Thermal Analysis
Glossary Of Terms
 Absorption

Absorption is defined as the process of converting radiation intercepted by matter to internal thermal energy. It is the fraction of the incident radiation absorbed by matter. This can be characterized by directional, hemispherical, spectral, and total absorptivity.
 Adiabatic condition

An adiabatic condition is a process or boundary condition in which there is no heat transfer. In the context of heat transfer, adiabatic conditions imply that there is no heat exchange between the system and its surroundings. This is significant because it allows for the analysis of heat transfer processes under specific conditions, such as when insulation prevents heat from entering or escaping a system.
Adiabatic can simplify the calculation of heat transfer rates and temperature distributions. By assuming adiabatic conditions, one can focus on specific aspects of heat transfer, such as conduction, convection, or radiation, without the added complexity of heat exchange with the surroundings.
 Advection

Advection refers to the transport of a substance by the bulk motion of a fluid. It is specifically associated with the movement of a large number of molecules collectively or as aggregates in the fluid. When there is a temperature gradient, this collective motion contributes to heat or mass transfer. In the context of heat transfer, advection is synonymous with the bulk fluid motion that contributes to the overall heat transfer in addition to the random molecular motion (diffusion).
 Biot number

The Biot number, denoted as Bi, is a dimensionless parameter that plays a fundamental role in conduction problems involving surface convection effects. It is defined as the ratio of the internal thermal resistance of a solid to the thermal resistance at the solidfluid interface. Mathematically, it is expressed as:
$$Bi = \frac{h L}{k}$$
Where:
 h is the convective heat transfer coefficient [W/m^{2}.K]
 L is a characteristic length (such as the thickness of the solid) [m]
 k is the thermal conductivity of the solid [W/m.K]
The Biot number provides a measure of the temperature drop in the solid relative to the temperature difference between the solid's surface and the fluid. It is used to evaluate the significance of convective heat transfer relative to conductive heat transfer in a given system.
 Black body

A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of wavelength and direction. It also emits the maximum amount of radiation possible for a given temperature and wavelength. The radiation emitted by a blackbody is a function of wavelength and temperature, but it is independent of direction, making it a diffuse emitter. No actual surface has precisely the properties of a black body, but the concept of a black body serves as a standard against which the radiative properties of real surfaces can be compared.
 Boiling

Boiling is the process in which evaporation occurs at a solidliquid interface. It happens when the temperature of the surface exceeds the saturation temperature corresponding to the liquid pressure. Heat is transferred from the solid surface to the liquid, leading to the formation of vapor bubbles, which grow and detach from the surface. Boiling can occur under various conditions such as pool boiling, forced convection boiling, subcooled boiling, and saturated boiling, each with unique characteristics. The boiling process is influenced by factors like excess temperature, the nature of the surface, thermophysical properties of the fluid, and buoyancy forces.
 Boundary layer

A boundary layer is a thin fluid layer that develops near a surface when a fluid flows over that surface. It is characterized by a gradual transition in fluid velocity (velocity boundary layer) and other properties, such as temperature (thermal boundary layer), from the surface to the outer free stream. The boundary layer is typically defined by its thickness, and it is important in problems involving convection transfer.
 Buoyancy

Buoyancy refers to the upward force exerted by a fluid on an object immersed in it. This force is a result of the difference in pressure between the top and bottom of the object, and it is directly related to the density of the fluid. When an object is placed in a fluid, it displaces some of the fluid, and the weight of the displaced fluid exerts an upward force on the object, which is known as the buoyant force.
In the context of fluid mechanics, buoyancy plays a significant role in driving fluid motion in free convection, where density variations due to temperature differences lead to the generation of buoyancy forces. These forces cause the fluid to move, leading to phenomena such as natural circulation and heat transfer in various engineering and environmental applications.
 Condensation

Condensation is the process in which a substance changes from its gaseous state to its liquid state. It is a crucial phenomenon in heat and mass transfer because it involves the transfer of thermal energy (heat) as well as the transfer of mass from the vapor phase to the liquid phase.
 Conduction

Conduction is the transfer of energy within a medium due to a temperature gradient, and it occurs as a result of random atomic or molecular activity. This type of heat transfer is governed by Fourier's law, which describes the heat flux in relation to the temperature distribution within the medium. Conduction can occur in various scenarios, including steadystate and transient conduction, and in both onedimensional and multidimensional systems. Additionally, conduction is the only mode of heat transfer in a medium in which the temperature distribution is governed by the heat equation.
 Convection

