Chapter 4

Incompressible Fluid Flow

• Chapter 4

• CLO2: Analyze problems related to internal and external fluid flow.

• CLO3: Describe fluid flow application in various Chemical Engineering unit operations.

Chapter Objectives

• At the end of this chapter, you should be able to:

  • Explain the difference between laminar and turbulent flow.

  • Analyze the flow in a pipeline system using the overall energy balance.

Key Concepts

• Liquid Flow in Conduits: Understanding the flow of liquids in pipes and channels.

• Reynolds Number: A dimensionless number used to predict flow patterns.

• Flow Characteristics:

  • Laminar Flow: Smooth and orderly, where particles move in parallel layers.

  • Turbulent Flow: Irregular and chaotic, characterized by eddies and vortices.

• Friction Factor: A critical parameter affecting the flow and pressure loss in conduits, derived from flow regime and pipe characteristics.

• Moody’s Diagram: Utilized to determine the friction factor based on Reynolds number and relative roughness.

• Energy Equation: A fundamental equation for analyzing liquid flow in pipeline systems.

Types of Conduits

Pipe

• Definition: A round cross-section closed conduit (e.g., water pipes, hydraulic hoses).

• Features:

  • Can withstand higher pressure differentials.

Duct

• Definition: A square cross-section closed conduit (e.g., heating and air-conditioning ducts).

• Features:

  • Lower pressure differentials across walls compared to pipes.

Channel

• Description: When the liquid does not fill the conduit, it is termed channel flow (gravity-driven).

Flow Terminologies

Streamline Flow

• Definition: Flow follows a specific path with continuous and smooth streamlines.

• Characteristics:

  • The speed of fluid is consistent along its path.

Uniform Flow

• Definition: The cross-sectional area and fluid velocity are constant throughout the flow.

• Example: Flow through a uniform pipe.

Steady Flow

• Definition: Fluid velocity varies from section to section but remains constant over time at each section.

• Example: Flow through tapering pipes.

Unsteady Flow

• Definition: Fluid velocity and area change over time at any cross section.

• Example: Waves traveling along a channel.

Flow Regimes and Reynolds Number

• Reynolds Number (Re):

  • Formula: ( Re = \frac{\mu \rho V D}{\mu} )

    • Where ( V ) = velocity, ( D ) = hydraulic diameter, ( \mu ) = dynamic viscosity, ( \rho ) = density.

  • Flow Types:

    • Laminar: ( Re < 2000 ) - orderly flow

    • Transitional: ( 2000 < Re < 4000 ) - fluctuating flow

    • Turbulent: ( Re > 4000 ) - chaotic flow

• Significant Findings: Flow type changes with varying velocity, viscosity, and conduit dimensions.

Energy Equation

• Bernoulli’s Equation: [ \frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_{loss} ]

• Head Losses: Due to friction, fittings, and other factors in the pipeline.

Friction Loss in Pipe Flow

Description

• Factors Influencing Friction Loss:

  • Pipe roughness, flow velocity, diameter, and fluid properties.

• Minor Losses: Losses at fittings and valves are considered small compared to losses in long pipes.

• Total Friction Loss Calculation: [ F_{Total} = F_{pipe} + F_{fittings} + F_{valves} + F_{expansion} + F_{contraction} ]

Pipe Characteristics

• Importance of inner diameter in calculating cross-sectional area for fluid flow.

• Material and surface roughness significantly affect the degree of friction loss in typical systems.

Example Problems

• For various flow scenarios (laminar vs. turbulent); determining Reynolds number, friction factor, and energy costs.