1.4 Pressure and hydrostatic pressure
9 min read•august 20, 2024
Pressure is a fundamental concept in fluid dynamics, quantifying force per unit area. It's crucial for understanding how fluids behave at rest and in motion. This topic explores pressure's definition, units, and its relationship to depth in fluids.
, caused by a fluid's weight, increases linearly with depth. The , P = ρgh, is key for calculating pressure at various depths. This concept is vital for designing fluid systems and analyzing submerged structures.
Definition of pressure
- Pressure is a fundamental concept in fluid dynamics that quantifies the force applied perpendicular to a surface per unit area
- Pressure is a scalar quantity, meaning it has magnitude but no direction, and it acts equally in all directions at a given point in a fluid
Pressure as force per unit area
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- Pressure is defined as the force applied perpendicular to a surface divided by the area over which the force acts
- Mathematically, pressure is expressed as , where is pressure, is force, and is area
- The force can be applied by a solid object, a liquid, or a gas
Units of pressure measurement
- The SI unit for pressure is the (Pa), which is equal to one newton per square meter ()
- Other common units of pressure include pounds per square inch (psi), atmospheres (atm), and
- Pressure units can be converted using appropriate conversion factors (e.g., )
Hydrostatic pressure
- Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above a given point
- Hydrostatic pressure is an important consideration in fluid dynamics, as it affects the behavior of fluids in various systems (e.g., pipes, tanks, and hydraulic devices)
Pressure at a depth in a fluid
- In a fluid at rest, the hydrostatic pressure at a given depth is determined by the weight of the fluid column above that point
- The pressure at a depth increases linearly with the depth, as the weight of the fluid column increases
Pressure vs depth relationship
- The relationship between hydrostatic pressure and depth is given by the equation , where is pressure, is the fluid density, is the acceleration due to gravity, and is the depth below the surface
- This relationship shows that pressure increases linearly with depth, with the rate of increase determined by the fluid density and the acceleration due to gravity
Hydrostatic pressure equation
- The hydrostatic pressure equation, , is a fundamental relationship in fluid dynamics
- This equation is used to calculate the pressure at a given depth in a fluid, and it is applicable to both liquids and gases
- The equation assumes that the fluid is at rest and that the density is constant throughout the fluid column
Pressure in different fluid states
- The behavior of pressure in fluids depends on the state of the fluid, whether it is a liquid or a gas
- Understanding the differences in pressure characteristics between liquids and gases is crucial for analyzing and designing fluid systems
Pressure in liquids
- In liquids, pressure is transmitted equally in all directions (Pascal's law) and increases linearly with depth (hydrostatic pressure)
- Liquids are generally considered incompressible, meaning that their density remains constant under normal conditions
- The pressure in a liquid is determined by the height of the liquid column above a given point and the density of the liquid
Pressure in gases
- In gases, pressure is also transmitted equally in all directions, but the relationship between pressure and depth is more complex than in liquids
- Gases are compressible, meaning that their density can change significantly with changes in pressure
- The pressure in a gas is related to the temperature and volume of the gas, as described by the ()
Incompressible vs compressible fluids
- Fluids can be classified as either incompressible or compressible based on how their density changes with pressure
- Incompressible fluids, such as liquids, have a constant density that does not change significantly with pressure under normal conditions
- Compressible fluids, such as gases, have a density that varies with pressure, and their behavior is governed by the laws of thermodynamics
Atmospheric pressure
- Atmospheric pressure is the pressure exerted by the weight of the Earth's atmosphere on a surface
- Understanding atmospheric pressure is important for various applications in fluid dynamics, such as weather forecasting, aviation, and vacuum systems
Definition of atmospheric pressure
- Atmospheric pressure is defined as the force per unit area exerted by the weight of the Earth's atmosphere on a surface
- At sea level, atmospheric pressure is caused by the weight of the air column extending from the surface to the top of the atmosphere
Standard atmospheric pressure
- is defined as the pressure exerted by the atmosphere at sea level under normal conditions
- The standard atmospheric pressure is 101,325 pascals (Pa) or 1 atmosphere (atm)
- This value is used as a reference for measuring and comparing pressures in various applications
Atmospheric pressure vs altitude
- Atmospheric pressure decreases with increasing altitude, as the weight of the air column above a given point decreases
- The relationship between atmospheric pressure and altitude is approximately exponential, with pressure decreasing more rapidly at higher altitudes
