Introduction to Momentum Transfer and Fluid Mechanics

Momentum transfer, also known as fluid mechanics, is a branch of engineering science that deals with the behavior of fluids (liquids, gases, and vapors) and the forces on them. It is fundamental in chemical engineering and related fields, providing the basis for understanding fluid flow, pressure, and related phenomena.

Objectives of Fluid Mechanics Study

• Explain the scope of fluid mechanics and define key concepts.

• Distinguish between fluid statics (fluids at rest) and fluid dynamics (fluids in motion).

• Define and explain fluid pressure, hydrostatic pressure, and pressure head.

• Analyze hydrostatic force on submerged surfaces.

• Understand and analyze different types of manometers and pressure differences in connected fluid columns.

Momentum Transfer: Definition

What is Momentum Transfer?

Momentum transfer refers to the study of the movement and interaction of fluids, encompassing both liquids and gases. It is commonly referred to as fluid mechanics.

Scope of Fluid Mechanics

• Fluid Statics: The study of fluids at rest, where there is no shear stress and the fluid is in equilibrium.

• Fluid Dynamics: The study of fluids in motion, where portions of the fluid are moving relative to each other.

Fluid Statics

Definition of a Fluid

A fluid is a substance that deforms continuously under the application of a shear stress, regardless of the magnitude of the applied stress. Fluids include both liquids and gases.

Basic Properties of Fluids

• Density (ρ): Mass per unit volume of a substance.

• Specific Gravity (SG): The ratio of the density of a fluid to the density of a reference substance (usually water for liquids).

• Viscosity: A measure of a fluid's resistance to deformation or flow.

• Surface Tension: The energy required to increase the surface area of a liquid due to intermolecular forces.

Classification of Fluids

• Incompressible Fluids: Fluids whose density does not change significantly with pressure (e.g., most liquids).

• Compressible Fluids: Fluids whose density changes appreciably with pressure and temperature (e.g., gases).

Key Concept: In a fluid at rest, the pressure at any point is independent of direction (isotropic).

Pressure vs. Force

Definitions

• Force: The push or pull exerted by a fluid against the walls of its container or on objects within the fluid.

• Pressure: The force exerted per unit area. In fluid statics, pressure is the primary variable of interest.

Formula for Force:

• Where F is force, m is mass, g is acceleration due to gravity, and gc is the gravitational conversion factor (32.174 lbf/lbm-s2 in English units).

• SI units: F in newtons (N = kg·m/s2), English units: F in lbf, CGS units: F in dynes.

Hydrostatics of Incompressible Fluids

Hydrostatic Pressure Gradient

For an incompressible fluid, the pressure variation with depth is given by:

• Where P is pressure, z is vertical position, γ is specific weight, ρ is density, and g is acceleration due to gravity.

Integrating between two points:

Pressure Head

The height of a fluid column that corresponds to a given pressure difference is called the pressure head:

• The pressure at a point in a homogenous, incompressible fluid at rest depends only on the vertical distance below the fluid surface, not on the shape or size of the container.

Hydrostatics of Compressible Fluids

Hydrostatic Pressure Gradient for Compressible Fluids

For compressible fluids (e.g., gases), density varies with pressure and temperature. The pressure gradient is:

• Where R is the specific gas constant and T is absolute temperature.

Integrating (assuming isothermal conditions):

Manometry: Measurement of Pressure

Manometers

Manometers are devices that measure pressure using columns of liquid in tubes. Types include:

• Open Tube Manometer: Measures pressure relative to atmospheric pressure using a vertical tube partially filled with liquid.

• U-Tube Manometer: Consists of a U-shaped tube containing a manometric fluid (e.g., mercury or water) to measure pressure differences.

• Differential U-Tube Manometer: Measures the pressure difference between two points in a system.

• Inclined Tube Manometer: Used for measuring small pressure differences with higher sensitivity.

Manometer Equations

• For a simple U-tube manometer:

  • If fluid A is a gas and its contribution is negligible:

  • For a general U-tube with two fluids:

• For an inclined manometer:

Buoyancy and Archimedes' Principle

Buoyant Force

The buoyant force is the upward force exerted by a fluid on a submerged or floating object. According to Archimedes' Principle:

• The magnitude of the buoyant force equals the weight of the fluid displaced by the object.

• If the buoyant force equals the object's weight, the object floats; otherwise, it sinks.

Formula for Buoyant Force:

• Where Vdisp is the volume of fluid displaced.

Fraction of Submergence for Floating Objects:

Summary Table: Types of Manometers

Example Problems

• Force Calculation: Calculate the force exerted by a 3 lb mass in English, CGS, and SI units using .

• Hydrostatic Pressure: Find the pressure at the interface of gasoline and water in a tank, given specific gravities and heights.

• Buoyancy: Determine the tension in a cable supporting a submerged object, or the specific gravity of a body from its weights in air and water.

Additional info: These notes provide foundational concepts for further study in fluid mechanics, including applications in chemical engineering, civil engineering, and related fields.