3.1 Boundary Layers and Convection Coefficients

Boundary layers are thin regions near solid surfaces where fluid velocity and temperature change rapidly. They control the temperature gradients that drive convection, so understanding them is essential for predicting heat transfer rates. This topic covers how boundary layers develop, how they differ in laminar vs. turbulent flow, how convection coefficients are defined and calculated, and the dimensionless parameters that tie it all together.

Boundary Layers in Convective Heat Transfer

Concept and Development of Boundary Layers

A boundary layer is the thin region near a solid surface where the fluid velocity transitions from zero at the surface to the free-stream velocity $U_\infty$ farther away. It exists because of the no-slip condition: fluid molecules in direct contact with the surface are held stationary (relative to the surface) by adhesive forces, so their velocity matches the surface velocity.

Starting from the leading edge of a surface, the boundary layer grows thicker in the flow direction. Viscous forces progressively slow down more and more fluid particles as you move downstream. The boundary layer thickness $\delta$ is conventionally defined as the distance from the surface where the local velocity reaches 99% of $U_\infty$.

Significance of Boundary Layers in Heat Transfer

The boundary layer determines the temperature gradient at the surface, which directly controls the heat flux. A thermal boundary layer develops alongside the velocity boundary layer, and its shape depends on both the velocity profile and the fluid's thermal properties.

• A thinner boundary layer produces a steeper temperature gradient at the wall, which means higher heat transfer rates.
• The convection coefficient $h$ is directly tied to boundary layer characteristics. Thinner layers and stronger velocity gradients near the surface both increase $h$.
• Anything that modifies the boundary layer (surface roughness, free-stream turbulence, pressure gradients) will also change the convection coefficient.

Laminar vs Turbulent Boundary Layers

Characteristics of Laminar Boundary Layers

In a laminar boundary layer, fluid moves in smooth, orderly layers with no cross-stream mixing. This regime occurs at low Reynolds numbers, below a critical value that depends on geometry and flow conditions (for a flat plate, the transition typically begins around $Re_x \approx 5 \times 10^5$).

• The velocity profile is approximately parabolic: zero at the wall, increasing smoothly to the free-stream value.
• The velocity gradient is steepest right at the surface and decreases with distance from the wall.
• Because there's no turbulent mixing, heat transfer relies more heavily on molecular conduction through the fluid near the wall, so convection coefficients tend to be lower than in turbulent flow.

Characteristics of Turbulent Boundary Layers

Turbulent boundary layers feature chaotic, three-dimensional fluid motion with intense mixing between layers. This regime occurs at high Reynolds numbers, above the critical transition value.

• The velocity profile is flatter than in laminar flow. There's a very thin region near the wall (the viscous sublayer) with an extremely steep velocity gradient, then a relatively uniform velocity distribution farther out.
• The steep near-wall gradient and vigorous mixing both enhance momentum and heat transfer, producing significantly higher convection coefficients than laminar flow.

Transition from laminar to turbulent flow depends on several factors:

• Surface roughness introduces disturbances that promote earlier transition.
• Adverse pressure gradients (pressure increasing in the flow direction) can also trigger transition.
• Free-stream turbulence intensity affects how quickly disturbances amplify.

Convection Coefficients for Different Flows

Definition and Units of Convection Coefficient

The convection coefficient $h$ quantifies the rate of convective heat transfer between a surface and a fluid. It appears in Newton's law of cooling:

$q = h \cdot (T_s - T_\infty)$

where $q$ is the heat flux in $W/m^2$, $T_s$ is the surface temperature, and $T_\infty$ is the fluid temperature. The units of $h$ are $W/(m^2 \cdot K)$. Higher $h$ means more effective heat transfer for a given temperature difference.

Convection Coefficients for External Flows

For external flows, empirical correlations relate the Nusselt number $Nu$ to the Reynolds number $Re$ and Prandtl number $Pr$. Once you find $Nu$, you extract $h$ from $Nu = h \cdot L / k$.

Flat plate:

• Laminar flow (Pohlhausen): $Nu_x = 0.332 \cdot Re_x^{0.5} \cdot Pr^{1/3}$ (local) or $\overline{Nu}_L = 0.664 \cdot Re_L^{0.5} \cdot Pr^{1/3}$ (average over length $L$)
• Turbulent flow: correlations such as $Nu_x = 0.0296 \cdot Re_x^{0.8} \cdot Pr^{1/3}$ apply in the turbulent region

Cylinder in cross-flow (Churchill-Bernstein):

$Nu = 0.3 + \frac{0.62 \cdot Re^{0.5} \cdot Pr^{1/3}}{[1 + (0.4/Pr)^{2/3}]^{1/4}} \cdot \left[1 + \left(\frac{Re}{282{,}000}\right)^{5/8}\right]^{4/5}$

Sphere (Whitaker):

$Nu = 2 + (0.4 \cdot Re^{1/2} + 0.06 \cdot Re^{2/3}) \cdot Pr^{0.4} \cdot (\mu_\infty / \mu_s)^{1/4}$

The viscosity ratio $\mu_\infty / \mu_s$ accounts for property variation between the free-stream and surface temperatures.

Convection Coefficients for Internal Flows

For flow inside pipes and ducts, the key distinction is again laminar vs. turbulent, and whether the flow is thermally/hydrodynamically developing or fully developed.

