FE Heat Transfer: Convection Review

Convection is often the dominant mode of heat transfer in engineering systems, from electronics cooling to HVAC design, making it a high-yield topic for the FE exam. Mastering convection requires understanding not just the underlying physics but also the empirical correlations and dimensionless numbers used to solve practical problems efficiently.

Newton's Law of Cooling and the Boundary Layer

All convective heat transfer analysis begins with Newton's law of cooling. This fundamental equation states that the convective heat transfer rate $q$ is proportional to the surface area $A$ and the temperature difference between the surface $T_s$ and the surrounding fluid bulk temperature $T_\infty$: $$q = h A (T_s - T_\infty)$$ Here, $h$ is the convective heat transfer coefficient, with units of W/m²·K. The central challenge in convection problems is determining the correct value of $h$, which is not a simple property of the fluid but depends on flow conditions, geometry, and fluid properties.

The reason $h$ is complex lies in the boundary layer concept. When a fluid flows over a surface, a thin region develops where fluid velocity changes from zero at the wall (the no-slip condition) to the free-stream velocity. This is the velocity boundary layer. Simultaneously, if there is a temperature difference, a thermal boundary layer forms where fluid temperature changes from the wall temperature to the bulk temperature. The thickness and behavior of these layers govern the rate of heat transfer. A thin thermal boundary layer typically indicates a high heat transfer coefficient, as the temperature gradient is steep.

Dimensionless Numbers: The Language of Convection

To correlate experimental data and create universal formulas, engineers use dimensionless numbers. You must be fluent in these for the FE exam:

• Reynolds Number ($Re$): The ratio of inertial to viscous forces. It determines the flow regime. $Re_L = \frac{\rho V L}{\mu}$, where $L$ is a characteristic length. For flow over a flat plate, $Re < 5 \times 10^5$ typically indicates laminar flow, while a higher value signifies turbulent flow.
• Nusselt Number ($Nu$): The primary output of most correlations. It represents the enhancement of heat transfer through convection relative to conduction across the same layer. $Nu_L = \frac{h L}{k_f}$, where $k_f$ is the fluid's thermal conductivity. Solving for $h$ is often a matter of finding $Nu$ from a correlation.
• Prandtl Number ($Pr$): A fluid property ratio comparing momentum diffusivity (kinematic viscosity) to thermal diffusivity. $Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k_f}$. It links the velocity and thermal boundary layers. For air, $Pr \approx 0.7$; for water, $Pr \approx 7$; for oils, $Pr \gg 1$.
• Grashof Number ($Gr$): Used in natural convection. It represents the ratio of buoyancy to viscous forces. $Gr_L = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$, where $g$ is gravity, $\beta$ is the volumetric thermal expansion coefficient.
• Rayleigh Number ($Ra$): The product of $Gr$ and $Pr$. It is the direct criterion for transition in natural convection. $Ra_L = Gr_L \cdot Pr$.

Forced Convection Correlations

Forced convection occurs when an external device (pump, fan, etc.) drives fluid motion. Correlations depend heavily on geometry and flow type.

External Flow refers to flow over bodies immersed in a fluid, like a flat plate, cylinder, or sphere. A classic FE exam problem involves flow over a flat plate. You must first calculate $Re_L$ to determine if the flow is laminar, turbulent, or mixed. For a laminar flat plate ($Re_L < 5 \times 10^5$), the average Nusselt number is given by: $$\overline{Nu}_L = 0.664 Re_L^{1/2} Pr^{1/3} \quad (Pr \gtrsim 0.6)$$ For a turbulent flat plate from the leading edge, a common correlation is $\overline{Nu}_L = 0.037 Re_L^{4/5} Pr^{1/3}$. For a mixed boundary layer, you often calculate laminar and turbulent portions separately.

Internal Flow refers to flow inside conduits like pipes or ducts. Here, the key parameter is the hydraulic diameter $D_h = \frac{4A_c}{P}$, especially for non-circular ducts. Flow regime is determined by the pipe Reynolds number $Re_D = \frac{\rho u_m D}{\mu}$, with transition around 2300. For fully developed, turbulent flow in a smooth pipe, the Dittus-Boelter equation is frequently used: $$Nu_D = 0.023 Re_D^{4/5} Pr^n$$ where $n=0.4$ for heating ($T_s > T_m$) and $n=0.3$ for cooling ($T_s < T_m$) of the fluid. Remember that for internal flow, the mean fluid temperature $T_m$, not $T_\infty$, is used in Newton's law.

Natural (Free) Convection Correlations

Natural convection is driven by density differences from temperature gradients in a body force field (like gravity). The key is using the $Gr$ or $Ra$ number. Correlations are typically of the form: $$\overline{Nu}_L = C (Gr_L Pr)^n = C Ra_L^n$$ Constants $C$ and $n$ depend on the geometry (vertical plate, horizontal cylinder, etc.) and the flow regime (laminar or turbulent). For example, for a vertical plate in a laminar flow regime ($10^4 < Ra_L < 10^9$), a common correlation is $\overline{Nu}_L = 0.59 Ra_L^{1/4}$. For the same plate in a turbulent regime ($10^9 < Ra_L < 10^{13}$), $\overline{Nu}_L = 0.1 Ra_L^{1/3}$. You must be able to calculate $Ra_L$, identify the correct geometry and regime, and select the right constants from a provided reference table, as is typical on the FE exam.

Common Pitfalls

1. Using the Wrong Temperature Difference: In Newton's law, for external flow, use $T_s - T_\infty$. For internal flow, you must use $T_s - T_m$ (the mean fluid temperature), which often changes along the pipe length. Using $T_\infty$ for an internal flow problem is a classic exam trap.
2. Misidentifying the Characteristic Length ($L$ or $D$): This is the most common calculation error. For a vertical plate, $L$ is the plate height. For flow over a horizontal plate, $L$ is the plate length in the flow direction for $Re$, but for $Nu$ in natural convection, it's often $A_s/P$. For internal flow in a non-circular duct, you must use the hydraulic diameter $D_h$. Always double-check the definition for your specific correlation.
3. Ignoring Flow Regime and Property Evaluation: Blindly plugging numbers into a formula without checking if the flow is laminar or turbulent (via $Re$ or $Ra$) will lead to the wrong correlation. Furthermore, fluid properties ($\mu, k, Pr$) are temperature-dependent. The FE exam will typically specify the reference temperature (e.g., "evaluate properties at the film temperature $T_f = (T_s + T_\infty)/2$"). Using the wrong reference temperature can subtly alter your answer.
4. Confusing Local vs. Average Quantities: Some correlations give a local Nusselt number $Nu_x$ at a distance $x$ from the leading edge. Most problems, however, ask for the total heat transfer rate from an entire surface, which requires the average heat transfer coefficient $\overline{h}$ and thus the average Nusselt number $\overline{Nu}_L$. Know which one your correlation provides.

Summary

• The core equation is Newton's law of cooling: $q = h A \Delta T$. Your primary task is finding $h$.
• Find $h$ by using the correct empirical correlation to calculate the Nusselt number ($Nu = hL/k_f$), then solve for $h$.
• Selecting the right correlation requires identifying the flow type (forced vs. natural), geometry (external vs. internal, flat plate vs. cylinder), and flow regime (laminar vs. turbulent) using $Re$ for forced flow or $Ra$ for natural convection.
• For forced internal flow, always use the mean fluid temperature $T_m$ in Newton's law and the hydraulic diameter $D_h$ for non-circular ducts.
• On the FE exam, meticulously verify your characteristic length, flow regime, and the reference temperature for fluid properties to avoid common traps.