3.1 Boundary Layers and Convection Coefficients

Boundary layers are crucial in convection heat transfer. They form near solid surfaces, affecting fluid velocity and temperature gradients. Understanding boundary layers helps explain how heat moves between surfaces and fluids in different flow conditions.

Convection coefficients quantify heat transfer rates in various flows. They depend on fluid properties, flow conditions, and surface characteristics. Knowing how to calculate and apply convection coefficients is key to solving real-world heat transfer problems.

Boundary Layers in Convective Heat Transfer

Concept and Development of Boundary Layers

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• A boundary layer is a thin region near a solid surface where the fluid velocity changes from zero at the surface to the free-stream velocity away from the surface
• The boundary layer develops as a result of the no-slip condition, which states that the fluid velocity at the surface is equal to the velocity of the surface
  • This condition arises due to the adhesive forces between the fluid molecules and the solid surface
  • The fluid particles in direct contact with the surface have zero velocity relative to the surface
• The thickness of the boundary layer increases in the flow direction, starting from the leading edge of the surface
  • The boundary layer grows as more fluid particles are slowed down by the viscous forces near the surface
  • The boundary layer thickness is typically defined as the distance from the surface where the velocity reaches 99% of the free-stream velocity

Significance of Boundary Layers in Heat Transfer

• The boundary layer plays a crucial role in convective heat transfer, as it determines the temperature gradient and heat flux at the surface
  • The temperature profile in the boundary layer is influenced by the velocity profile and the thermal properties of the fluid
  • A thinner boundary layer results in a steeper temperature gradient and higher heat transfer rates
• The convective heat transfer coefficient is directly related to the boundary layer characteristics, such as thickness and velocity profile
  • A thinner boundary layer and higher velocity gradients near the surface lead to higher convection coefficients
  • Factors that affect the boundary layer, such as surface roughness and flow turbulence, also influence the convective heat transfer coefficient

Laminar vs Turbulent Boundary Layers

Characteristics of Laminar Boundary Layers

• Laminar boundary layers are characterized by smooth, parallel flow with no mixing between fluid layers
  • The fluid particles move in orderly, parallel paths without significant cross-flow or mixing
  • Laminar flow occurs at low Reynolds numbers, typically below a critical value that depends on the geometry and flow conditions
• In laminar boundary layers, the velocity profile is parabolic, with the maximum velocity at the center and zero velocity at the surface
  • The velocity gradient is highest near the surface and decreases towards the center of the flow
  • The parabolic velocity profile is a result of the balance between viscous forces and pressure gradients in the flow

Characteristics of Turbulent Boundary Layers

• Turbulent boundary layers are characterized by chaotic, fluctuating flow with significant mixing between fluid layers
  • The fluid particles have random, three-dimensional motion with intense mixing and cross-flow
  • Turbulent flow occurs at high Reynolds numbers, typically above a critical value that depends on the geometry and flow conditions
• In turbulent boundary layers, the velocity profile is flatter compared to laminar flow, with a sharp gradient near the surface and a more uniform velocity distribution away from the surface
  • The velocity gradient near the surface is much higher than in laminar flow, resulting in enhanced momentum and heat transfer
  • The flatter velocity profile is a result of the increased mixing and momentum exchange between fluid layers
• The transition from laminar to turbulent flow occurs at a critical Reynolds number, which depends on the surface roughness, pressure gradient, and other factors
  • Surface roughness promotes early transition to turbulence by introducing disturbances in the flow
  • Adverse pressure gradients (increasing pressure in the flow direction) can also trigger the transition to turbulence

Convection Coefficients for Different Flows

Definition and Units of Convection Coefficient

• The convection coefficient $h$ is a measure of the rate of heat transfer between a surface and a fluid, expressed in units of $W/(m^2·K)$
  • It quantifies the ability of a fluid to transfer heat by convection, considering the fluid properties, flow conditions, and surface characteristics
  • Higher values of $h$ indicate more effective heat transfer, while lower values suggest less efficient heat transfer

Convection Coefficients for External Flows

• For external flow over a flat plate, the convection coefficient can be determined using empirical correlations based on the Nusselt number $Nu$, Reynolds number $Re$, and Prandtl number $Pr$
  • The Dittus-Boelter correlation is commonly used for turbulent flow: $Nu = 0.023 · Re^{0.8} · Pr^{0.4}$
  • For laminar flow, the Pohlhausen correlation can be applied: $Nu = 0.664 · Re^{0.5} · Pr^{1/3}$
• The convection coefficient for flow over cylinders and spheres can be estimated using correlations that account for the shape and orientation of the object
  • The Churchill-Bernstein correlation is used for flow over a cylinder: $Nu = 0.3 + \frac{0.62 · Re^{0.5} · Pr^{1/3}}{[1 + (0.4/Pr)^{2/3}]^{1/4}} · [1 + (\frac{Re}{282,000})^{5/8}]^{4/5}$
  • For flow over a sphere, the Whitaker correlation is commonly employed: $Nu = 2 + (0.4 · Re^{1/2} + 0.06 · Re^{2/3}) · Pr^{0.4} · (\mu_∞/\mu_s)^{1/4}$

