed in a dish of mercury, and as the air pressure pushes down on the surface of the mercury in the dish, the mercury in the tube rises.

The most commonly used unit of atmospheric pressure is the millibar (mb) or hectopascal (hPa). One millibar is equal to one-thousandth of a bar, or 100 pascals.

Atmospheric pressure plays an important role in weather patterns and the behavior of air masses. It is affected by temperature, humidity, and wind patterns, and can change rapidly in response to these factors. Understanding atmospheric pressure is therefore essential for predicting and understanding weather patterns and their impact on our daily lives.

 

There are several devices used for measuring pressure, including:

1. Simple barometer: A simple barometer is a device that measures atmospheric pressure. It typically consists of a glass tube filled with mercury or other fluid and inverted in a dish of mercury. The weight of the atmosphere presses down on the mercury in the dish, causing the level in the tube to rise or fall.
2. Manometer: A manometer is a device used to measure pressure of gases and liquids. It consists of a U-shaped tube filled with liquid, such as water or mercury, and connected to the system being measured. The pressure difference between the two ends of the tube causes the liquid to rise or fall in one of the arms, indicating the pressure.
3. Siphon: A siphon is a device used to measure the pressure difference between two points in a fluid system. It consists of a U-shaped tube with one end placed in the fluid at the higher pressure and the other end placed in the fluid at the lower pressure. The difference in pressure causes the fluid to flow from the higher pressure side to the lower pressure side, allowing the pressure difference to be measured.
4. Syringe: A syringe can be used to measure pressure in a closed system. By connecting the syringe to the system and pulling back on the plunger, the pressure inside the system can be measured by the resistance of the plunger.
5. Pump: A pump can also be used to measure pressure in a closed system. By pumping air or liquid into the system, the pressure inside the system can be increased until it reaches a desired level, allowing the pressure to be measured.

 

The relative density of a liquid can be determined using either a U-tube or Hare’s apparatus. Both of these methods are based on the principle of Archimedes’ buoyancy, which states that the upward force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

Here are the steps to determine the relative density of a liquid using a U-tube:

1. Fill a U-tube with the liquid whose density is known. The liquid should be at the same level in both arms of the U-tube.
2. Use a dropper to add a small amount of the liquid whose density is to be determined to one arm of the U-tube. The liquid should be added slowly and carefully so as not to create bubbles or disturb the equilibrium of the system.
3. Measure the difference in height between the two levels of the liquid in the U-tube. This is the difference in the pressure exerted by the two liquids.
4. Use the formula D = h / H, where D is the relative density of the liquid being measured, h is the difference in height between the two levels of the liquid, and H is the height of the liquid whose density is known.

 

Here are the steps to determine the relative density of a liquid using Hare’s apparatus:

1. Fill the lower part of the Hare’s apparatus with the liquid whose density is known. The liquid should be at the same level as the bottom of the capillary tube.
2. Place a small amount of the liquid whose density is to be determined in the capillary tube using a dropper.
3. Adjust the height of the reservoir so that the meniscus of the liquid in the capillary tube is at the same level as the meniscus of the liquid in the lower part of the apparatus.
4. Measure the height of the liquid in the capillary tube above the level of the liquid in the lower part of the apparatus.
5. Use the formula D = (h / H) + 1, where D is the relative density of the liquid being measured, h is the height of the liquid in the capillary tube above the level of the liquid in the lower part of the apparatus, and H is the height of the liquid whose density is known.

 

Calculations

Example of calculating pressure at the bottom of a tank

1) Suppose you have a tank with a height of 5 meters, and it is filled with water to a depth of 4 meters. What is the pressure at the bottom of the tank?

Solution: First, we need to find the weight of the water column above the bottom of the tank. The weight of water is equal to its density multiplied by its volume and the acceleration due to gravity. The density of water is approximately 1000 kg/m^3, and the volume of the water column is (5 – 4) = 1 m^3. The acceleration due to gravity is 9.8 m/s^2.

Weight of water column = density x volume x acceleration due to gravity = 1000 kg/m^3 x 1 m^3 x 9.8 m/s^2 = 9800 N

The pressure at the bottom of the tank is equal to the weight of the water column divided by the area of the bottom of the tank. Assuming the tank has a square base with sides of 2 meters, the area of the bottom of the tank is 2m x 2m = 4 m^2.

Pressure = weight of water column / area of the bottom of the tank = 9800 N / 4 m^2 = 2450 Pa

Therefore, the pressure at the bottom of the tank is 2450 Pa.

 

Example of calculating pressure in a closed container

2) Suppose you have a sealed container with a volume of 1 cubic meter and filled with air at a temperature of 20 degrees Celsius and a pressure of 1 atm. What is the pressure in the container if the temperature is increased to 30 degrees Celsius?

Solution: Assuming that the volume of the container does not change, we can use the ideal gas law to calculate the pressure of the air in the container.

PV = nRT

where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin.

First, we need to convert the temperature from Celsius to Kelvin by adding 273.15.

Initial conditions: P1 = 1 atm V = 1 m^3 T1 = 20 + 273.15 = 293.15 K

n = ? (unknown)

We can solve for n using the ideal gas law:

n = PV / RT

where R = 8.31 J/(mol*K) is the gas constant.

n = (1 atm x 1 m^3) / (8.31 J/(mol*K) x 293.15 K) = 0.0408 mol

Now we can use the ideal gas law again to find the final pressure of the air in the container:

Final conditions: V = 1 m^3 T2 = 30 + 273.15 = 303.15 K n = 0.0408 mol

P2 = nRT2 / V = (0.0408 mol x 8.31 J/(mol*K) x 303.15 K) / 1 m^3 = 101750 Pa

Therefore, the pressure in the container at 30 degrees Celsius is approximately 101750 Pa, which is greater than the initial pressure of 1 atm.