Hydraulic Fluid Statics
The science of studying the mechanical laws of equilibrium fluids and their applications, with a particular focus on hydraulic pressure and its characteristics in stationary conditions.
Introduction to Fluid Statics
Fluid statics is the science that studies the mechanical laws of equilibrium fluids and their applications. The so-called equilibrium (or静止) refers to the absence of relative motion between macroscopic fluid particles, achieving relative equilibrium. Therefore, the static state of fluids includes two forms:
- One is that the fluid has no relative motion with respect to the Earth, called absolute静止, also known as fluid equilibrium in a gravitational field, such as liquid contained in a stationary container.
- The other is that the fluid as a whole has relative motion with respect to the Earth, but the fluid has no relative motion with respect to the moving container, and there is no relative motion between fluid particles. This kind of静止 is called relative静止 or relative equilibrium of fluids, such as liquid contained in a container undergoing uniform accelerated linear motion or uniform angular velocity rotational motion.
Understanding these states is crucial for analyzing hydraulic pressure distribution in various hydraulic systems. Whether in absolute or relative静止 conditions, the principles governing hydraulic pressure remain consistent, forming the foundation for hydraulic engineering applications.
Forces Acting on Static Fluids
In fluid statics, we analyze two main categories of forces that act on fluids in equilibrium. These forces determine the distribution of hydraulic pressure within the fluid and are essential for understanding fluid behavior in various applications.
| Type | Description |
|---|---|
| Body Forces |
Forces that act on every particle of the fluid, with magnitudes proportional to the mass of the fluid. In homogeneous fluids, body forces are proportional to the volume of the fluid, hence they are also called volume forces.
Common body forces include:
The magnitude of body forces is measured using unit mass force, which is the body force acting on a unit mass of fluid. For a homogeneous fluid with mass m, volume V, and subject to body force F, we have: F = ma = m(fₓi + fᵧj + f_z k)
where the acceleration a = F/m = (fₓi + fᵧj + f_z k) represents the unit mass force, numerically equal to acceleration a. The components fₓ, fᵧ, f_z represent the unit mass force components in the x, y, z coordinate axes, respectively, and are numerically equal to the acceleration components in those axes.
In a gravitational field, fluids are only subject to the Earth's gravitational pull. Taking the z-axis as vertically upward and the xoy plane as horizontal, the components of the unit mass force in the x, y, z axes are: fₓ = 0, fᵧ = 0, f_z = -mg/m = -g
The negative sign indicates that gravitational acceleration g is opposite to the direction of the z-axis.
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| Surface Forces |
Forces that act on the outer surface of the studied fluid, with magnitudes proportional to the surface area. Surface forces include normal forces and tangential forces.
Normal forces are pressures in the direction of the inner normal of the surface. The normal force per unit area is called the fluid's normal stress. Tangential forces are frictional forces along the tangential direction of the surface. The tangential force per unit area is the shear stress caused by fluid viscosity. The mechanism of surface forces is actually the macroscopic manifestation of molecular forces from surrounding fluid molecules or solid molecules on the surface of the studied fluid. In the context of hydraulic pressure, these surface forces are critical for understanding how pressure is transmitted through a fluid. Hydraulic pressure, as a surface force, plays a fundamental role in hydraulic systems, where it is transmitted equally in all directions according to Pascal's principle, enabling the operation of various hydraulic components. |
The interaction between body forces and surface forces determines the equilibrium conditions of fluids. In hydraulic systems, understanding how these forces balance is crucial for predicting hydraulic pressure distribution and designing efficient hydraulic components. The relationship between these forces directly influences the measurement and application of hydraulic pressure in engineering practice.
Fluid Static Pressure and Its Characteristics
Hydraulic pressure is a fundamental concept in fluid statics, representing the normal force per unit area in a stationary fluid. Understanding its characteristics is essential for analyzing and designing hydraulic systems.
| Item | Description |
|---|---|
| Pressure |
In a stationary or relatively stationary fluid, the internal normal surface force per unit area is called pressure. In hydraulic transmission, it is customarily referred to as "hydraulic pressure." This pressure arises from the molecules of the fluid colliding with each other and with the walls of the container.