Convection refers to the mode of heat transfer that involves both energy transfer by the bulk fluid motion (advection) and the random motion of fluid molecules (conduction or diffusion). It occurs when a fluid in motion comes into contact with a surface at a different temperature, resulting in the transfer of heat. The interaction between the fluid and the surface leads to the development of a boundary layer in the fluid, where the velocity varies from zero at the surface to a finite value associated with the flow.
 Diffusion

Diffusion is the movement of molecules from an area of high concentration to an area of low concentration. It is a process by which molecules spread out and mix with other molecules due to their random motion. In the context of mass transfer, diffusion refers to the movement of a species in a mixture as a result of a concentration difference. This process is governed by Fick's law, which describes the diffusive flux of a species in terms of the concentration gradient and the binary diffusion coefficient.
 Diffusivity

Diffusivity is a measure of how quickly a substance will spread or diffuse. It can be defined as the proportionality constant between the flux of a substance and the concentration gradient of that substance. In the context of mass transfer, diffusivity specifically refers to the rate at which a particular species within a mixture will spread or diffuse.
 Dynamic viscosity

Dynamic viscosity, denoted by the symbol μ, is defined as the measure of a fluid's resistance to shear or angular deformation. It quantifies the internal friction within the fluid as it flows, and it is a key property that determines the fluid's behavior under shear stress. In the context of a Newtonian fluid, the dynamic viscosity is directly proportional to the rate of shear deformation. It is measured in units of [Pa·s] in SI units. Alternative units include [lbm/ft·h] and [lbf·h/ft^{2}].
 Electromagnetic spectrum

The electromagnetic spectrum encompasses all forms of electromagnetic radiation, ranging from highenergy gamma rays and Xrays to ultraviolet (UV) radiation, visible light, and infrared radiation, to longer wavelength microwaves and radio waves. It includes wavelengths from approximately 0.1 to 100 μm. This spectrum is of interest to various fields such as highenergy physics, nuclear engineering, electrical engineering, and atmospheric science due to the diverse properties and interactions of the different types of radiation.
 Enthalpy

Enthalpy H is a thermodynamic property of a system. It represents the total internal energy of a system, including the amount of energy associated with the pressure and volume of the system. Enthalpy is defined as H = U + PV, where U is the internal energy of the system, P is the pressure, and V is the volume. In the context of fluid mechanics and heat transfer, enthalpy is often used to describe the total energy of a fluid, including its internal energy and the energy associated with its pressure and volume.
 Emission

Emission refers to the process of a material releasing thermal energy in the form of radiation. This radiation is emitted in the form of electromagnetic waves, and the intensity and spectral distribution of the emitted radiation depend on the temperature and properties of the emitting material. Emission can be characterized by parameters such as total emissive power, spectral emissive power, emissivity, and directional emissivity, which describe the amount and characteristics of the radiation emitted by a surface.
 Evaporation

Evaporation is the process by which a liquid turns into a gas or vapor. This occurs when the molecules of a liquid gain enough energy to break free from the liquid phase and enter the gaseous phase. The energy required for evaporation usually comes from the surroundings, and the process leads to a cooling effect on the remaining liquid. In the context of heat and mass transfer, evaporation is a key aspect of processes such as evaporative cooling, where the transfer of energy and mass between a liquid and its surroundings is crucial.
 Forced convection

Forced convection is the heat transfer process in which the fluid motion is induced by external means, such as a fan, a pump, or the wind. An example of forced convection is the use of a fan to provide air cooling for hot electrical components on printed circuit boards. In forced convection, the flow is driven by external forces rather than natural buoyancy forces.
 Forced convection boiling

In forced convection boiling, the majority of the heat transfer occurs due to direct transfer from the hot surface to the liquid. This is similar to a type of liquid phase forced convection, where the fluid motion is induced by the rising bubbles. The forced convection boiling process significantly impacts the heat transfer characteristics, and correlations are used to understand and predict these effects.
 Fouling

