3.2 Forced Convection: External Flow

External Flow Characteristics

Forced convection in external flow describes what happens when a fluid moves over a surface like a flat plate, cylinder, or sphere. Understanding how boundary layers develop and how flow transitions between regimes is the foundation for predicting heat transfer rates in applications ranging from electronics cooling to heat exchanger design.

Velocity and Thermal Boundary Layers

When fluid flows over a surface, two distinct boundary layers develop simultaneously.

The velocity boundary layer is the region near the surface where fluid velocity changes from zero at the surface (the no-slip condition) to the free-stream velocity $U_\infty$ at the outer edge. The thermal boundary layer is the analogous region where fluid temperature transitions from the surface temperature $T_s$ to the free-stream temperature $T_\infty$.

Both boundary layers start with zero thickness at the leading edge and grow thicker as you move downstream. The relative thickness of these two layers depends on the Prandtl number of the fluid:

• For $Pr > 1$ (most liquids), the thermal boundary layer is thinner than the velocity boundary layer
• For $Pr < 1$ (liquid metals), the thermal boundary layer is thicker
• For $Pr \approx 1$ (most gases), the two layers have roughly the same thickness

Flow Regimes and Heat Transfer

The local heat transfer coefficient is inversely related to the thermal boundary layer thickness. A thinner thermal boundary layer means a steeper temperature gradient at the surface, which drives higher heat transfer.

For flow over a flat plate, the boundary layer starts laminar at the leading edge and transitions to turbulent flow at a critical distance. That transition is governed by the local Reynolds number:

$Re_x = \frac{U_\infty x}{\nu}$

where $x$ is the distance from the leading edge and $\nu$ is the kinematic viscosity. The critical Reynolds number for a flat plate is typically around $Re_{x,cr} \approx 5 \times 10^5$, though surface roughness and free-stream turbulence can trigger earlier transition.

For flow over cylinders and spheres, the flow separates from the surface, creating a wake region on the downstream side. This separation and wake formation significantly affect the local heat transfer distribution around the object. The relevant Reynolds number uses the diameter as the characteristic length:

$Re_D = \frac{U_\infty D}{\nu}$

The Nusselt number ties everything together. It represents the ratio of convective to conductive heat transfer and is the dimensionless form of the heat transfer coefficient:

$Nu = \frac{hL}{k}$

For external flows, $Nu$ is generally a function of $Re$ and $Pr$.

Heat Transfer Coefficients for External Flow

Empirical Correlations

Since analytical solutions exist only for simple cases, engineers rely on empirical correlations to estimate average heat transfer coefficients. Choosing the right correlation depends on the geometry and flow regime.

Flat plate (laminar, $Re_L < 5 \times 10^5$):

$\overline{Nu_L} = 0.664 \, Re_L^{1/2} \, Pr^{1/3}$

This is the classic Blasius-type result, not the Sieder-Tate correlation (which is actually for internal pipe flow).

Flat plate (turbulent, $Re_L > 5 \times 10^5$):

$\overline{Nu_L} = 0.037 \, Re_L^{4/5} \, Pr^{1/3}$

For a plate with a mixed boundary layer (laminar leading section transitioning to turbulent), a combined correlation accounts for both regions.

Cylinder in cross-flow (Churchill-Bernstein correlation):

$\overline{Nu_D} = 0.3 + \frac{0.62 \, Re_D^{1/2} \, Pr^{1/3}}{\left[1 + (0.4/Pr)^{2/3}\right]^{1/4}} \left[1 + \left(\frac{Re_D}{282000}\right)^{5/8}\right]^{4/5}$

Valid for $Re_D \, Pr > 0.2$. Properties are evaluated at the film temperature.

Sphere (Whitaker correlation):

$\overline{Nu_D} = 2 + (0.4 \, Re_D^{1/2} + 0.06 \, Re_D^{2/3}) \, Pr^{0.4} \left(\frac{\mu_\infty}{\mu_s}\right)^{1/4}$

The leading "2" corresponds to the conduction limit for a sphere in a stagnant fluid. Properties are evaluated at $T_\infty$ except $\mu_s$, which is evaluated at the surface temperature.

