🌬️Heat and Mass Transport Unit 11 – Boundary Layer Theory

What's Boundary Layer Theory?

• Boundary Layer Theory describes the behavior of fluid flow near a solid surface
• Focuses on the thin layer of fluid directly adjacent to the surface where viscous effects are significant
• Explains the transition from zero velocity at the surface (no-slip condition) to the free-stream velocity away from the surface
• Provides a framework for understanding and predicting heat, mass, and momentum transfer in fluid flows
• Developed by Ludwig Prandtl in 1904, revolutionizing the field of fluid mechanics
• Enables the simplification of complex fluid flow problems by dividing the flow into two regions: the boundary layer and the inviscid outer flow
• Applies to various fields, including aerodynamics, heat exchanger design, and chemical processing

Key Concepts and Definitions

• Boundary layer: The thin layer of fluid adjacent to a solid surface where viscous effects are significant
• No-slip condition: The assumption that the fluid velocity at the surface is equal to the velocity of the surface (zero for a stationary surface)
• Viscosity: A measure of a fluid's resistance to deformation and flow
• Shear stress: The force per unit area acting parallel to the surface, caused by the velocity gradient in the boundary layer
• Displacement thickness: The distance by which the boundary layer displaces the inviscid outer flow
• Momentum thickness: A measure of the momentum deficit in the boundary layer compared to the inviscid outer flow
• Thermal boundary layer: The region where temperature gradients exist due to heat transfer between the fluid and the surface
• Concentration boundary layer: The region where concentration gradients exist due to mass transfer between the fluid and the surface

Types of Boundary Layers

• Laminar boundary layer: Characterized by smooth, parallel streamlines and minimal mixing between fluid layers
  • Occurs at low Reynolds numbers (< 5 × 10^5 for a flat plate)
  • Velocity profile is parabolic, with a gradual increase from zero at the surface to the free-stream velocity
• Turbulent boundary layer: Characterized by chaotic, fluctuating motion and enhanced mixing between fluid layers
  • Occurs at high Reynolds numbers (> 5 × 10^5 for a flat plate)
  • Velocity profile is fuller, with a rapid increase near the surface followed by a more gradual approach to the free-stream velocity
• Transitional boundary layer: The region where the flow transitions from laminar to turbulent
  • Influenced by factors such as surface roughness, pressure gradient, and free-stream turbulence
• Thermal boundary layer: The region where temperature gradients exist due to heat transfer between the fluid and the surface
• Concentration boundary layer: The region where concentration gradients exist due to mass transfer between the fluid and the surface

Governing Equations

• Continuity equation: Represents the conservation of mass in the boundary layer
  • For incompressible flow: ∂u/∂x + ∂v/∂y = 0
• Momentum equation (Navier-Stokes equations): Represents the conservation of momentum in the boundary layer
  • Simplified for boundary layer flow: u∂u/∂x + v∂u/∂y = -1/ρ ∂p/∂x + ν ∂^2u/∂y^2
• Energy equation: Represents the conservation of energy in the boundary layer
  • Simplified for boundary layer flow: u∂T/∂x + v∂T/∂y = α ∂^2T/∂y^2
• Species conservation equation: Represents the conservation of species in the boundary layer
  • Simplified for boundary layer flow: u∂C/∂x + v∂C/∂y = D ∂^2C/∂y^2
• Boundary conditions: Specify the flow conditions at the surface and the edge of the boundary layer
  • No-slip condition at the surface: u(y=0) = 0
  • Free-stream condition at the edge of the boundary layer: u(y→∞) = U∞

Boundary Layer Development

• Boundary layer starts as a thin layer at the leading edge of a surface and grows in thickness downstream
• Growth is influenced by factors such as surface geometry, free-stream velocity, and fluid properties
• Laminar boundary layer: Thickness grows proportionally to the square root of the distance from the leading edge (δ ∝ √(νx/U∞))
• Turbulent boundary layer: Thickness grows more rapidly than the laminar boundary layer
  • Empirical correlations, such as the 1/7th power law, are used to estimate the turbulent boundary layer thickness (δ ∝ x^(4/5))
• Transition from laminar to turbulent flow occurs at a critical Reynolds number (Re_cr ≈ 5 × 10^5 for a flat plate)
  • Transition point moves upstream with increasing surface roughness, pressure gradient, and free-stream turbulence
• Separation: When the boundary layer detaches from the surface due to adverse pressure gradients
  • Occurs when ∂u/∂y = 0 at the surface
  • Can lead to increased drag, reduced lift, and flow instabilities

