Thermal Engineering

Direct contact condensation

Similarly as in the previous chapter, in this chapter we will discuss heat transfer with phase change, but in this case we will discuss condensation of gas phase (vapor-to-liquid phase change).

In general, condensation is the change of the physical state of matter from gas phase into liquid phase, and is the reverse of vaporisation. Flow processes associated with condensation on a solid surface are almost a mirror image of those involved in boiling. Condensation occurs when the temperature of a vapor is reduced below its saturation temperature or when the pressure of a vapor is increased above its saturation parameters (see phase diagram of water).

Direct contact condensation, DCC, occurs when vapor is brought into contact with a cold liquid. As in jet condensers, the cooling water is sprayed on the exhaust steam and there is direct contact between the exhaust steam and cooling water. The process of condensation is very fast and efficient, but here cooling water and condensed steam are mixed up. The advantages of direct contact condensation over the conventional processes using metallic transfer surfaces are due to the relative simplicity of design, less corrosion and scaling problems, lower maintenance costs, higher specific transfer areas, and higher transfer rates. Despite these advantages jet condensers are not usual in thermal power plants especially due to the loss of condensate.

Condensation in Pressurizer

A pressurizer is a component of a pressurized water reactor. Pressure in the primary circuit of PWRs is maintained by a pressurizer, a separate vessel that is connected to the primary circuit (hot leg) and partially filled with water which is heated to the saturation temperature (boiling point) for the desired pressure by submerged electrical heaters. Temperature in the pressurizer can be maintained at 350 °C (662 °F), which gives a subcooling margin (the difference between the pressurizer temperature and the highest temperature in the reactor core) of 30 °C. Subcooling margin is very important safety parameter of PWRs, since the boiling in the reactor core must be excluded. The basic design of the pressurized water reactor includes such requirement that the coolant (water) in the reactor coolant system must not boil. To achieve this, the coolant in the reactor coolant system is maintained at a pressure sufficiently high that boiling does not occur at the coolant temperatures experienced while the plant is operating or in an analyzed transient.

Functions

Pressure in the pressurizer is controlled by varying the temperature of the coolant in the pressurizer. For these purposes two systems are installed. Water spray system and electrical heaters system. Volume of the pressurizer (tens of cubic meters) is filled with water on saturation parameters and steam. The water spray system (relatively cool water – from cold leg) can decrease the pressure in the vessel by condensing the steam on water droplets sprayed in the vessel. Since vapor is brought into contact with a cold liquid, in this case, we are talking about direct contact condensation. On the other hand the submerged electrical heaters are designed to increase the pressure by evaporation the water in the vessel. Water pressure in a closed system tracks water temperature directly; as the temperature goes up, pressure goes up.

See also:

Boiling and Condensation

Radiation Heat Transfer

In preceding chapters, we have discussed convection and conduction, which require the presence of matter as a medium to carry the heat from the hotter to the colder region. But a third type of heat transfer, radiation heat transfer, occurs without any medium at all. In general, the radiation heat transfer from one surface to another is the radiation leaving the first surface for the other minus that arriving from the second surface. Radiation heat transfer is mediated by electromagnetic radiation, known as thermal radiation, that arises due to the temperature of a body. Any material that has a temperature above absolute zero gives off some radiant energy. Most energy of this type is in the infra-red region of the electromagnetic spectrum although some of it is in the visible region. One of most important examples of radiation heat transfer is the Earth’s absorption of solar radiation, followed by its outgoing thermal radiation. These processes determine the temperature and climate of the Earth.

Thermal Radiation

Thermal radiation is electromagnetic radiation in the infra-red region of the electromagnetic spectrum although some of it is in the visible region. The term thermal radiation is frequently used to distinguish this form of electromagnetic radiation from other forms, such as radio waves, x-rays, or gamma rays. It is generated by the thermal motion of charged particles in matter and therefore any material that has a temperature above absolute zero gives off some radiant energy. Thermal radiation does not require any medium for energy transfer. In fact, energy transfer by radiation is fastest (at the speed of light) and it suffers no attenuation in a vacuum.

In contrast to heat transfer by conduction or convection, which take place in the direction of decreasing temperature, thermal radiation heat transfer can occur between two bodies separated by a medium colder than both bodies. For example, solar radiation reaches the surface of the earth after passing through cold layers of atmosphere at high altitudes.

