Fluid Statics: Pressure Measurement

Fluid statics, also known as hydrostatics, is the branch of fluid mechanics that studies fluids at rest. In a fluid at rest, there are no relative motions between fluid layers, and therefore, no shear stresses exist. The primary force acting on fluid elements is pressure. Understanding pressure and how to measure it is fundamental in many engineering applications.

Pressure

Pressure (P) is defined as the force (F) applied perpendicular to the surface of an object per unit area (A) over which that force is distributed. Mathematically, it is expressed as:

P = F / A

In fluids, pressure at a point is the same in all directions. This is due to the fluid's inability to sustain shear stress. The standard unit of pressure is Pascal (Pa), which is equivalent to one Newton per square meter (N/m²). Other common units include kilopascal (kPa), megapascal (MPa), bar, atmosphere (atm), pounds per square inch (psi), and millimeters of mercury (mmHg).

Pressure Variation in a Static Fluid

In a static fluid, pressure varies with depth. Consider a small rectangular prism of fluid of height 'dh' and base area 'dA'. The weight of this fluid element is its density (ρ) times its volume (d V = dA * dh) times the acceleration due to gravity (g).

Weight = ρ * g * dA * dh

For the fluid element to be in equilibrium, the pressure difference between the top and bottom surfaces must balance this weight. Let P be the pressure at the bottom and P + dP be the pressure at the top. The net upward force due to pressure is (P - (P + dP)) * dA = -dP * dA.

Equating the forces: -dP * dA = ρ * g * dA * dh

This simplifies to: dP = -ρ * g * dh

Integrating this equation gives the hydrostatic pressure equation:

P = P₀ - ρ * g * h

Where P₀ is the pressure at the reference level (h=0). If we consider the pressure at depth 'h' from the surface where the pressure is atmospheric (P_atm), then:

P_h = P_atm + ρ * g * h

The term (ρ * g * h) is known as the gauge pressure, which is the pressure relative to atmospheric pressure. Absolute pressure is the sum of gauge pressure and atmospheric pressure.

Pressure Measurement Devices: Manometers

Manometers are instruments used to measure pressure, particularly gauge pressure, by balancing a column of liquid against the unknown pressure. They are based on the principle of hydrostatic pressure. The most common types are U-tube manometers and inclined-tube manometers.

U-Tube Manometer

A U-tube manometer consists of a U-shaped tube, partially filled with a liquid of known density (manometric fluid), such as mercury or colored water. One end of the U-tube is connected to the point where the pressure is to be measured, and the other end is either open to the atmosphere or connected to a reference pressure.

Working Principle: The fluid in the pipe exerts a pressure P₁ on the surface of the manometric fluid in one arm of the U-tube. The other arm is exposed to a known pressure P₂ (e.g., atmospheric pressure). The difference in the liquid levels in the two arms creates a pressure difference that balances the applied pressure difference.

Let's consider a U-tube manometer measuring the gauge pressure in a pipe. The pipe is connected to the left limb of the U-tube. The right limb is open to the atmosphere. The manometric fluid has a density ρ_m. The pressure in the pipe is P_pipe. The atmospheric pressure is P_atm.

Let 'h' be the difference in the liquid levels between the two limbs. The pressure at the level of the interface in the left limb is P_pipe + ρ_m * g * h₁, where h₁ is the height of the fluid column in the left limb above the interface. The pressure at the same horizontal level in the right limb is P_atm + ρ_m * g * h₂, where h₂ is the height of the fluid column in the right limb above the interface.

Since both points are at the same horizontal level within the same fluid, their pressures must be equal:

P_pipe + ρ_m * g * h₁ = P_atm + ρ_m * g * h₂

Rearranging this equation:

P_pipe - P_atm = ρ_m * g * (h₂ - h₁)

The term (h₂ - h₁) is the total difference in levels, 'h'. So, the gauge pressure is:

P_gauge = P_pipe - P_atm = ρ_m * g * h

Types of U-Tube Manometers:

• Simple U-Tube Manometer: Used to measure gauge pressure. One end is connected to the pressure source, and the other is open to the atmosphere.
• Differential U-Tube Manometer: Used to measure the pressure difference between two points. Both ends are connected to different pressure sources.
• Inverted U-Tube Manometer: Used for measuring pressure differences of light liquids (like air or gases). The manometric fluid is lighter than the fluid in the pipe, and the manometer is inverted.

Selection of Manometric Fluid: The choice of manometric fluid is critical.

• For measuring positive gauge pressures, a fluid denser than the fluid in the pipe (e.g., mercury) is used.
• For measuring negative gauge pressures (suction), a fluid lighter than the fluid in the pipe (e.g., water or oil) is used.
• The density of the manometric fluid determines the sensitivity. A lighter fluid will show a larger deflection for a given pressure difference, making it more sensitive for low-pressure measurements.

Example: A U-tube manometer measures the pressure of water flowing in a pipe. The water level in the manometer limb connected to the pipe is 10 cm below the level in the other limb, which is open to the atmosphere. The manometric fluid is mercury (density = 13600 kg/m³). The density of water is 1000 kg/m³.

In this case, the pressure in the pipe is higher than atmospheric pressure. Let 'h' be the difference in mercury levels. The diagram shows that the mercury level in the limb connected to the pipe is lower. This means the pressure in the pipe is pushing the mercury down.

Let's assume the interface in the left limb is at a certain level. The pressure at this level in the left limb is P_pipe + ρ_water * g * h_w, where h_w is the height of the water column above the interface in the left limb. The pressure at the same horizontal level in the right limb is P_atm + ρ_m * g * h_m, where h_m is the height of the mercury column in the right limb.

