Fluid Dynamics: Shell Balance and Momentum

Shell Balance Overview

  • Shell balance is a method to analyze fluid motion in a certain volume (shell) using mass and momentum conservation principles.

  • Fluid components are incompressible and cannot accumulate within the shell.

Mass Balance Concept

  • Input - Output = 0 (conservation of mass) in terms of mass flow rate.

  • Mass flow rate calculation: Mass=Density×Volume\text{Mass} = \text{Density} \times \text{Volume}.

  • Flow rate Q=ρ′qQ = \rho' q where qq represents the volume flow rate.

Integration and Velocity Profiles

  • Velocity varies across the boundary layer; at velocity boundary layer edges, velocity is nearly zero.

  • Average velocity and surface area considerations are crucial for calculating flow.

  • Flow rate expressions involve integrating instantaneous velocity across the boundary layer: Q=∫0δu(y)×surface areaQ = \int_0^{\delta}{u(y) \times \text{surface area}}.

Boundary Layer Dynamics

  • The boundary layer where momentum transfer occurs has varying velocities, with uu at the center and 00 at the edge.

  • The thickness of the boundary layer (δ\delta) influences flow dynamics and is critical in applications like desalination and laminar flow analysis.

Momentum Balance Principles

  • Momentum conservation is treated as a vector quantity; must specify direction.

  • Analyze momentum changes at different shell edges:

    • Left/right: flow in/out momentum.

    • Top/bottom: effects of wall interaction (friction, shear stress).

Formulation of Momentum Changes

  • Utilize shear stress and flow rate equations to describe momentum changes depending on the flow direction.

  • Momentum loss at the bottom wall attributed to shear stress, while top edge momentum loss relates to the outgoing flow.

Boundary Layer Growth Relations

  • Boundary layer thickness influences flow characteristics, inversely related to Reynolds number: δ∼1Re\delta \sim \frac{1}{\sqrt{Re}}.

  • For a flat plate in motion, transition from laminar to turbulent flow occurs around a Reynolds number of 300,000, directly related to flow characteristics.