Fouling refers to the accumulation of unwanted material on solid surfaces, particularly in heat exchangers. This accumulation can decrease the efficiency of heat transfer and increase the resistance to fluid flow, leading to reduced performance of the heat exchanger. Fouling can be caused by various factors such as deposition of solids, precipitation of dissolved salts, biological growth, or chemical reactions. It is a common issue in industrial processes and can be mitigated through regular cleaning and maintenance of heat exchanger surfaces.
 Fouling factor

The fouling factor, denoted as R_{f}, is a measure of the resistance to heat transfer caused by the accumulation of deposits on a heat transfer surface. It represents the additional thermal resistance due to fouling and is typically expressed in [m^{2} · K/W]. Higher fouling factors indicate greater resistance to heat transfer due to fouling.
Fouling factors can vary depending on the fluid and its temperature. For example, seawater and treated boiler feedwater have different fouling factors based on their temperature ranges. The fouling factor is an important consideration in the design and operation of heat exchangers, as it affects the overall heat transfer coefficient and efficiency of the system.
 Fourier number

The Fourier number, often denoted as Fo, is a dimensionless time parameter used to characterize transient conduction problems. It is defined as the ratio of the rate of heat conduction within a solid to the rate of heat storage within the solid. Mathematically, it is expressed as:
$$Fo = \frac{\alpha \cdot t}{L^2}$$
 α is the thermal diffusivity of the solid
 t is time
 L is a characteristic length of the solid
The Fourier number is used to determine the relative importance of conduction within a solid over a given period of time. It helps in understanding the behavior of temperature distribution within the solid during transient conduction. For example, a small Fourier number indicates that the conduction process is relatively fast compared to the rate of heat storage, and as a result, the temperature distribution within the solid changes rapidly with time. On the other hand, a large Fourier number suggests that the rate of heat storage within the solid is significant compared to the rate of heat conduction, leading to a slower change in the temperature distribution over time.
 Fourier's law

Fourier's law is a fundamental principle governing heat conduction, expressing the relationship between the heat flux (the rate of heat transfer per unit area) and the temperature gradient (the rate of change of temperature with distance) in a material. It is given by the equation:
$$q = k \nabla T$$
Where:
 q is the heat flux [W/m^{2}]
 k is the thermal conductivity of the material [W/m.K]
 ∇T represents the temperature gradient [K/m]
Fourier's law indicates that the heat flux is proportional to the negative of the temperature gradient, and it applies to all forms of matter, including solids, liquids, and gases. The law provides a means to calculate the heat transfer within a material based on the temperature distribution and is essential for understanding and analyzing conduction heat transfer in various physical systems.
 Free convection

Free convection is a type of fluid motion that occurs within a fluid due to buoyancy forces, without any external forcing conditions. It originates when a body force, typically gravity, acts on a fluid with density gradients, resulting in the induction of free convection currents. This type of convection is driven by the presence of a temperature gradient, leading to buoyancy forces and the development of free convection boundary layers. Free convection plays a significant role in various systems and applications, influencing heat transfer, temperature distributions, and environmental processes.
 Froude number

The Froude number Fr is a dimensionless number used to quantify the influence of gravity on the motion of a fluid. It is defined as the ratio of the inertia force to the gravitational force and is given by the equation:
$$Fr = \frac{V}{\sqrt{gL}}$$
 V is the characteristic velocity
 g is the acceleration due to gravity
 L is the characteristic length
In the context of fluid flow, the Froude number is particularly important in analyzing open channel flow and in determining the type of flow regime, such as subcritical, critical, or supercritical flow.
 Grashof number

The Grashof number is a dimensionless parameter that relates buoyancy forces to viscous forces in free convection heat transfer. It is defined as:
$$Gr = \frac{g \cdot \beta \cdot \Delta T \cdot L^3}{\nu^2}$$
Where:
 g is the acceleration due to gravity
 β is the coefficient of volume expansion
 ΔT is the temperature difference between the surface and the surrounding fluid
 L is a characteristic length
 ν is the kinematic viscosity
The Grashof number plays a crucial role in free convection, similar to the role played by the Reynolds number in forced convection. It is a measure of the influence of buoyancy forces on the heat transfer process.
 Gray body