Stefan–Boltzmann Law

Radiation heat transfer rate, q [W/m²], from a body (e.g. a black body) to its surroundings is proportional to the fourth power of the absolute temperature and can be expressed by the following equation:

q =  εσT⁴

where σ is a fundamental physical constant called the Stefan–Boltzmann constant, which is equal to 5.6697×10^-8 W/m²K⁴. The Stefan–Boltzmann constant is named after Josef Stefan (who discovered the Stefan-Boltzman law experimentally in 1879) and Ludwig Boltzmann (who derived it theoretically soon after). As can be seen, radiation heat transfer is important at very high temperatures and in a vacuum.

Emissivity

The emissivity, ε, of the surface of a material is its effectiveness in emitting energy as thermal radiation and varies between 0.0 and 1.0.

By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies. To take into account the fact that real objects are gray bodies, the Stefan-Boltzmann law must include emissivity. Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Absorptivity

Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, value of absorptivity is in the range 0 < α < 1.

From its definition, a blackbody, which is an idealized physical body, absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect absorber. Since for real objects the absorptivity is less than unity, a real object can not absorb all incident light. The incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

Q = εσA_1-2(T⁴₁ −T⁴₂)  [J/s]

q =  εσ(T⁴₁ −T⁴₂) [J/m²s]

The area factor A_1-2, is the area viewed by body 2 of body 1, and can become fairly difficult to calculate.

Blackbody Radiation

It is known that the amount of radiation energy emitted from a surface at a given wavelength depends on the material of the body and the condition of its surface as well as the surface temperature. Therefore, various materials emit different amounts of radiant energy even whhen they are at the same temperature. A body that emits the maximum amount of heat for its absolute temperature is called a blackbody.

A blackbody is an idealized physical body, that has specific properties. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Since the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, a blackbody is also a perfect absorber of electromagnetic radiation.

Kirchhoff’s Law of thermal radiation:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

A blackbody absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Its absorptivity is therefore equal to unity, which is also the highest possible value. That is, a blackbody is a perfect absorber (and a perfect emitter).

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

See also: Ultraviolet Catastrophe

Blackbody Emissive Power

The blackbody emissive power, E_b [W/m²], from a blackbody to its surroundings is proportional to the fourth power of the absolute temperature and can be expressed by the following equation:

E_b =  σT⁴

where σ is a fundamental physical constant called the Stefan–Boltzmann constant, which is equal to 5.6697×10^-8 W/m²K⁴ and T is the absolute temperature of the surface in K.

The term blackbody was introduced by a German physicist Gustav Kirchhoff in 1860. Blackbody radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation. Three following laws are associated with blackbody radiation:

• Kirchhoff’s law. This law gives the relationship between the emissivity and absorptivity of an object.
• Planck’s law. This law describes the spectrum of blackbody radiation, which depends only on the object’s temperature.
• Wien’s displacement law. This law determines the most likely frequency of the emitted radiation.
• Stefan–Boltzmann law. This law gives the radiant intensity.

All bodies above absolute zero temperature radiate some heat. The sun and earth both radiate heat toward each other. This seems to violate the Second Law of Thermodynamics, which states that heat cannot spontaneously flow from cold system to hot system without external work being performed on the system. The paradox is resolved by the fact that each body must be in direct line of sight of the other to receive radiation from it. Therefore, whenever the cool body is radiating heat to the hot body, the hot body must also be radiating heat to the cool body. Moreover, the hot body will radiate more energy than cold body. The case of different emissivities is solved by the Kirchhoff’s Law of thermal radiation, which states that object with low emissivity have also low absorptivity. As a result, heat cannot spontaneously flow from cold system to hot system and the second law is still satisfied.

See also:

Heat Transfer

Thermal Radiation – Radiant Heat

Thermal radiation is electromagnetic radiation in the infra-red region of the electromagnetic spectrum although some of it is in the visible region. The term thermal radiation is frequently used to distinguish this form of electromagnetic radiation from other forms, such as radio waves, x-rays, or gamma rays. It is generated by the thermal motion of charged particles in matter and therefore any material that has a temperature above absolute zero gives off some radiant energy. Thermal radiation does not require any medium for energy transfer. In fact, energy transfer by radiation is fastest (at the speed of light) and it suffers no attenuation in a vacuum.

In contrast to heat transfer by conduction or convection, which take place in the direction of decreasing temperature, thermal radiation heat transfer can occur between two bodies separated by a medium colder than both bodies. For example, solar radiation reaches the surface of the earth after passing through cold layers of atmosphere at high altitudes.