- At an altitude of about 5,500 meters (18,000 feet), the atmospheric pressure is approximately half of its value at sea level
Gauge pressure vs absolute pressure
- Pressure measurements can be expressed in either or , depending on the reference point used
- Understanding the difference between gauge pressure and absolute pressure is crucial for properly interpreting pressure measurements and performing calculations
Definition of gauge pressure
- Gauge pressure is the pressure measured relative to the local atmospheric pressure
- Gauge pressure is zero-referenced against atmospheric pressure, meaning that a gauge pressure of zero corresponds to the local atmospheric pressure
- Gauge pressure can be positive (above atmospheric pressure) or negative (below atmospheric pressure)
Definition of absolute pressure
- Absolute pressure is the pressure measured relative to a perfect vacuum
- Absolute pressure is always positive and includes the local atmospheric pressure
- Absolute pressure is used in thermodynamic calculations and when dealing with vacuum systems
Relationship between gauge and absolute pressure
- Absolute pressure is equal to the sum of gauge pressure and atmospheric pressure
- Mathematically, this relationship is expressed as , where is absolute pressure, is gauge pressure, and is atmospheric pressure
- To convert between gauge pressure and absolute pressure, the local atmospheric pressure must be known
Pascal's law
- Pascal's law, also known as the principle of transmission of fluid-pressure, is a fundamental principle in fluid dynamics
- Pascal's law states that pressure applied to a confined fluid is transmitted undiminished in all directions and acts with equal force on equal areas
Statement of Pascal's law
- Pascal's law can be stated as follows: pressure applied to a confined fluid is transmitted undiminished in all directions and acts with equal force on equal areas
- This means that when pressure is applied to a fluid in a closed system, the pressure is transmitted equally throughout the fluid, regardless of the shape or size of the container
Applications of Pascal's law
- Pascal's law has numerous applications in fluid systems, such as hydraulic lifts, hydraulic brakes, and hydraulic presses
- In these systems, a small force applied to a small area can be used to generate a large force over a larger area, thanks to the transmission of pressure through the fluid
Hydraulic systems and pressure transmission
- rely on Pascal's law to transmit pressure and force through a fluid medium
- In a hydraulic system, a fluid (usually oil) is used to transmit pressure from one point to another
- The pressure is generated by a pump and is transmitted through pipes and hoses to actuators, such as cylinders or motors, which convert the fluid pressure into mechanical force
Hydrostatic paradox
- The is a counterintuitive phenomenon in fluid dynamics that demonstrates the independence of hydrostatic pressure from the shape of the container
- The hydrostatic paradox has important implications for the design and analysis of fluid systems
Explanation of hydrostatic paradox
- The hydrostatic paradox states that the pressure at the bottom of a fluid-filled container depends only on the height of the fluid column and the density of the fluid, not on the shape or volume of the container
- This means that a tall, narrow container and a short, wide container with the same fluid height will have the same pressure at the bottom, even though the tall container has less fluid volume
Pressure independence of container shape
- The hydrostatic paradox arises from the fact that hydrostatic pressure is determined by the weight of the fluid column above a given point
- The weight of the fluid column depends only on its height and the fluid density, not on the cross-sectional area of the container
- As a result, the pressure at a given depth is independent of the shape of the container
Implications for fluid system design
- The hydrostatic paradox has important implications for the design of fluid storage tanks, dams, and other structures that contain fluids
- Engineers must consider the height of the fluid column when designing these structures, rather than just the volume of fluid they contain
- Failing to account for the hydrostatic paradox can lead to structural failures or leaks in fluid systems
Measuring pressure
- Measuring pressure is essential for monitoring and controlling fluid systems in various applications
- Several types of pressure measurement devices are used, depending on the specific requirements of the application
Types of pressure gauges
- Pressure gauges are devices used to measure and display pressure readings
- Common types of pressure gauges include Bourdon tube gauges, diaphragm gauges, and piezoresistive gauges
- Each type of gauge has its own advantages and limitations, such as accuracy, durability, and pressure range
Manometers and pressure measurement
- Manometers are simple devices used to measure pressure by balancing the pressure of a fluid against a column of liquid, usually water or mercury
- U-tube manometers consist of a U-shaped tube partially filled with a liquid, with one end connected to the fluid system being measured and the other end open to the atmosphere
- The difference in height between the liquid columns in the two legs of the manometer indicates the pressure difference between the system and the atmosphere
