UO1-Topic 1 (Part 5) Particle Motion AY2021

Particle Motion 1

Objectives

  • Define drag force.

  • Calculate drag force for flow past various solid bodies such as spheres, cylinders, and discs.

What is Drag Force?

  • Definition: When a solid body moves through a fluid or a fluid flows past a solid body, the fluid resists the motion of the object.

  • Opposing Force: The force opposing the motion of the object is termed drag force, which acts to retard the object's motion.

Parameters Affecting Drag Force

  • Velocity: The speed of the object or fluid.

  • Size: The dimensions of the object.

  • Shape: The geometry of the object affects the flow around it.

  • Density of Fluid: The mass density of the fluid influences drag.

  • Viscosity of Fluid: The fluid's resistance to flow affects drag; higher viscosity usually results in higher drag.

Types of Flow

  • Laminar Flow: Smooth and orderly fluid motion.

  • Turbulent Flow: Chaotic and irregular fluid motion.

Flow Past A Cylinder

  • Conditions: If the fluid is incompressible and the cylinder is small, the kinetic and pressure energy remains constant across the surface of the cylinder.

Bernoulli's Equation

  • Represents the conservation of energy in fluid flow:

    ( PA + \frac{1}{2} \rho U^2 + \rho gh = constant )

Energy Change

  • Kinetic Energy: Maximum at points B and C, zero at A and D.

  • Pressure Variation: Pressure decreases from A to B and A to C; it rises from B to D and C to D, remaining equal at A and D.

Calculation of Drag Force

  • Formula: [ F_D = C_D \cdot \rho \cdot A_P \cdot \frac{U_0^2}{2} ]

    • Where:

      • ( F_D ): Drag force

      • ( C_D ): Drag coefficient

      • ( A_P ): Projected area

      • ( U_0 ): Velocity of the flow

Finding Drag Coefficient

  • Common Values:

    • Spheres: ( C_D = 0.5 )

    • Cylinders: ( C_D = 0.33 )

    • Disks: ( C_D = 1.12 )

Flow Characteristics

  • Flow past a Sphere: ( T_{ap} = 4 \pi )

  • Flow Parallel to Cylinder: ( T_{A2} = \frac{P}{4} )

  • Flow Perpendicular to Cylinder: ( dp L A = dp \times L )

Example Problem 1.11

  • Scenario: Air at 37.8 °C and 101.3 kPa flows past a 45 mm diameter sphere at 25 m/s.

  • Drag Coefficient Calculation:

    • From CD chart for spheres, ( C_D = 0.5 )

  • Calculated Parameters:

    • Diameter: ( dp = 0.045 m )

    • Velocity: ( U_0 = 25 m/s )

Projected Area Calculation for Sphere

  • [ A_P = \frac{\pi (0.045^2)}{4} \approx 1.59 \times 10^{-3} m^2 ]

Example Problem 1.12

  • Scenario: A cylindrical steam boiler stack with diameter 1.0 m and height 30 m is exposed to air.

  • Parameters given:

    • Temperature: 37.8 °C, Velocity: 180 km/h = 50 m/s

    • Density: 1.137 kg/m³, Viscosity: 1.9 x 10⁻⁵ Pa·s

  • Drag Force Calculation: [ F_D = C_D \cdot p \cdot (D_P imes L) \cdot \frac{U_0^2}{2} ]

    • With calculated drag force ( F = 14070 N ) at ( C_D = 0.33 )