Hydraulic pressure is a scalar quantity, meaning it has magnitude but no direction, although the force it exerts acts in a specific direction. The unit of hydraulic pressure in the International System of Units is the Pascal (Pa), which is equivalent to one newton per square meter (N/m²). |
| Characteristics of Fluid Static Pressure |
Fluid static pressure exhibits two key characteristics:
These characteristics of hydraulic pressure form the basis for many hydraulic devices and systems, enabling the transmission of force through fluids in various applications from heavy machinery to precision instruments. |
The characteristics of hydraulic pressure are fundamental to the operation of hydraulic systems. The fact that hydraulic pressure acts equally in all directions allows for the transmission of force through pipes and hoses in any orientation. Additionally, because hydraulic pressure acts perpendicular to surfaces, it enables efficient transfer of force to pistons and other components in hydraulic machinery.
Understanding these properties helps engineers design systems that effectively utilize hydraulic pressure for various applications, from heavy lifting to precise control mechanisms. The consistent behavior of hydraulic pressure in static fluids provides a reliable foundation for hydraulic system design and analysis.
Fundamental Equation of Fluid Statics
The fundamental equation of fluid statics describes how hydraulic pressure varies with depth in a stationary fluid. This equation is crucial for understanding the distribution of hydraulic pressure in various hydraulic systems and applications.
The fundamental equation of fluid statics explains the generation of pressure. As shown in Figure 1-5, when a liquid of density ρ in a container is in a stationary state, consider a vertical small liquid column with base area A and height h, where the top surface of the small liquid column coincides with the liquid surface. Since the small liquid column is in equilibrium under the action of gravity and the pressure of the surrounding liquid, from the mechanical equilibrium equation of the small liquid column (p₀A = pA + ρghA), we derive the basic equation of liquid statics:
Where:
- p is the hydraulic pressure at depth h
- p₀ is the pressure at the liquid surface
- ρ is the density of the liquid
- g is the acceleration due to gravity
- h is the depth below the liquid surface
In hydraulic technology, the surface pressure p₀ caused by external forces is often very large, generally ranging from several megapascals to tens of megapascals. The pressure ρgh caused by the weight of the liquid is relatively small compared to p₀. For example, the average density of hydraulic oil is 8829 N/m³, and the height of hydraulic equipment generally does not exceed 10 m. In this case, the static pressure generated by the oil's own weight generally does not exceed 0.088 MPa, so it can be ignored.
This is an important simplification in hydraulic system design, as it allows engineers to focus primarily on the externally applied pressure when analyzing system performance. While the hydraulic pressure due to fluid weight is negligible in most cases, it becomes significant in very tall systems or when dealing with very dense fluids.
Methods of Expressing Pressure
There are several ways to express hydraulic pressure, as shown in Table 1-13:
| Category | Description |
|---|---|
| Absolute Pressure |
The pressure measured with absolute vacuum (where no gas exists) as the reference is called absolute pressure, as shown in Figure 1-6. This is the true hydraulic pressure at a point, including the pressure from the atmosphere.
Absolute pressure is always positive and represents the total pressure exerted on a surface, including atmospheric pressure. It is essential for certain calculations, such as determining boiling points or analyzing vacuum systems. |
| Gauge Pressure |
With atmospheric pressure pₐ as the reference zero line, the pressure above this reference line is called gauge pressure, which is the pressure measured by pressure gauges, hence also known as gauge pressure.
In hydraulic transmission, the pressure p generally refers to gauge pressure. The relationship between absolute pressure and gauge pressure is: Absolute pressure = Gauge pressure + Atmospheric pressure (pₐ)
Most pressure measuring devices in hydraulic systems are calibrated to read gauge pressure, making this the most commonly used pressure reference in hydraulic engineering practice. |
| Vacuum Pressure |
If the absolute pressure at a point in the liquid is less than atmospheric pressure, the value by which this absolute pressure is less than atmospheric pressure is called vacuum pressure. That is, with atmospheric pressure pₐ as the reference zero line, the pressure below this zero line is called vacuum pressure, which is also a gauge pressure.
Vacuum pressure = Atmospheric pressure - Absolute pressure (Pa)
As shown in Figure 1-7, the pump's suction chamber must form a vacuum of h = (pₐ - p₁)/ρg so that atmospheric pressure pₐ can push the oil in the tank up to the pump's suction port installed at height h, allowing the pump to suck in oil. The vacuum pressure in the pump's suction chamber is therefore critical for proper pump operation. |
Understanding the different ways to express hydraulic pressure is essential for proper communication and calculation in hydraulic engineering. The choice of pressure reference depends on the specific application and the type of measurement being made. Gauge pressure is most commonly used in hydraulic system design and operation, while absolute pressure is necessary for certain thermodynamic calculations. Vacuum pressure is particularly important in pump design and suction line analysis.
