Manometry and Pressure Measurement

Manometry and Pressure Measurement#

Manometry is a technique used to measure pressure by utilizing the hydrostatic principles of fluids. A manometer typically involves a column of fluid(s), and the pressure at various points can be related by applying the hydrostatic equation. In this guide, we explore how to solve manometry exercises systematically, including transitioning between surfaces and handling multiple fluids.

1. Hydrostatic Equation#

The pressure at any point in a static fluid varies with depth according to:

\[ \frac{dp}{dz} = -\rho g, \]

where:

  • \(p\) is the pressure,

  • \(z\) is the vertical coordinate,

  • \(\rho\) is the fluid density,

  • \(g\) is the acceleration due to gravity.

Integrating this equation between two points in the same fluid:

\[ p_B = p_A + \rho g (z_A - z_B), \]

where:

  • \(p_A\) and \(p_B\) are pressures at points \(A\) and \(B\),

  • \(z_A\) and \(z_B\) are their vertical coordinates.

2. Manometry Principles#

a. Transitioning Between Surfaces#

To solve manometer problems:

  1. Start at a Known Pressure: Typically, this is the open surface of a fluid column or a reference point.

  2. Apply the Hydrostatic Equation: Move step-by-step through the system, adding or subtracting \(\rho g h\) as the fluid column ascends or descends.

    • Pressure increases when moving downward in a fluid column.

    • Pressure decreases when moving upward in a fluid column.

  3. Account for Fluid Interfaces: When transitioning between fluids of different densities, ensure continuity of pressure.

b. Key Rules#

  • Same horizontal plane: Pressure remains constant.

  • Across fluid interfaces: Pressure is continuous, but the density may change.

3. Common Manometer Configurations#

a. Simple Manometer#

A single fluid column connects two points, one of which is open to the atmosphere. The pressure difference is:

\[ p_A - p_B = \rho g h, \]

where \(h\) is the height difference.

b. Differential Manometer#

Used to measure the pressure difference between two points in a system. With two connected columns:

\[ p_A - p_B = \rho_1 g h_1 - \rho_2 g h_2, \]

where:

  • \(\rho_1\), \(\rho_2\) are the densities of fluids in the two columns,

  • \(h_1\), \(h_2\) are the heights of the columns.

4. Solving Manometry Problems#

Example 1: Single-Fluid U-Tube Manometer#

Two points, \(A\) and \(B\), are connected by a U-tube filled with a fluid of density \(\rho = 1000 \, \text{kg/m}^3\). The height difference between the two sides is \(h = 0.5 \, \text{m}\). Point \(A\) is exposed to the atmosphere (\(p_A = p_{\text{atm}}\)). Find \(p_B\).

Solution:#

  1. Starting at \(A\):

    \[ p_A = p_{\text{atm}}. \]

  2. Moving downward:

    \[ p_A + \rho g h = p_B. \]

  3. Substituting values:

    \[ p_B = p_{\text{atm}} + (1000)(9.81)(0.5) = p_{\text{atm}} + 4905 \, \text{Pa}. \]

Example 2: Multi-Fluid Manometer#

A U-tube contains two immiscible fluids:

  • Oil (\(\rho_{\text{oil}} = 850 \, \text{kg/m}^3\)) on top,

  • Water (\(\rho_{\text{water}} = 1000 \, \text{kg/m}^3\)) below.

The heights are:

  • Oil column: \(h_{\text{oil}} = 0.3 \, \text{m}\),

  • Water column: \(h_{\text{water}} = 0.4 \, \text{m}\).

Determine the pressure difference \(p_A - p_B\), where \(A\) is at the oil surface, and \(B\) is at the bottom.

Solution:#

  1. Starting at \(A\):

    \[ p_A. \]

  2. Moving downward through the oil column:

    \[ p_A + \rho_{\text{oil}} g h_{\text{oil}}. \]

  3. Transitioning into water (pressure is continuous):

    \[ p_A + \rho_{\text{oil}} g h_{\text{oil}} + \rho_{\text{water}} g h_{\text{water}} = p_B. \]

  4. Rearrange for \(p_A - p_B\):

    \[ p_A - p_B = -\rho_{\text{oil}} g h_{\text{oil}} - \rho_{\text{water}} g h_{\text{water}}. \]

  5. Substituting values:

    \[ p_A - p_B = -(850)(9.81)(0.3) - (1000)(9.81)(0.4). \]
    \[ p_A - p_B = -2493.15 - 3924 = -6417.15 \, \text{Pa}. \]

Thus, \(p_B > p_A\) by \(6417.15 \, \text{Pa}\).

5. Applications of Manometry#

  1. Pressure Measurement in Pipelines
    Manometers are commonly used to measure gas or liquid pressures relative to the atmosphere.

  2. Differential Pressure Sensors
    In engineering applications, differential manometers help determine flow rates by measuring pressure drops across components.

  3. Fluid Property Determination
    Manometers can be used to deduce fluid densities by analyzing height differences.

6. Tips for Solving Manometry Problems#

  • Draw a Diagram: Clearly mark pressure changes, fluid densities, and height differences.

  • Work Systematically: Always move step-by-step, ensuring continuity of pressure.

  • Check Units: Ensure consistent use of SI units for density, height, and gravitational acceleration.

Manometry is an essential tool in fluid mechanics, providing direct insights into pressure variations in static fluids and enabling practical measurements in engineering systems.

Contents