Pressure transducers and sensors
- Pressure transducers and sensors are electronic devices that convert pressure into an electrical signal
- Common types of pressure transducers include strain gauge, capacitive, and piezoelectric transducers
- Pressure transducers offer advantages such as high accuracy, fast response times, and the ability to interface with electronic control systems
Pressure forces on submerged surfaces
- When a surface is submerged in a fluid, it experiences pressure forces due to the hydrostatic pressure distribution
- Understanding the pressure forces on submerged surfaces is crucial for designing and analyzing structures such as dams, tanks, and underwater vehicles
Pressure distribution on submerged planes
- The pressure distribution on a submerged plane surface varies linearly with depth, following the hydrostatic pressure equation ()
- The pressure at any point on the surface depends on the depth of that point below the fluid surface
- The pressure distribution is always perpendicular to the surface, regardless of the orientation of the plane
Resultant force and center of pressure
- The resultant force on a submerged plane surface is the sum of all the pressure forces acting on the surface
- The magnitude of the resultant force depends on the area of the surface and the average pressure acting on it
- The point of application of the resultant force is called the center of pressure, which is the point where the resultant force acts as if it were concentrated
Buoyancy and Archimedes' principle
- is the upward force exerted by a fluid on an object immersed in it
- Archimedes' principle states that the buoyant force acting on an object is equal to the weight of the fluid displaced by the object
- The buoyant force acts through the center of buoyancy, which is the centroid of the displaced fluid volume
- Understanding buoyancy and Archimedes' principle is essential for analyzing the stability and equilibrium of floating and submerged objects
Key Terms to Review (19)
Absolute pressure: Absolute pressure is the total pressure exerted on a system, measured relative to a perfect vacuum. This measurement includes the atmospheric pressure acting on the surface of a liquid or gas, which makes it essential in understanding fluid behavior under varying conditions. Absolute pressure is crucial in calculations related to buoyancy, hydrostatic forces, and fluid dynamics, as it provides a baseline from which changes in pressure can be measured.
Bar: A bar is a unit of pressure defined as 100,000 pascals, which is commonly used in various scientific and engineering applications. This measurement connects to the behavior of fluids under different conditions and plays a crucial role in understanding pressure variations, especially in the context of hydrostatics, where pressure changes with depth in a fluid due to the weight of the fluid above.
Bernoulli's equation: Bernoulli's equation is a principle in fluid dynamics that describes the conservation of energy in a flowing fluid, relating the pressure, velocity, and height of the fluid at different points along a streamline. This equation reveals how changes in velocity and elevation affect pressure within the fluid, establishing a key connection between pressure and fluid flow, and has wide-ranging applications from hydrostatics to aerodynamics.
Buoyancy: Buoyancy is the upward force exerted by a fluid that opposes the weight of an object submerged in it. This force is a critical concept that relates to how objects behave in liquids and gases, affecting everything from floating boats to the movement of particles in a fluid. Understanding buoyancy helps explain phenomena such as hydrostatic pressure, Archimedes' principle, and how different densities influence fluid behavior in natural systems.
Cavitation: Cavitation is the formation and collapse of vapor-filled bubbles in a liquid, which occurs when local pressure drops below the vapor pressure of the liquid. This phenomenon can lead to significant damage in hydraulic systems and affects fluid flow, particularly in pumps and propellers. The process is closely related to the concepts of pressure dynamics and bubble behavior, making it crucial for understanding fluid behavior under varying pressure conditions.
Compressibility: Compressibility refers to the measure of how much a substance can decrease in volume under pressure. It plays a crucial role in understanding the behavior of fluids, particularly gases, under varying pressure conditions and is essential for analyzing phenomena like sound propagation and flow characteristics at different speeds. The concept of compressibility connects to various fluid dynamics aspects, including how changes in pressure influence fluid behavior, the speed of sound in a medium, and how compressibility effects become significant at high velocities.
Gauge pressure: Gauge pressure is the pressure measured relative to atmospheric pressure, meaning it excludes the atmospheric pressure from its value. This measurement is crucial in various applications, particularly in fluid dynamics, where understanding the behavior of fluids under different pressures is essential. Gauge pressure is important for determining the performance of hydraulic systems, understanding buoyancy in fluids, and analyzing the behavior of submerged objects in liquids.
