Heat Transfer: Conduction, Convection, Radiation, and Heat Exchangers
Key Takeaways
- Choose the heat-transfer mode before calculating: conduction through material, convection between surface and fluid, or radiation between surfaces.
- Thermal-resistance networks are the fastest FE method for composite walls, cylinders, contact resistance, and combined convection boundaries.
- Convection coefficients are not material properties; they depend on flow condition, geometry, fluid properties, and temperature scale used in correlations.
- Radiation calculations require absolute temperature to the fourth power, emissivity, area, and sometimes view-factor judgment.
- Use LMTD when terminal temperatures are known; use effectiveness-NTU when outlet temperatures are unknown but exchanger size or UA is known.
- Heat-exchanger signs and heat-capacity rates must be tracked so hot-side heat loss equals cold-side heat gain after losses are considered.
Pick the mode
Heat transfer asks how energy crosses a temperature difference. Conduction is energy transfer through a solid or stationary medium. Convection is energy transfer between a surface and a moving or mixed fluid. Radiation is electromagnetic exchange and does not require a material medium. Many FE problems combine modes, such as conduction through a wall with convection on both sides.
The first decision is geometry. A plane wall uses L/(kA) as conduction resistance. A cylinder uses a logarithmic radius relationship. A sphere has its own resistance form. If the geometry does not match the formula, the answer can be wrong even with correct numbers.
Thermal resistance
Thermal-resistance networks make heat-transfer problems look like circuit problems. Heat rate equals temperature difference divided by total thermal resistance. Resistances in series add directly. Parallel paths require reciprocal addition, just like electrical resistances.
| Element | Resistance idea | Trap |
|---|---|---|
| Plane wall conduction | L/(kA) | Using thickness in mm while area is m^2 |
| Cylindrical wall conduction | logarithmic radius form | Using outside area for every layer without checking formula |
| Convection surface | 1/(hA) | Treating h as a solid conductivity |
| Contact resistance | added at interface | Forgetting it in layered assemblies |
| Fouling resistance | added in heat exchangers | Mixing inside-area and outside-area bases |
For steady one-dimensional conduction, the heat rate is constant through all series layers. Temperature drops are larger across larger resistances. That logic helps catch arithmetic errors: a highly insulating layer should have a larger temperature drop than a thin metal wall.
Convection and fins
Newton's law of cooling is q = h A (T_s - T_infinity). The challenge is usually finding or using h. In FE-style questions, h may be given directly, or a Nusselt-number correlation may be supplied. A Nusselt number converts to h through Nu = h L_c/k, where L_c is the characteristic length used by the correlation. Reynolds, Prandtl, and Grashof numbers help decide forced versus natural convection and laminar versus turbulent behavior.
Fins increase area but are not perfectly isothermal. Fin efficiency accounts for temperature drop along the fin. Very long fins eventually add little extra heat transfer because the far end approaches the surrounding fluid temperature. Do not assume doubling length always doubles heat transfer.
Radiation
Thermal radiation uses absolute temperature. For a small surface radiating to large surroundings, the heat rate often follows epsilon sigma A (T_s^4 - T_sur^4). Celsius or Fahrenheit temperatures cannot be raised to the fourth power in radiation formulas. Emissivity ranges from 0 to 1, with a blackbody at 1. View factors matter when finite surfaces exchange with each other, but many FE items simplify the surroundings as large and isothermal.
Radiation and convection may act in parallel from the same surface to the same surroundings. In that case, heat rates add. Do not combine temperatures first; compute each mode with its own relationship or use an equivalent coefficient only when justified.
Heat exchangers
Heat exchangers transfer energy between two fluid streams. The hot stream loses heat and the cold stream gains heat, with q = m_dot c_p Delta T for single-phase streams when c_p is appropriate. The heat-capacity rate is C = m_dot c_p; the stream with smaller C experiences the larger temperature change for the same heat rate.
Use the log mean temperature difference method when inlet and outlet temperatures are known or can be determined from an energy balance. Counterflow and parallel flow have different end temperature differences, so draw the exchanger before computing LMTD. Use effectiveness-NTU when outlet temperatures are not known but UA, flow arrangement, and heat-capacity rates are available.
Exam checks
Check watts versus kilowatts, m^2 versus mm^2, and kelvin temperature differences versus absolute kelvin temperatures. Temperature differences may be in C or K for conduction and convection, but radiation and thermodynamic ratios require absolute temperature. For heat exchangers, confirm that the hot outlet remains hotter than physically possible limits and that heat lost by one stream matches heat gained by the other unless losses are stated.
A composite wall has inside convection, two solid layers, and outside convection. What is the most direct FE method for steady one-dimensional heat rate?
A surface radiates to large surroundings. Which temperature scale must be used in the T^4 radiation terms?
A heat exchanger problem gives both inlet temperatures, flow rates, heat capacities, and UA, but neither outlet temperature. Which method is usually most appropriate?