A gray body is a theoretical concept used in thermal radiation and heat transfer. It refers to a surface that emits and absorbs radiation with an emissivity and absorptivity that are independent of the wavelength of the radiation. This means that a gray body emits and absorbs radiation equally well at all wavelengths. In other words, it has a constant emissivity and absorptivity across the entire electromagnetic spectrum.
 Heat flux

Heat flux, denoted by the symbol q'', is the rate of heat transfer per unit area. It is a measure of the amount of heat energy that passes through a surface per unit time. Mathematically, it is defined as the heat transfer per unit time per unit area and is expressed in units of watts per square meter [W/m²].
 Ideal gas

An ideal gas is a theoretical gas composed of a large number of molecules with no volume and no attractive forces between them. In an ideal gas, the molecules are in constant, random motion and undergo perfectly elastic collisions with each other and with the walls of the container. The behavior of an ideal gas is described by the ideal gas law, which relates the pressure, volume, temperature, and the number of moles of gas. This law is represented by the equation:
$$P V = n R T$$
where:
 P is the pressure
 V is the volume
 n is the number of moles
 R is the ideal gas constant
 T is the temperature
 Irradiation

Irradiation refers to the radiant energy that is incident on a surface per unit area. It is related to radiative heat transfer as it determines the amount of radiation that is absorbed by a surface. The irradiation received by a surface affects the rate at which the surface gains or loses thermal energy through radiation. In the context of radiative heat transfer, irradiation plays a crucial role as it influences the heat exchange between surfaces and the surrounding environment.
 Kinematic viscosity

Kinematic viscosity, denoted by the symbol ν, is a fluid property defined as the ratio of the dynamic viscosity, measured in [Pa⋅s], to the mass density, measured in [kg/m^{3}]. It has units of [m^{2}/s]. Kinematic viscosity is a measure of a fluid's resistance to flow under the influence of gravity, and it provides insight into the fluid's internal friction and ability to deform. In the context of fluid dynamics, kinematic viscosity plays a crucial role in determining the behavior of boundary layers, diffusion, and the overall flow characteristics of a fluid.
 Laminar flow

Laminar flow is a type of fluid flow where the fluid moves in parallel layers with minimal mixing between the layers. In laminar flow, the fluid moves smoothly in an orderly fashion, and the velocity of the fluid at any point remains constant over time. This type of flow is characterized by welldefined streamlines and low fluid velocity gradients. Laminar flow is typically observed at lower flow rates and is distinguished by its organized and predictable behavior.
 Latent heat

Latent heat refers to the energy associated with a phase change of a substance, such as from solid to liquid, liquid to gas, or vice versa. This energy is not related to a change in temperature, but rather to the change in state of the substance. For example, when a liquid evaporates and turns into a gas, or when a gas condenses into a liquid, the energy involved in this phase change is known as latent heat. It's important to note that this energy is not directly related to the temperature of the substance, but rather to the change in its physical state.
 Lumped capacitance method

The lumped capacitance method is a simplified approach used to analyze transient conduction problems in which a solid experiences a sudden change in its thermal environment. The method assumes that the temperature of the solid is spatially uniform at any instant during the transient process, meaning that temperature gradients within the solid are considered negligible.
To determine the validity of the lumped capacitance method, a criterion is used to assess whether it can be applied with reasonable accuracy. This criterion involves calculating the Biot number, which compares the convection heat transfer at the solid's surface to the heat conduction within the solid. If the Biot number is less than a certain value (typically 0.1), the lumped capacitance method is considered valid for the given problem.
The characteristic length, denoted as L_{c}, plays a crucial role in determining the validity of the method. It is defined as the ratio of the solid's volume to its surface area, and it facilitates the calculation of L_{c} for solids of different shapes. Additionally, the Fourier number, another dimensionless parameter, is used to characterize transient conduction problems in conjunction with the Biot number.
Overall, the lumped capacitance method provides a simple and convenient way to analyze transient heating and cooling problems under specific conditions, and it is often used as a first approximation for such scenarios
 Newton's law of cooling

Newton's law of cooling is a fundamental principle in heat transfer. It states that the rate of heat transfer between a surface and a fluid is proportional to the temperature difference between the surface and the fluid. Mathematically, it can be expressed as:
$$q'' = h \cdot \left( T_s  T_{\infty} \right)$$
Where:
 q'' is the heat flux [W/m^{2}]
 h is the convection heat transfer coefficient [W/m^{2}⋅K]
 T_{s} is the surface temperature [K]
 T_{∞} is the fluid temperature away from the surface [K]
This law is essential for understanding the heat transfer process between a surface and a fluid, and it provides a basis for analyzing convective heat transfer in various engineering applications.
 Phase change