Dimensionless Numbers and Fluid Properties

Three dimensionless groups appear repeatedly in convection analysis:

• Reynolds number $Re = \frac{U_\infty L}{\nu}$: ratio of inertial to viscous forces. Determines whether the flow is laminar or turbulent.
• Prandtl number $Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$: ratio of momentum diffusivity to thermal diffusivity. It characterizes how quickly thermal effects diffuse relative to velocity effects. Air has $Pr \approx 0.71$; water at room temperature has $Pr \approx 7$.
• Grashof number $Gr = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$: ratio of buoyancy to viscous forces. This is primarily relevant in natural convection, not forced convection. In forced convection problems, you can generally neglect buoyancy unless $Gr/Re^2$ is significant (mixed convection).

Fluid properties vary with temperature, so most correlations specify where to evaluate them. The film temperature is the most common reference:

$T_f = \frac{T_s + T_\infty}{2}$

Some correlations (like Whitaker's) use a viscosity ratio instead, so always check the specific correlation's requirements.

Calculating Heat Transfer Rate

Once you have the average Nusselt number from the appropriate correlation, extract the heat transfer coefficient:

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

Then apply Newton's law of cooling to find the convective heat transfer rate:

$Q = hA(T_s - T_\infty)$

where $A$ is the surface area exposed to the flow, $T_s$ is the surface temperature, and $T_\infty$ is the free-stream temperature.

Surface Roughness Impact on Convective Heat Transfer

Roughness Effects on Boundary Layer and Turbulence

Surface roughness disrupts the velocity boundary layer by introducing disturbances that promote turbulent mixing near the wall. This enhanced mixing brings more high-temperature (or low-temperature) fluid into contact with the surface, increasing heat transfer.

Roughness is quantified using the equivalent sand-grain roughness $k_s$, a standardized measure based on Nikuradse's experiments with sand-coated surfaces. The parameter that matters for heat transfer is the relative roughness, defined as the ratio $k_s / L$ (where $L$ is the characteristic length of the surface).

Roughness Effects in Different Flow Regimes

The impact of roughness depends strongly on the flow regime:

• Turbulent flow: Roughness elements protrude through the thin viscous sublayer, directly enhancing turbulent mixing. The Nusselt number and heat transfer coefficient increase with relative roughness, up to a limit where the roughness elements are fully exposed to the turbulent core.
• Laminar flow: Roughness has minimal direct effect on heat transfer in a fully laminar boundary layer. However, it can trigger an earlier transition to turbulence (lowering the critical Reynolds number), which then increases heat transfer.

The Dipprey-Sabersky correlation is one approach for estimating the effect of roughness on heat transfer in turbulent flow. It accounts for the increased friction and heat transfer associated with rough surfaces. Note that roughness always increases drag, so there's a trade-off between enhanced heat transfer and increased pumping power.

Convective Heat Transfer Problem Solving

Problem-Solving Steps

Follow this systematic approach for external convection problems:

1. Identify the geometry and flow configuration. Is it a flat plate, cylinder, sphere, or other shape? What is the characteristic length?
2. Gather fluid properties. Start by estimating the film temperature $T_f = (T_s + T_\infty)/2$ and look up $\rho$, $\mu$, $\nu$, $k$, $Pr$, and $c_p$ at that temperature.
3. Calculate the Reynolds number using the appropriate characteristic length and free-stream velocity. Compare to the critical value to determine the flow regime.
4. Select the correct empirical correlation based on geometry and flow regime. Double-check the correlation's validity range ($Re$ range, $Pr$ range).
5. Compute the average Nusselt number from the correlation, then extract the average heat transfer coefficient: $h = \overline{Nu} \cdot k / L$.
6. Apply Newton's law of cooling to find the heat transfer rate: $Q = hA(T_s - T_\infty)$.

Accounting for Surface Roughness

When the problem involves a rough surface:

1. Calculate the relative roughness $k_s / L$.
2. Apply the appropriate roughness correction (such as the Dipprey-Sabersky correlation) to adjust the smooth-surface Nusselt number.
3. If the heat transfer rate changes the surface temperature significantly, you may need to iterate: recalculate the film temperature, update fluid properties, and repeat until the solution converges.

Iteration is especially important when the surface temperature isn't fixed (e.g., constant heat flux conditions), since the calculated heat transfer rate feeds back into the temperature used for property evaluation.