If the mercury level in the right limb is 0.1 m higher than in the left limb:

P_pipe - P_atm = ρ_m * g * h_m - ρ_water * g * h_w

However, a simpler U-tube manometer problem usually relates the pressure difference directly to the manometer fluid difference. If the mercury level difference is 'h', and the pressure in the pipe is higher, then:

P_pipe (at interface level) = P_atm + ρ_m * g * h

This equation measures the pressure difference based on the mercury column height. If the fluid in the pipe was mercury, the calculation would be straightforward. When the fluid in the pipe is different from the manometric fluid, we must account for the height difference of the fluid in the pipe as well.

Let's re-evaluate the example more precisely: Pipe contains water. U-tube contains mercury. Right limb open to atmosphere. Mercury level in right limb is 0.1m higher than in left limb. This means the pressure in the pipe is higher.

Consider the horizontal level of the mercury-water interface in the left limb.

Pressure in left limb at interface level = P_pipe + ρ_water * g * h_w

Pressure in right limb at the same horizontal level = P_atm + ρ_m * g * h_m

Where h_w is the height of water column above the interface in the left limb, and h_m is the height of mercury column in the right limb above the interface level.

Let the mercury interface in the left limb be at datum. The mercury interface in the right limb is at h_m = 0.1 m above the datum. The water level in the left limb is some h_w above the datum.

P_pipe + ρ_water * g * h_w = P_atm + ρ_m * g * h_m

P_gauge = P_pipe - P_atm = ρ_m * g * h_m - ρ_water * g * h_w

If the problem states "water level in the manometer limb connected to the pipe is 10 cm below the level in the other limb", this refers to the water column *above* the mercury. This implies the left limb has water, then mercury. The right limb has mercury.

Let's assume the U-tube is filled such that the mercury is at the bottom. The pipe is connected to the left limb.

Case 1: Pressure in pipe is higher. Mercury level in right limb is higher. P_pipe + ρ_water * g * h_w = P_atm + ρ_m * g * h_m If h_m = 0.1 m (mercury difference) and the water column difference is negligible or not given, then P_gauge = ρ_m * g * h_m. P_gauge = 13600 kg/m³ * 9.81 m/s² * 0.1 m = 13341.6 Pa.

Case 2: Pressure in pipe is lower (suction). Mercury level in left limb is higher. P_atm + ρ_m * g * h_m = P_pipe + ρ_water * g * h_w P_gauge = P_pipe - P_atm = ρ_m * g * h_m - ρ_water * g * h_w (This will be negative).

Mnemonic for U-tube manometer calculation:

Inclined-Tube Manometer

An inclined-tube manometer is a variation of the U-tube manometer designed to measure very low pressures, such as those found in gas flow measurements. It consists of a reservoir connected to a long, narrow tube inclined at a small angle (θ) to the horizontal.

Working Principle: The inclination of the tube increases the length of the fluid column that needs to be displaced to achieve a certain vertical height difference. This magnified horizontal movement of the fluid meniscus makes it easier to read small pressure differences accurately.

Let 'h' be the vertical difference in the liquid levels between the reservoir and the inclined tube. Let 'L' be the distance the liquid meniscus moves along the inclined tube. The angle of inclination is θ.

From trigonometry, the vertical height 'h' is related to the distance 'L' along the incline by:

h = L * sin(θ)

If a U-tube manometer measures a vertical head 'h', an inclined manometer with the same manometric fluid and pressure difference will have the meniscus move a distance 'L' along the incline, where L = h / sin(θ). Since sin(θ) is less than 1 for θ < 90°, L will be greater than h. This amplification of the reading (L) allows for the detection of smaller pressure variations.

The pressure difference measured by an inclined manometer is given by:

P_gauge = ρ_m * g * h = ρ_m * g * L * sin(θ)

Where:

• ρ_m is the density of the manometric fluid.
• g is the acceleration due to gravity.
• L is the distance moved along the inclined tube.
• θ is the angle of inclination of the tube with the horizontal.

Advantages:

• Increased sensitivity for measuring low pressures.
• Easier to read small pressure changes due to magnified movement.

Disadvantages:

• Only suitable for measuring low pressures.
• More prone to errors due to surface tension effects and parallax error in reading the meniscus.
• Requires careful calibration and stable installation.

Example: An inclined manometer is used to measure the pressure drop across a filter. The manometer fluid is oil with a density of 850 kg/m³. The tube is inclined at an angle of 30° to the horizontal. The pressure difference causes the oil to move 5 cm (0.05 m) along the inclined tube. Calculate the pressure difference.

Given:

• ρ_m = 850 kg/m³
• g = 9.81 m/s²
• L = 0.05 m
• θ = 30°
• sin(30°) = 0.5

Pressure difference (P_gauge) = ρ_m * g * L * sin(θ)

P_gauge = 850 kg/m³ * 9.81 m/s² * 0.05 m * 0.5

P_gauge = 2084.625 Pa

This pressure difference is equivalent to a vertical head of h = L * sin(θ) = 0.05 m * 0.5 = 0.025 m of oil. The inclined tube magnifies the reading by a factor of 1/sin(θ) = 1/0.5 = 2.

Summary of Pressure Measurement using Manometers

Manometers are essential tools for measuring fluid pressure, particularly gauge pressure.

• U-tube Manometers: Versatile for measuring moderate pressures. Simple U-tubes measure gauge pressure, while differential U-tubes measure pressure differences between two points.
• Inclined-tube Manometers: Highly sensitive for measuring low pressures by magnifying the displacement of the manometric fluid along an inclined tube.

Key factors in using manometers include selecting the appropriate manometric fluid (density and wetting properties) and ensuring accurate reading of the fluid levels, considering any fluid in the connected pipe that might be different from the manometric fluid.