A phase change refers to the transition of a substance from one state of matter to another, such as from solid to liquid, liquid to gas, or vice versa. During a phase change, the energy associated with the substance can be divided into sensible energy, which is mainly related to changes in temperature, and latent energy (or latent heat), which is associated with the change in phase. For example, when a material changes from solid to liquid (melting) or from liquid to vapor (vaporization, evaporation, boiling), the latent energy increases. Conversely, when the phase change is from vapor to liquid (condensation) or from liquid to solid (solidification, freezing), the latent energy decreases.
 Pool boiling

Pool boiling is a process that occurs when a solid surface's temperature exceeds the saturation temperature corresponding to the liquid pressure. Heat is transferred from the solid surface to the liquid, leading to the formation of vapor bubbles that subsequently detach from the surface. Pool boiling can occur under various conditions, such as when the liquid is quiescent and its motion near the surface is due to free convection and mixing induced by bubble growth and detachment. It can also be classified as subcooled or saturated, depending on the temperature of the liquid. In subcooled boiling, most of the liquid's temperature is below the saturation temperature, while in saturated boiling, the liquid's temperature slightly exceeds the saturation temperature.
 Prandtl number

The Prandtl number, denoted as Pr, is a dimensionless number that represents the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity in a fluid. It provides a measure of the relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers, respectively. For laminar boundary layers, the value of Prandtl number strongly influences the relative growth of the velocity boundary layer and thermal boundary layer. Specifically, the Prandtl number affects the relative growth of the velocity and thermal boundary layers, with higher Prandtl numbers leading to greater thermal boundary layer growth compared to the velocity boundary layer.
 Radiation

See thermal radiation.
 Radiation intensity

Radiation intensity refers to the directional distribution of radiation leaving a surface or the radiation incident upon a surface from different directions. It is mathematically defined using the spherical coordinate system, where the differential solid angle is used to measure the directional distribution of radiation. The intensity of radiation emitted or incident upon a surface is typically expressed in terms of the rate at which radiant energy is emitted or incident at a specific wavelength, direction, and per unit area of the emitting or receiving surface. This concept is crucial for understanding the directional effects of radiation and for determining the net radiative heat transfer rate.
 Reflection

Reflection is the process by which incident radiation is redirected back from the surface of a material. In the context of radiative heat transfer, reflection plays a significant role in determining the amount of thermal energy absorbed by a material. The reflectivity of a surface, which is the ratio of reflected radiation to incident radiation, affects the amount of energy that is absorbed by the material. A surface with high reflectivity will reflect a larger portion of the incident radiation, leading to lower absorption of thermal energy. Conversely, a surface with low reflectivity will absorb more of the incident radiation, thus increasing the thermal energy of the material.
 Reynolds number

The Reynolds number, denoted as Re, is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It represents the ratio of the inertial forces to the viscous forces within a fluid. The Reynolds number is calculated using the formula:
Re = (ρ * u * L) / μ
Where:
 ρ is the density of the fluid [kg/m^{3}]
 u is the velocity of the fluid [m/s]
 L is a characteristic length (such as the diameter of a pipe or the distance from the leading edge for flow over a flat plate) [m]
 μ is the dynamic viscosity of the fluid [Pa.s].
When the Reynolds number is small, the viscous forces dominate, and the flow remains laminar. Conversely, when the Reynolds number is large, inertial forces dominate, leading to turbulent flow. The critical value at which the transition from laminar to turbulent flow occurs varies depending on the specific flow situation.
 Saturated boiling

Saturated boiling occurs when the temperature of the liquid slightly exceeds the saturation temperature. Bubbles formed at the surface are then propelled through the liquid by buoyancy forces, eventually escaping from a free surface. This type of boiling is characterized by the presence of bubbles and the movement of these bubbles through the liquid due to buoyancy forces.
 Solar radiation

