Heat Transfer for the FE Exam
Heat transfer problems appear on the FE Mechanical, Chemical, and other discipline exams. Heat always flows from hot to cold, and the rate of transfer depends on the mode (conduction, convection, or radiation) and the conditions. Understanding the governing equations for each mode is essential.
Conduction
Conduction is heat transfer through a solid material (or a stationary fluid) due to molecular interactions. The rate of heat conduction is governed by Fourier's law:
q = -k * A * dT/dx
Where q is the heat transfer rate (W), k is the thermal conductivity (W/(m*K)), A is the cross-sectional area perpendicular to heat flow, and dT/dx is the temperature gradient.
Steady-State Conduction Through a Plane Wall
For a flat wall of thickness L with temperatures T_1 and T_2 on opposite faces:
q = k * A * (T_1 - T_2) / L
This can be written in terms of thermal resistance: q = delta_T / R, where R_cond = L / (k * A).
Composite Walls
For walls made of multiple layers in series, the total thermal resistance is the sum of individual resistances:
R_total = R_1 + R_2 + R_3 + ...
This thermal resistance concept is analogous to electrical resistances in series. The heat flow rate is: q = (T_hot - T_cold) / R_total.
Radial Conduction (Cylindrical Coordinates)
For heat conduction through a cylindrical wall (such as insulation around a pipe):
q = 2 * pi * k * L * (T_i - T_o) / ln(r_o / r_i)
Where r_i and r_o are the inner and outer radii, and L is the length. The thermal resistance is:
R_cyl = ln(r_o / r_i) / (2 * pi * k * L)
Convection
Convection is heat transfer between a surface and a moving fluid. Newton's law of cooling gives the rate:
q = h * A * (T_s - T_inf)
Where h is the convection heat transfer coefficient (W/(m^2*K)), T_s is the surface temperature, and T_inf is the fluid temperature far from the surface.
The convection thermal resistance is: R_conv = 1 / (h * A).
Types of Convection
- Forced convection: Fluid motion is driven by external means (fan, pump). Higher h values.
- Natural (free) convection: Fluid motion is driven by buoyancy forces from temperature differences. Lower h values.
The Nusselt number relates the convection coefficient to fluid properties and geometry: Nu = h * L / k_fluid. Various correlations give Nu as a function of Reynolds and Prandtl numbers for different geometries.
Overall Heat Transfer Coefficient
For heat transfer from a hot fluid through a wall to a cold fluid, the overall heat transfer coefficient U combines convection on both sides and conduction through the wall:
q = U * A * (T_hot_fluid - T_cold_fluid)
1 / (U * A) = 1/(h_1 * A_1) + L/(k * A_wall) + 1/(h_2 * A_2)
Radiation
Radiation is heat transfer through electromagnetic waves. All objects above absolute zero emit thermal radiation. The Stefan-Boltzmann law gives the total emissive power of a blackbody:
E_b = sigma * T^4
Where sigma = 5.67 x 10^(-8) W/(m^2*K^4) is the Stefan-Boltzmann constant and T is in Kelvin.
For a real (non-black) surface with emissivity epsilon:
E = epsilon * sigma * T^4
Emissivity ranges from 0 to 1. A blackbody has epsilon = 1.
Radiation Exchange Between Two Surfaces
The net radiation heat transfer between two surfaces depends on their temperatures, emissivities, and geometric relationship (view factor):
q_12 = epsilon * sigma * A * (T_1^4 - T_2^4) (for a small body completely enclosed by a large surrounding surface)
Fins and Extended Surfaces
Fins increase the surface area available for convection, enhancing heat transfer. Fin effectiveness and fin efficiency are used to evaluate fin performance:
- Fin efficiency: The ratio of actual heat transfer from the fin to the heat transfer if the entire fin were at the base temperature.
- Fin effectiveness: The ratio of heat transfer with the fin to heat transfer without the fin.
Heat Exchangers
Heat exchangers transfer heat between two fluids. Common types include parallel flow, counterflow, and cross flow. The log mean temperature difference (LMTD) method is used for design:
q = U * A * LMTD
Where LMTD = (delta_T_1 - delta_T_2) / ln(delta_T_1 / delta_T_2)
For counterflow exchangers, delta_T_1 and delta_T_2 are the temperature differences at each end. Counterflow gives a higher LMTD (better performance) than parallel flow for the same inlet and outlet temperatures.
Practice Tips
- Master the thermal resistance concept; it simplifies complex composite wall and combined convection/conduction problems
- Practice unit conversions between Celsius and Kelvin for radiation problems (T must be absolute)
- Know the difference between parallel flow and counterflow heat exchangers
- Practice LMTD calculations
Heat transfer problems on the FE exam follow predictable patterns. Identify the mode (conduction, convection, radiation, or combination), apply the appropriate equation, and solve systematically.