The principle of hydrostatic equilibrium is useful to determine the pressure exerted by the fluid on the wall of a column or container.
Consider a column filled with a stationary mass of static fluid. Take any cross-section of the column parallel to the earth’s surface. The pressure is the same on this cross-section, but if we take another cross-section of the column, the pressure is different than the first cross-section. So, the pressure is changing from height to height.
Hydrostatic Equilibrium Equation Derivation
Assume the following dimension of the column:
- The cross-sectional area of the column is
- Pressure is at height from the bottom of the column.
- The density of the fluid is
Now, we will analyze a small portion of the column with height and area
The three forces acting on this small portion of the column are as follows:
- Pressure is acting in the upward direction =
- The force due to pressure acts in a downward direction =
- The force due to gravity acting downward =
Force due to gravity is equal to mass times acceleration =
Mass = Volume × Density =
Volume = Area × Height =
Therefore,
And force due to gravity =
The resultant force on a small portion must be zero, and forces acting upward are taken as positive, and forces acting downward are taken as negative.
(1)
Divide equation (1) by
(2)
Equation (2) is the desired basic equation that can be used for obtaining the pressure
at any height.
Incompressible Fluid
For incompressible fluids, density is independent of pressure.
Integrating Equation (2), we get
(3)
Equation no. 3 shows that the pressure is at its maximum at the base of the column
or container of the fluid, decreasing as we move up the column.
Consider that the pressure at the bottom of the column is where , and the pressure at any height above the base is such that , then
(4)
Integrating the above equation (4), we get
(5)
Where and are expressed in , in , in , and in in SI Unit.
With the help of Equation (5), the pressure difference in a fluid between any two points can be obtained by measuring the height of the vertical column of the fluid.
Compressible Fluids
For compressible fluids, density changes with pressure.
For an ideal gas, the density is given by the following relation:
(6)
Here,
= absolute pressure
= molecular weight of the gas
= universal gas constant
= absolute temperature.
Putting the value of from Equation (6) into Equation (2),
(7)
Rearranging Equation (7),
(8)
Integrating Equation (8), we get
(9)
Integrating the above equation between two heights and where the pressures acting are and , we get
(10)
Equation (10) is known as the Barometric Equation, giving us the idea of pressure distribution within an ideal gas for isothermal conditions.
Hydrostatic Equilibrium in a Centrifugal Field
In a rotating centrifuge, a layer of liquid is thrown outward at the axis of rotation and held against the bowl’s wall by centrifugal force.
The surface of the liquid is shaped like the paraboloid of the revolution. Still, in industrial centrifuges, the rotational speed is so high and the centrifugal force is so much greater than the force of gravity that the liquid surface is virtually cylindrical and coaxial with the axis of rotation.
This situation is shown in Figure 2: Hydrostatic Equilibrium in a Centrifugal Field.
Here,
= Radial distance from the axis of rotation to the free liquid surface.
= The radius of the centrifugal bowl.
The entire mass of the liquid indicated in the figure is rotating like a rigid body, with no sliding of one layer of liquid over another.
Under these conditions, the pressure distribution in the liquid may be determined by the principle of fluid statics.
The pressure drop over any ring of rotating liquid is calculated as follows:
considering the ring of liquid shown in the figure above and the volume element of thickness at radius
Where,
= Centrifugal Force.
= Mass of liquid in an element.
= angular velocity
If is the density of liquid and “b” is the breadth of the ring,
Eliminating gives,
The change in pressure over the element is the force exerted by the element of the liquid divided by the area of the ring
The pressure drop over the entire ring is
Assuming the density is constant and integrating gives
Summary: Hydrostatic Equilibrium and Pressure Variation in Fluids
The concept of hydrostatic equilibrium delves into the behavior of fluids within columns or containers and how pressure changes with height. In a stationary fluid column, pressure remains constant along horizontal cross-sections parallel to the earth’s surface but varies from one cross-section to another as you ascend.
For incompressible fluids, the pressure decreases with increasing height, expressed as
p + zρg = constant, where 'z'
is pressure, 'z'
is height, ‘ρ
‘ is density, and '
g’ is the acceleration due to gravity.
In contrast, compressible fluids like ideal gases exhibit a pressure-height relationship defined by the Barometric Equation: ln(p) + g(M/RT)dz = constant
, where M
represents molecular weight, R
is the universal gas constant, and T
is the absolute temperature.
The concept extends to hydrostatic equilibrium in rotating centrifuges, where a layer of liquid is pushed outward by centrifugal force. The pressure distribution follows fluid statics principles, and the pressure drop over a rotating liquid ring can be calculated using dp = ω²ρrdr
.
These insights into hydrostatic equilibrium, pressure variations in fluids, and their application in rotating systems provide a comprehensive understanding of fluid behavior in various contexts.
FAQ’s
The hydrostatic equilibrium involves understanding the pressure exerted by a fluid in a column or container, particularly how it varies with height.
In a column filled with stationary fluid, pressure remains constant on any horizontal cross-section parallel to the Earth’s surface. However, as you move to a different cross-section, the pressure changes with height.
The basic equation for hydrostatic equilibrium is derived as dp + ρdzg = 0
, where p
is pressure, ρ
is density, dz
is the height change, and g
is the acceleration due to gravity.
For incompressible fluids, pressure decreases with increasing height. This relationship is described by the equation p + zρg = constant
.
In compressible fluids like ideal gases, the pressure-height relationship is given by ln(p) + g(M/RT)dz = constant
, which is known as the Barometric Equation.
In a rotating centrifuge, a layer of liquid is pushed outward by centrifugal force, creating a unique shape. The pressure distribution in this situation follows the principles of fluid statics.
The pressure drop over a rotating liquid ring is calculated using the equation dp = ω²ρrdr
, where ω
is the angular velocity, ρ
is the density, r
is the radial distance, and dr
is the change in radius
The pressure difference across a rotating liquid ring is given by p₂ - p₁ = (ω²ρ(r₂² - r₁²))/2
, where p₁
and p₂
are pressures at different radial distances, and r₁
and r₂
are the corresponding radius.
Hydrostatic equilibrium in the atmosphere refers to the balanced state of forces and pressures that exist vertically in the Earth’s atmosphere. In simple terms, it’s the condition where the upward force due to air pressure at any given height is equal to the downward force due to the weight of the air above that height.
The principle of hydrostatic equilibrium, often simply called hydrostatic equilibrium. It pertains to the equilibrium or balance of forces within a fluid, such as a liquid or gas, that is at rest or in a state of constant motion (i.e., not accelerating).
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