Laminar flow in circular pipes:

• Fully developed with constant wall heat flux: $Nu = 4.36$
• Fully developed with constant wall temperature: $Nu = 3.66$
• Developing flow depends on the Graetz number: $Gz = (D/L) \cdot Re \cdot Pr$

Turbulent flow in smooth circular pipes (Dittus-Boelter):

$Nu = 0.023 \cdot Re^{0.8} \cdot Pr^n$

where $n = 0.4$ for heating the fluid and $n = 0.3$ for cooling. This correlation is valid for $Re > 10{,}000$, $0.6 < Pr < 160$, and $L/D > 10$.

Non-circular ducts use the hydraulic diameter $D_h = 4A_c / P$ (where $A_c$ is the cross-sectional area and $P$ is the wetted perimeter) in place of $D$.

• Laminar flow (Sieder-Tate): $Nu = 1.86 \cdot \left(Re \cdot Pr \cdot \frac{D_h}{L}\right)^{1/3} \cdot \left(\frac{\mu_b}{\mu_w}\right)^{0.14}$
• Turbulent flow (Gnielinski, valid for $2300 < Re < 5 \times 10^6$):

$Nu = \frac{(f/8)(Re - 1000) \cdot Pr}{1 + 12.7 \cdot (f/8)^{1/2} \cdot (Pr^{2/3} - 1)}$

where $f$ is the Darcy friction factor (obtainable from the Moody chart or the Petukhov relation $f = (0.790 \ln Re - 1.64)^{-2}$).

Convection Coefficients for Natural Convection

Natural (free) convection is driven by buoyancy rather than an external pump or fan. The driving dimensionless groups are the Grashof number and the Rayleigh number.

$Gr = \frac{g \cdot \beta \cdot (T_s - T_\infty) \cdot L^3}{\nu^2}$

• $g$: gravitational acceleration
• $\beta$: volumetric thermal expansion coefficient (for an ideal gas, $\beta = 1/T_f$ where $T_f$ is the film temperature in Kelvin)
• $L$: characteristic length
• $\nu$: kinematic viscosity

$Ra = Gr \cdot Pr$

Vertical plates (Churchill-Chu, valid for all $Ra$):

$Nu = \left(0.825 + \frac{0.387 \cdot Ra^{1/6}}{[1 + (0.492/Pr)^{9/16}]^{8/27}}\right)^2$

Horizontal plates (McAdams):

• Hot surface facing up: $Nu = 0.54 \cdot Ra^{1/4}$ for $10^4 < Ra < 10^7$; $Nu = 0.15 \cdot Ra^{1/3}$ for $10^7 < Ra < 10^{11}$
• Hot surface facing down: $Nu = 0.27 \cdot Ra^{1/4}$ for $10^5 < Ra < 10^{11}$

The hot-surface-down case gives lower $Nu$ because the buoyant plume is trapped against the plate, suppressing convective motion.

Dimensional Analysis for Convective Heat Transfer

Buckingham Pi Theorem and Dimensionless Parameters

Convection problems involve many variables: velocity, viscosity, density, conductivity, specific heat, characteristic length, and so on. Dimensional analysis reduces this complexity by grouping variables into a smaller set of dimensionless parameters.

The Buckingham Pi theorem states: if a physical problem involves $n$ variables and $k$ independent dimensions (mass, length, time, temperature), the problem can be described by $p = n - k$ dimensionless groups $\Pi_1, \Pi_2, \ldots, \Pi_p$. This is why convection correlations are written in terms of $Nu$, $Re$, and $Pr$ rather than listing every individual property.

Key Dimensionless Parameters in Convective Heat Transfer

Reynolds number $Re$: ratio of inertial forces to viscous forces.

$Re = \frac{\rho \cdot v \cdot L}{\mu} = \frac{v \cdot L}{\nu}$

• Determines flow regime. For internal pipe flow: laminar below $Re \approx 2300$, turbulent above $Re \approx 4000$, transitional in between.
• For a flat plate: transition near $Re_x \approx 5 \times 10^5$.

Nusselt number $Nu$: ratio of convective to conductive heat transfer across the boundary layer.

$Nu = \frac{h \cdot L}{k}$

A $Nu$ of 1 would mean heat crosses the fluid layer by conduction alone. Values much greater than 1 indicate that convection significantly enhances heat transfer.

Prandtl number $Pr$: ratio of momentum diffusivity to thermal diffusivity.

$Pr = \frac{\nu}{\alpha} = \frac{c_p \cdot \mu}{k}$

• $Pr < 1$ (e.g., liquid metals, $Pr \approx 0.01$): the thermal boundary layer is thicker than the velocity boundary layer. Heat diffuses faster than momentum.
• $Pr \approx 1$ (e.g., gases like air, $Pr \approx 0.71$): both boundary layers have similar thickness.
• $Pr > 1$ (e.g., oils, $Pr \approx 100{-}1000$): the velocity boundary layer is thicker. Momentum diffuses faster than heat.

Grashof number $Gr$ and Rayleigh number $Ra$: govern natural convection.

$Gr = \frac{g \cdot \beta \cdot (T_s - T_\infty) \cdot L^3}{\nu^2}, \qquad Ra = Gr \cdot Pr$

• $Gr$ compares buoyancy forces to viscous forces.
• $Ra$ combines buoyancy and thermal diffusion effects. It determines whether natural convection is laminar or turbulent and which correlation to use.