Convection Coefficients for Internal Flows

• For internal flow in pipes and ducts, the convection coefficient depends on the flow regime (laminar or turbulent), the geometry of the cross-section, and the surface roughness
  • In laminar flow, the Nusselt number is a function of the Graetz number $Gz = (D/L) · Re · Pr$, where $D$ is the pipe diameter and $L$ is the pipe length
  • For turbulent flow in smooth pipes, the Dittus-Boelter correlation can be used: $Nu = 0.023 · Re^{0.8} · Pr^n$, where $n = 0.4$ for heating and $n = 0.3$ for cooling
• The convection coefficient for flow in non-circular ducts can be estimated using correlations that account for the shape and aspect ratio of the cross-section
  • The Sieder-Tate correlation is used for laminar flow in rectangular ducts: $Nu = 1.86 · (Re · Pr · \frac{D_h}{L})^{1/3} · (\frac{\mu_b}{\mu_w})^{0.14}$, where $D_h$ is the hydraulic diameter, $\mu_b$ is the bulk fluid viscosity, and $\mu_w$ is the viscosity at the wall temperature
  • For turbulent flow in rectangular ducts, the Gnielinski correlation can be applied: $Nu = \frac{(f/8) · (Re - 1000) · Pr}{1 + 12.7 · (f/8)^{1/2} · (Pr^{2/3} - 1)}$, where $f$ is the friction factor

Convection Coefficients for Natural Convection

• In natural convection, the convection coefficient is a function of the Grashof number $Gr$ and the Prandtl number $Pr$, which account for the buoyancy-driven flow and the fluid properties
  • The Grashof number represents the ratio of buoyancy forces to viscous forces: $Gr = \frac{g · \beta · (T_s - T_∞) · L^3}{\nu^2}$, where $g$ is the gravitational acceleration, $\beta$ is the thermal expansion coefficient, $T_s$ is the surface temperature, $T_∞$ is the ambient temperature, $L$ is the characteristic length, and $\nu$ is the kinematic viscosity
  • The Rayleigh number $Ra$ is the product of the Grashof and Prandtl numbers: $Ra = Gr · Pr$
• Empirical correlations for the Nusselt number in natural convection are based on the geometry and orientation of the surface
  • For vertical plates, the Churchill-Chu correlation can be used: $Nu = (0.825 + \frac{0.387 · Ra^{1/6}}{[1 + (0.492/Pr)^{9/16}]^{8/27}})^2$
  • For horizontal plates with the hot surface facing upward, the McAdams correlation is applicable: $Nu = 0.54 · Ra^{1/4}$ for $10^4 < Ra < 10^7$ and $Nu = 0.15 · Ra^{1/3}$ for $10^7 < Ra < 10^{11}$

Dimensional Analysis for Convective Heat Transfer

Buckingham Pi Theorem and Dimensionless Parameters

• Dimensional analysis is a technique used to simplify complex problems by identifying the relevant dimensionless parameters that govern the system's behavior
  • It helps in reducing the number of variables and obtaining generalized relationships between the parameters
  • Dimensional analysis is particularly useful in fluid mechanics and heat transfer, where the governing equations are often complex and involve many variables
• The Buckingham Pi theorem states that any physically meaningful equation involving $n$ variables can be rewritten in terms of $p = n - k$ dimensionless parameters, where $k$ is the number of independent dimensions
  • The dimensionless parameters, denoted as $\Pi_1, \Pi_2, ..., \Pi_p$, are formed by combining the original variables in a way that eliminates their dimensions
  • The theorem provides a systematic approach to determine the minimum number of dimensionless parameters needed to describe a physical problem

Key Dimensionless Parameters in Convective Heat Transfer

• The Reynolds number $Re$ is a dimensionless parameter that represents the ratio of inertial forces to viscous forces in a fluid
  • It is defined as $Re = \frac{\rho · v · L}{\mu}$, where $\rho$ is the fluid density, $v$ is the velocity, $L$ is the characteristic length, and $\mu$ is the dynamic viscosity
  • The Reynolds number is used to characterize the flow regime (laminar or turbulent) and the boundary layer behavior
  • Laminar flow typically occurs at low Reynolds numbers $(Re < 2300)$, while turbulent flow occurs at high Reynolds numbers $(Re > 4000)$
• The Nusselt number $Nu$ is a dimensionless parameter that represents the ratio of convective heat transfer to conductive heat transfer in a fluid
  • It is defined as $Nu = \frac{h · L}{k}$, where $h$ is the convection coefficient, $L$ is the characteristic length, and $k$ is the thermal conductivity of the fluid
  • The Nusselt number is used to quantify the effectiveness of convective heat transfer
  • Higher Nusselt numbers indicate more effective convective heat transfer compared to conductive heat transfer
• The Prandtl number $Pr$ is a dimensionless parameter that represents the ratio of momentum diffusivity to thermal diffusivity in a fluid
  • It is defined as $Pr = \frac{\nu}{\alpha} = \frac{c_p · \mu}{k}$, where $\nu$ is the kinematic viscosity, $\alpha$ is the thermal diffusivity, $c_p$ is the specific heat capacity, $\mu$ is the dynamic viscosity, and $k$ is the thermal conductivity
  • The Prandtl number is used to characterize the relative thickness of the velocity and thermal boundary layers
  • For $Pr < 1$, the thermal boundary layer is thicker than the velocity boundary layer, while for $Pr > 1$, the velocity boundary layer is thicker than the thermal boundary layer
• Other dimensionless parameters, such as the Grashof number $Gr$ and the Rayleigh number $Ra$, are used to describe natural convection and the onset of convective instabilities
  • The Grashof number represents the ratio of buoyancy forces to viscous forces: $Gr = \frac{g · \beta · (T_s - T_∞) · L^3}{\nu^2}$
  • The Rayleigh number is the product of the Grashof and Prandtl numbers: $Ra = Gr · Pr$
  • These parameters are used to characterize the flow regime and heat transfer in natural convection problems