Pressure Measurement Standards and Measurement
Pressure is the normal force per unit area at various points inside a fluid, also known as "intensity of pressure." The unit of pressure is "Pa." According to different pressure zeros, there are three ways to express it, as shown in Table 1-14.
| Category | Description |
|---|---|
| Absolute Pressure |
With absolute vacuum as the zero point. This represents the total hydraulic pressure at a point, including atmospheric pressure.
Symbol: pₐᵦₛ |
| Gauge Pressure |
Gauge pressure (or manometric pressure) with atmospheric pressure as the zero point. This is the hydraulic pressure measured relative to the surrounding atmosphere.
Symbol: p₉ |
| Vacuum Pressure |
When the absolute pressure is less than atmospheric pressure, the value by which it is less than atmospheric pressure is called vacuum pressure, also known as negative pressure.
Symbol: pᵥₐc Where: p₉ = pₐᵦₛ - pₐ
pᵥₐc = pₐ - pₐᵦₛ
p₉ — gauge pressure (manometric pressure), Pa; pₐᵦₛ — absolute pressure, Pa; pₐ — atmospheric pressure, Pa; pᵥₐc — vacuum pressure, Pa. |
Pressure Measuring Instruments
There are three main types of instruments for measuring hydraulic pressure: metal elastic pressure gauges, electrical pressure gauges, and liquid column pressure gauges.
Metal Elastic Pressure Gauges
These utilize the deformation of metal elastic elements caused by the hydraulic pressure of the liquid being measured. They have a relatively large measuring range and are mostly used in hydraulic transmission systems.
Common types include Bourdon tube gauges, diaphragm gauges, and bellows gauges, which convert pressure into mechanical displacement to indicate the measured hydraulic pressure.
Electrical Pressure Gauges
These convert the deformation of elastic elements into electrical signals, facilitating remote measurement and dynamic measurement of hydraulic pressure.
They are particularly useful in automated systems where pressure data needs to be transmitted to control systems for monitoring and adjustment purposes.
Liquid Column Pressure Gauges
These have high measurement accuracy but a small range, and are generally used in low-pressure experimental settings. They operate based on the height of a liquid column supported by the measured hydraulic pressure.
Common examples include U-tube manometers, piezometers, and inclined-tube manometers for very low pressure measurements.
When the difference between the pressure of the measured fluid and atmospheric pressure is very small, an inclined micro-manometer is often used to improve measurement accuracy. The working principle of the micro-manometer is shown in Figure 1-8.
The communicating vessel is filled with a liquid of density ρᵐ. The tube on the right can rotate around a pivot to form a small acute angle. The original liquid surface of the container is 0-0. When the pressure p of the fluid to be measured is greater than atmospheric pressure pₐ and is introduced into the micro-manometer, the liquid surface in the micro-manometer drops by Δh, while the liquid surface in the measuring tube rises by h, forming equilibrium.
According to the equipressure surface equation:
Where z₁ and z₂ are the elevations of the liquid surfaces. Rearranging gives:
From the geometry of the inclined tube, we know that h = l sinα, where l is the length of the liquid column along the inclined tube and α is the angle of inclination.
According to the principle of equal volume:
Where A is the cross-sectional area of the reservoir and a is the cross-sectional area of the inclined tube. Therefore:
When A >> a, the term Δh can be neglected, and the gauge pressure of the measured fluid is approximately:
The inclined design increases the length of the liquid column for a given pressure difference, making it easier to measure small changes in hydraulic pressure. This principle is widely used in laboratories and in applications where precise measurement of small pressure differences is required.
Total Force Exerted by Static Fluid on Solid Surfaces
Calculating the total force exerted by a static fluid on solid surfaces is essential for designing hydraulic system components, storage tanks, and other hydraulic structures. The magnitude and distribution of hydraulic pressure determine these forces, which must be considered in structural design.
| Category | Description |
|---|---|
| Total Pressure on Plane Surfaces |
Consider an arbitrarily shaped flat plate with area A, placed in a stationary liquid (density ρ), as shown in Figure 1-9. The pressure p at any point in the liquid is proportional to the depth h and acts perpendicularly toward the plate. The total force exerted by the liquid on the plate is equivalent to finding the resultant of a parallel force system.