Heat and Mass Transfer in Boundary Layers

• Heat transfer in the boundary layer is governed by the energy equation
  • Temperature profile develops within the thermal boundary layer
  • Heat transfer rate is proportional to the temperature gradient at the surface (q" = -k ∂T/∂y|_(y=0))
• Mass transfer in the boundary layer is governed by the species conservation equation
  • Concentration profile develops within the concentration boundary layer
  • Mass transfer rate is proportional to the concentration gradient at the surface (j" = -D ∂C/∂y|_(y=0))
• Analogy between heat, mass, and momentum transfer: Similar mathematical formulations allow for the use of analogies (Reynolds, Prandtl, Chilton-Colburn) to relate heat and mass transfer coefficients to friction coefficients
• Nusselt number (Nu): Dimensionless parameter representing the ratio of convective to conductive heat transfer
  • For laminar flow over a flat plate: Nu_x = 0.332 Re_x^(1/2) Pr^(1/3)
  • For turbulent flow over a flat plate: Nu_x = 0.0296 Re_x^(4/5) Pr^(1/3)
• Sherwood number (Sh): Dimensionless parameter representing the ratio of convective to diffusive mass transfer
  • For laminar flow over a flat plate: Sh_x = 0.332 Re_x^(1/2) Sc^(1/3)
  • For turbulent flow over a flat plate: Sh_x = 0.0296 Re_x^(4/5) Sc^(1/3)

Applications in Engineering

• Aerodynamics: Boundary layer theory is used to predict drag and lift forces on airfoils, wings, and other aerodynamic surfaces
  • Laminar flow control techniques (suction, shaping) are used to reduce drag and increase efficiency
• Heat exchangers: Understanding boundary layer development and heat transfer is crucial for the design and optimization of heat exchangers
  • Techniques such as surface roughening and flow disruption are used to enhance heat transfer
• Cooling of electronic devices: Boundary layer theory is applied to predict and optimize convective cooling of electronic components
  • Jet impingement and micro-channel cooling are examples of boundary layer-based cooling techniques
• Chemical processing: Mass transfer in boundary layers is important for processes such as absorption, adsorption, and catalytic reactions
  • Packed bed reactors and falling film reactors rely on efficient mass transfer in the boundary layer
• Environmental engineering: Boundary layer theory is used to model the dispersion of pollutants in the atmosphere and water bodies
  • Atmospheric boundary layer and ocean surface boundary layer are important for understanding the transport and mixing of pollutants

Problem-Solving Techniques

• Integral methods: Solve the boundary layer equations by integrating them across the boundary layer thickness
  • Von Kármán momentum integral equation: Relates the boundary layer thickness to the surface shear stress
  • Thwaites' method: Approximate solution for laminar boundary layers with pressure gradients
• Similarity solutions: Exploit the similarity between velocity, temperature, and concentration profiles in certain boundary layer flows
  • Blasius solution: Exact solution for laminar flow over a flat plate with zero pressure gradient
  • Falkner-Skan solution: Similarity solution for laminar flow over a wedge with pressure gradients
• Numerical methods: Discretize the boundary layer equations and solve them using computational techniques
  • Finite difference methods: Approximate derivatives using finite differences and solve the resulting algebraic equations
  • Finite element methods: Divide the domain into elements and solve the weak form of the boundary layer equations
• Experimental techniques: Measure velocity, temperature, and concentration profiles in the boundary layer to validate theoretical predictions
  • Hot-wire anemometry: Measures velocity profiles using the cooling effect of the fluid on a heated wire
  • Laser Doppler velocimetry: Measures velocity profiles using the Doppler shift of laser light scattered by particles in the fluid
  • Particle image velocimetry: Measures velocity fields by tracking the displacement of seeded particles in the fluid