Heat Conduction
Fourier’s Law
The rate of heat transfer by conduction is expressed as:
where:
= rate of heat transfer (W) = thermal conductivity of the material (W/m·K) = area of the cross-section (m²) = temperature gradient (K/m)
Transient Conduction Equation
For a transient state, the one-dimensional heat conduction equation can be used:
where:
= temperature (K) = time (s) = density (kg/m³) = specific heat capacity (J/kg·K)
Heat Convection
Newton’s Cooling Law
The rate of heat transfer by convection is expressed as:
where:
= rate of heat transfer (W) = convective heat transfer coefficient (W/m²·K) = surface area (m²) = surface temperature (K) = temperature of the fluid far from the surface (K)
Heat Radiation
Stefan-Boltzmann Law
The rate of heat transfer by radiation is expressed as:
where:
= rate of heat transfer (W) = emissivity of the surface (dimensionless) = Stefan-Boltzmann constant ( ) = surface area (m²) = surface temperature (K) = surrounding temperature (K)
Heat Transfer in Composite Systems
Thermal Resistance
The total thermal resistance
where each
with:
= thickness of the layer (m) = thermal conductivity of the layer (W/m·K)
Heat Transfer in Parallel
The rate of heat transfer in a parallel system can be calculated as:
where each
Heat Transfer by Evaporation and Condensation
Latent Heat
The heat transferred during evaporation or condensation is calculated as:
where:
= heat transferred (J) = mass of the fluid (kg) = latent heat of evaporation or condensation (J/kg)
Heat Transfer Efficiency
Efficiency of a Heat Exchanger
Efficiency can be defined as:
where:
= actual rate of heat transfer (W) = maximum rate of heat transfer (W)
Dimensionless Numbers
Nusselt Number (Nu)
The Nusselt number is a dimensionless number that characterizes heat transfer by convection compared to conduction. It is defined as:
where:
= convective heat transfer coefficient (W/m²·K) = characteristic length (m), which can be the diameter of a pipe, the height of a plate, etc. = thermal conductivity of the fluid (W/m·K)
Interpretation:
- A high Nusselt number indicates that convection is significant compared to conduction. This is often the case in turbulent flows or in systems where high temperatures are applied.
- A low Nusselt number suggests that conduction is the predominant mode of heat transfer, as in laminar flow situations.
Reynolds Number (Re)
The Reynolds number is a dimensionless number that describes the relationship between inertial forces and viscous forces in a moving fluid. It is defined as:
where:
= density of the fluid (kg/m³) = velocity of the fluid (m/s) = characteristic length (m) = dynamic viscosity of the fluid (Pa·s)
Interpretation:
: laminar flow. : turbulent flow. : transitional regime.
Prandtl Number (Pr)
The Prandtl number is a dimensionless number that relates momentum diffusion (viscosity) to heat diffusion. It is defined as:
where:
= dynamic viscosity (Pa·s) = specific heat capacity (J/kg·K) = thermal conductivity (W/m·K)
Interpretation:
- A
indicates that heat diffusion is faster than momentum diffusion (light fluid). - A
suggests that momentum diffusion is faster than heat diffusion (heavy fluid).
Schmidt Number (Sc)
The Schmidt number is a dimensionless number that relates mass diffusion to momentum diffusion in a fluid. It is defined as:
where:
= mass diffusion coefficient (m²/s)
Interpretation:
- A
indicates that mass diffusion is faster than momentum diffusion. - A
indicates that momentum diffusion is faster than mass diffusion.
Grashof Number (Gr)
The Grashof number is a dimensionless number that measures the importance of buoyancy forces in a fluid due to temperature differences. It is defined as:
where:
= acceleration due to gravity (m/s²) = thermal expansion coefficient (1/K) = surface temperature (K) = surrounding temperature (K) = characteristic length (m) = kinematic viscosity (m²/s)
Interpretation:
- A high
indicates that buoyancy forces dominate the flow, as in natural convection. - A low
suggests that the flow is dominated by viscosity.
Physical Constants
Table with some of the main physical constants and properties that are useful in the analysis of heat transfer and fluid mechanics.
| Constant/Property | Symbol | Value | Units | Description |
|---|---|---|---|---|
| Stefan-Boltzmann Constant | W/m²·K⁴ | Constant that relates thermal radiation to temperature. | ||
| Thermal Conductivity of Air | W/m·K | Thermal conductivity of air at 25 °C. | ||
| Thermal Conductivity of Water | W/m·K | Thermal conductivity of water at 25 °C. | ||
| Specific Heat Capacity of Water | J/kg·K | Specific heat capacity of water. | ||
| Density of Water | kg/m³ | Density of water at 4 °C. | ||
| Dynamic Viscosity of Water | Pa·s | Dynamic viscosity of water at 25 °C. | ||
| Dynamic Viscosity of Air | Pa·s | Dynamic viscosity of air at 25 °C. | ||
| Density of Air | kg/m³ | Density of air at 25 °C and 1 atm. | ||
| Gravity | m/s² | Acceleration due to gravity on Earth. | ||
| Latent Heat of Vaporization of Water | J/kg | Heat required to vaporize 1 kg of water at 100 °C. | ||
| Emissivity of Steel | dimensionless | Emissivity of the steel surface. | ||
| Emissivity of Water | dimensionless | Emissivity of the water surface. |
Notes on Constants:
- The provided values are approximate and may vary with temperature and pressure. It is important to consult specific tables or technical literature for more precise values under particular conditions.
- Properties such as density and viscosity of air and water change with temperature, so be sure to consider the specific conditions of the problem you are analyzing.
Properties of Substances
Table with some important properties of the main materials and substances commonly used in heat transfer analysis.
| Material/Substance | Thermal Conductivity | Specific Heat Capacity | Density | Viscosity | Emissivity | Description |
|---|---|---|---|---|---|---|
| Water | Liquid commonly used in heating and cooling systems. | |||||
| Air | Gas used as a heat transfer medium in convection systems. | |||||
| Copper | Metal with high thermal conductivity, used in electrical and thermal applications. | |||||
| Aluminum | Lightweight metal efficient in heat transfer, used in heat exchangers. | |||||
| Steel | Material used in structures and thermal equipment components. | |||||
| Glass | Material used in windows and containers, with low thermal conductivity. | |||||
| Polypropylene (PP) | Plastic used in low-temperature applications. | |||||
| Polystyrene (PS) | Common insulating material, used in packaging and insulation. | |||||
| Concrete | Construction material with good heat storage capacity. |
Notes on Properties:
- The provided values are approximate and may vary depending on temperature, pressure, and purity of materials. It is always recommended to consult specific sources or technical literature for accurate values.
- Thermal conductivity is essential for determining how a material can transfer heat.
- Specific heat capacity indicates the amount of energy needed to change the temperature of a unit mass of material.
- Density is important for calculating the weight and volume of a material in engineering applications.
- Viscosity affects the flow of liquids and gases and, therefore, heat transfer in convection systems.
- Emissivity is a key factor in thermal radiation, affecting how a material emits and absorbs thermal radiation.