Solar radiation refers to the electromagnetic radiation emitted by the sun. It is essential for sustaining life on Earth, as it is the primary source of energy for processes such as photosynthesis, which provides food, fiber, and fuel for humans. Solar radiation also has the potential to meet significant demands for heat and electricity through thermal and photovoltaic processes. The solar radiation flux at the outer surface of the Earth's atmosphere has been measured to be 1368 W/m^{2} and varies based on factors such as geographic latitude, time of day, and year. This radiation carries energy and plays a crucial role in determining the temperature of the Earth's surface and atmosphere.
 Subcooled boiling

Subcooled boiling occurs when the temperature of most of the liquid is below the saturation temperature. In this mode of boiling, bubbles formed at the surface may condense in the liquid. The heat flux in subcooled boiling increases typically as $\left( T_s  T_l\right)^n$ or $\left( \Delta T_e + \Delta T_{sub} \right)^n$, where 5/4 ≤ n ≤ 4/3, depending on the geometry of the hot surface. The influence of subcooling is considered to be negligible in the nucleate boiling regime, where the maximum heat flux occurs. The process of subcooled boiling involves free convection and mixing induced by bubble growth and detachment, and it has implications for the heat transfer coefficient.
 Thermal boundary layer

A thermal boundary layer develops when there is a temperature difference between the fluid free stream and the surface over which the fluid flows. This boundary layer is fundamental to problems involving heat transfer and convection transport. It is of significance to engineers due to its relation to surface shear stress, surface frictional effects, and heat transfer rates. The thickness of the thermal boundary layer is influenced by various factors such as the Prandtl number and the distance from the leading edge of the surface.
 Thermal radiation

Thermal radiation refers to the electromagnetic energy emitted by matter as a result of its temperature. It is concentrated in the spectral region from approximately 0.1 to 100 μm. The emission of thermal radiation is associated with the oscillations or transitions of the many electrons that constitute matter, which are sustained by the internal energy and temperature of the matter. All forms of matter emit thermal radiation, and it is a key factor in the transfer of heat energy between surfaces and their surroundings. This process is crucial in understanding the cooling of hot solids, energy balance in surfaces, and the environmental radiation balance, including the impact of solar radiation on the Earth's atmosphere.
 Transmission

Transmission refers to the process of transferring thermal energy due to a spatial temperature difference. Whenever there is a temperature variation within a medium or between different media, heat transfer occurs. This transfer of energy can take place through conduction, convection, or radiation, each with its own physical origins and rate equations.
 Turbulent flow

Turbulent flow is a chaotic and disordered fluid motion characterized by irregular fluctuations in velocity and pressure. It is distinguished by the presence of turbulent eddies and vortices, leading to a mixing of fluid particles. In turbulent flow, the transport of momentum, energy, and species is enhanced by the turbulent mixing, resulting in increased transfer rates. The transition from laminar to turbulent flow is influenced by triggering mechanisms such as natural flow structures or disturbances within the fluid. Within a turbulent boundary laeyer number, three regions can be delineated based on the distance from the surface: a viscous sublayer, a buffer layer and a turbulent zone. These regions exhibit different characteristics in terms of transport mechanisms and velocity profiles.
 Vaporization

Vaporization is the conversion of a substance from a liquid or solid state into a vapor or gas phase. This can occur through boiling, which is a rapid vaporization that happens at the boiling point of the liquid, or through evaporation, which is a slower process that occurs at the surface of a liquid below its boiling point.
 Velocity boundary layer

A velocity boundary layer develops when there is fluid flow over a surface. This boundary layer is characterized by a thin layer of fluid in which velocity gradients and shear stresses are large, while outside this layer, velocity gradients and shear stresses are negligible. The boundary layer thickness, denoted as δ, is the distance from the surface at which the fluid velocity reaches 0.99 times the free stream velocity. As the distance from the leading edge increases, the effects of viscosity penetrate farther into the free stream, causing the boundary layer to grow. The velocity boundary layer is of fundamental importance to problems involving convection transport and is related to surface shear stress and frictional effects.
 View factor

View factors, also known as configuration or shape factors, are a concept used to quantify the exchange of radiation between surfaces. They represent the fraction of radiation leaving one surface that is intercepted by another surface. View factors are defined by the geometry of the surfaces and are used to calculate the heat transfer by radiation between surfaces. They are important in the context of thermal radiation and play a crucial role in determining the energy exchange in various engineering and physical systems.