Hydraulic systems: Hydraulic systems are technologies that use liquid under pressure to transmit power and perform work. These systems rely on the principles of fluid mechanics, particularly pressure and hydrostatic pressure, to move components and exert force in various applications, from heavy machinery to vehicle braking systems. They also illustrate the conservation of energy, as the work done in these systems is equal to the energy transferred through the fluid.
Hydrostatic Equilibrium: Hydrostatic equilibrium refers to the state in which the forces acting on a fluid are balanced, resulting in a stable condition where there is no net movement. In this state, the gravitational force pulling the fluid downwards is exactly countered by the pressure gradient force pushing upwards, leading to a uniform pressure distribution within the fluid at rest. Understanding hydrostatic equilibrium is crucial for analyzing fluid behavior in various contexts, such as natural bodies of water and atmospheric conditions.
Hydrostatic Paradox: The hydrostatic paradox refers to the principle that the pressure at a given depth in a fluid is determined solely by the height of the fluid column above it, regardless of the shape or volume of the container holding the fluid. This means that two containers with different shapes can exert the same pressure at the bottom if they have the same height of liquid. This principle highlights the relationship between fluid pressure and depth, emphasizing that hydrostatic pressure is uniform in a static fluid environment.
Hydrostatic Pressure: Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity, typically measured at a specific depth within the fluid. It plays a crucial role in understanding how fluids behave under different conditions and affects various phenomena such as buoyancy and fluid statics. The relationship between hydrostatic pressure and depth is linear, meaning that as you go deeper into a fluid, the pressure increases proportionally.
Hydrostatic Pressure Equation: The hydrostatic pressure equation describes the pressure exerted by a fluid at rest due to the force of gravity. It is expressed mathematically as $$P =
ho g h$$, where $$P$$ is the hydrostatic pressure, $$\rho$$ is the fluid density, $$g$$ is the acceleration due to gravity, and $$h$$ is the height of the fluid column above the point of measurement. This equation is crucial for understanding how pressure changes with depth in a fluid and connects directly to the concepts of pressure and fluid statics.
Ideal Gas Law: The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation $$PV = nRT$$. This law allows for the prediction of the behavior of gases under various conditions and connects concepts like pressure and speed of sound by providing a basis for understanding how gas molecules interact and move.
Pascal: The Pascal is a unit of pressure defined as one newton per square meter, symbolized as Pa. It serves as a fundamental measure of pressure in various fields, including fluid dynamics, where it helps quantify forces exerted by fluids at rest and in motion, and is crucial in understanding hydrostatic pressure and buoyancy effects.
Pascal's Principle: Pascal's Principle states that when pressure is applied to a confined fluid, the pressure change is transmitted undiminished throughout the fluid in all directions. This principle is crucial in understanding how fluids behave under pressure and is foundational for concepts like hydrostatic pressure, where the pressure in a fluid at rest increases with depth due to the weight of the fluid above it.
Pressure vs Depth Relationship: The pressure vs depth relationship describes how the pressure in a fluid increases with increasing depth due to the weight of the fluid above. This concept is fundamental to understanding hydrostatic pressure, where pressure at a given depth is directly proportional to the height of the fluid column above that point, and it highlights how fluid weight affects pressure within liquids.
Standard Atmospheric Pressure: Standard atmospheric pressure is the pressure exerted by the weight of the air above a given point at sea level, defined as 101.325 kPa or 1 atm. This value serves as a reference point in fluid dynamics for various calculations involving fluid behavior, hydrostatics, and the design of equipment that operates within different pressure conditions. It is a critical benchmark that helps in understanding how pressure variations impact fluid systems and contributes to calculations related to buoyancy and hydrostatic forces.
Surface Tension: Surface tension is a physical property of liquids that describes the elastic-like force existing at the surface of a liquid, caused by the cohesive forces between liquid molecules. This phenomenon allows liquids to resist external forces and is responsible for behaviors such as the formation of droplets and the ability of small objects to float on the surface without sinking. Understanding surface tension is crucial for analyzing how pressure changes affect liquids and how bubbles behave in fluids.
Viscosity: Viscosity is a measure of a fluid's resistance to deformation or flow, indicating how thick or sticky it is. It plays a crucial role in determining how fluids behave under various conditions, affecting everything from pressure changes to momentum conservation and fluid dynamics equations.