On the pressure-receiving surface of the plate, take an infinitesimal area dA, where the pressure can be considered uniformly distributed: p = p₀ + ρgh = p₀ + ρgy sinα
Therefore, the infinitesimal force exerted by the liquid on the infinitesimal area dA is:
dF = pdA = (p₀ + ρgy sinα)dA
Integrating the above equation gives the total pressure exerted by the fluid on the plate A:
F = ∫dF = ∫pdA = ∫(p₀ + ρgy sinα)dA = p₀A + ρg sinα∫ydA
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Since ∫ydA is the moment of area of plane A about the ox axis passing through point o, i.e., ∫ydA = ycA, where yc is the distance from the centroid c of the plate to the ox axis, and yc sinα = hc, the total pressure is:
F = p₀A + ρgA yc sinα = p₀A + ρghcA
The point of application of the total pressure is called the center of pressure, denoted as point d. The moment of the total pressure F about the ox axis should be equal to the sum of the moments of the infinitesimal pressures dF about the ox axis:
y_d F = ∫pydA = ∫(p₀ + ρgy sinα)ydA = p₀A yc + ρg sinα∫y²dA
Where ∫y²dA is the moment of inertia of area A about the ox axis, Iₓ, and Iₓ = Iₓc + yc²A, where Iₓc is the moment of inertia of plane A about the axis passing through point c and parallel to the ox axis.
When the liquid surface is at atmospheric pressure, the formula for the center of pressure is: y_d = yc + Iₓc/(ycA)
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| Total Pressure on Curved Surfaces |
Calculating the force exerted by a fluid on a curved surface involves finding the resultant of a spatial force system. Since the directions of the forces at different points on the curved surface are different, it is common to decompose the pressure dF on each infinitesimal area into components and then sum them up.
Consider a curved surface ab with area A, submerged in a liquid, as shown in Figure 1-10. Assuming the liquid surface is at atmospheric pressure, take an infinitesimal area dA on the curved surface ab (with corresponding submerged depth h), the force acting on it is: dF = ρghdA |
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Horizontal Component:
Decompose dF into horizontal component dFₓ and vertical component dFᵧ, then integrate over the entire curved surface A: Fₓ = ∫dFₓ = ∫dF cosθ = ∫ρghdA cosθ = ρg∫hdAₓ = ρghcAₓ
Where dAₓ = dA cosθ is the projection of area dA onto the zox coordinate plane, and ∫hdAₓ = hcAₓ is the moment of area of the projected area Aₓ about the ox axis (the x-axis is perpendicular to the paper).
The horizontal component of the total pressure is: Fₓ = ρghcAₓ
Its line of action passes through the center of pressure of Aₓ.
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Vertical Component:
Fᵧ = ∫dFᵧ = ∫dF sinθ = ∫ρghdA sinθ = ρg∫hdAᵧ = ρgV
Where Aᵧ is the projected area of area A onto the yox coordinate plane; ∫hdAᵧ is the volume of liquid above the curved surface, commonly referred to as the pressure prism. Thus:
The vertical component of the total pressure is: Fᵧ = ρgV
That is, the vertical component of the total force on the curved surface is equal to the weight of the pressure prism, and its line of action passes through the center of gravity of the pressure prism.
For cylindrical curved surfaces, the horizontal component Fₓ and vertical component Fᵧ of the total force must be coplanar, and the resultant total force is: F = √(Fₓ² + Fᵧ²)
θ = arctan(Fₓ/Fᵧ)
The line of action of the total pressure must pass through the intersection of the vertical and horizontal components.
It should be noted that the pressure prism is the enclosed space formed by the curved surface under consideration, the vertical plane passing through the boundary of the curved surface, and the free surface of the liquid or its extension. Regardless of whether this volume is filled with liquid or not, the calculation formula for the vertical component Fᵧ = ρgV remains unchanged. However, the direction of the vertical component differs depending on whether the pressure prism is on the same side or the opposite side of the pressure-receiving surface. |
The calculation of forces exerted by static fluids on solid surfaces is fundamental to hydraulic engineering design. Whether dealing with plane or curved surfaces, understanding how hydraulic pressure distributes and results in resultant forces is essential for ensuring the structural integrity and proper functioning of hydraulic components and systems.
Engineers use these principles to design everything from hydraulic cylinders and valves to storage tanks and dams, ensuring they can withstand the hydraulic pressure forces they will encounter in operation. The ability to accurately calculate these forces allows for efficient and safe hydraulic system design.