Stefan–Boltzmann Law

Radiation heat transfer rate, q [W/m²], from a body (e.g. a black body) to its surroundings is proportional to the fourth power of the absolute temperature and can be expressed by the following equation:

q =  εσT⁴

where σ is a fundamental physical constant called the Stefan–Boltzmann constant, which is equal to 5.6697×10^-8 W/m²K⁴. The Stefan–Boltzmann constant is named after Josef Stefan (who discovered the Stefan-Boltzman law experimentally in 1879) and Ludwig Boltzmann (who derived it theoretically soon after). As can be seen, radiation heat transfer is important at very high temperatures and in a vacuum.

Emissivity

The emissivity, ε, of the surface of a material is its effectiveness in emitting energy as thermal radiation and varies between 0.0 and 1.0.

By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies. To take into account the fact that real objects are gray bodies, the Stefan-Boltzmann law must include emissivity. Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Absorptivity

Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, value of absorptivity is in the range 0 < α < 1.

From its definition, a blackbody, which is an idealized physical body, absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect absorber. Since for real objects the absorptivity is less than unity, a real object can not absorb all incident light. The incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

Q = εσA_1-2(T⁴₁ −T⁴₂)  [J/s]

q =  εσ(T⁴₁ −T⁴₂) [J/m²s]

The area factor A_1-2, is the area viewed by body 2 of body 1, and can become fairly difficult to calculate.

Blackbody Radiation

It is known that the amount of radiation energy emitted from a surface at a given wavelength depends on the material of the body and the condition of its surface as well as the surface temperature. Therefore, various materials emit different amounts of radiant energy even whhen they are at the same temperature. A body that emits the maximum amount of heat for its absolute temperature is called a blackbody.

A blackbody is an idealized physical body, that has specific properties. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Since the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, a blackbody is also a perfect absorber of electromagnetic radiation.

Kirchhoff’s Law of thermal radiation:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

A blackbody absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Its absorptivity is therefore equal to unity, which is also the highest possible value. That is, a blackbody is a perfect absorber (and a perfect emitter).

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

See also: Ultraviolet Catastrophe

Spectrum – Thermal Radiation

The Stefan–Boltzmann law determines the total blackbody emissive power, E_b, which is the sum of the radiation emitted over all wavelengths. Planck’s law describes the spectrum of blackbody radiation, which depends only on the object’s temperature and relates the spectral blackbody emissive power, E_bλ. This law is named after a German theoretical physicist Max Planck, who proposed it in 1900. Planck’s law is a pioneering result of modern physics and quantum theory. Planck’s hypothesis that energy is radiated and absorbed in discrete “quanta” (or energy packets) precisely matched the observed patterns of blackbody radiation and resolved the ultraviolet catastrophe.

Using this hypothesis, Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by:

where

• B_ν(v,T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T
• h is the Planck constant
• c is the speed of light in a vacuum
• k_B is the Boltzmann constant
• ν is the frequency of the electromagnetic radiation
• T is the absolute temperature of the body

The Planck’s law has the following important features:

• The emitted radiation varies continuously with wavelength.
• At any wavelength the magnitude of the emitted radiation increases with increasing temperature.
• The spectral region in which the radiation is concentrated depends on temperature, with comparatively more radiation appearing at shorter wavelengths as the temperature increases (Wien’s Displacement Law).

See also:

Radiation

Conduction – Convection – Radiation

Heat transfer is an engineering discipline that concerns the generation, use, conversion, and exchange of heat (thermal energy) between physical systems.  In power engineering it determines key parameters and materials of heat exchangers. Heat transfer is usually classified into various mechanisms, such as:

• Heat Conduction. Heat conduction, also called diffusion, occurs within a body or between two bodies in contact. It is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings
• Heat Convection. Heat convection depends on motion of mass from one region of space to another. Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid.
• Thermal Radiation. Radiation is heat transfer by electromagnetic radiation, such as sunshine, with no need for matter to be present in the space between bodies.

Conduction

Thermal conduction, also called heat conduction, occurs within a body or between two bodies in contactwithout the involvement of mass flow and mixing. It is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. Heat transfer by conductionis dependent upon the driving “force” of temperature difference and the thermal conductivity (or the resistance to heat transfer). The thermal conductivity is dependent upon the nature and dimensions of the heat transfer medium. All heat transfer problems involve the temperature difference, the geometry, and the physical propertiesof the object being studied. In conduction heat transfer problems, the object being studied is usually a solid.

Microscopically this mode of energy transfer is attributed to free electron flow from higher to lower energy levels, lattice vibration and molecular collision. Consider a block of stone at high temperature, that consists of atoms that are oscillating intensely around their average positions. At low temperatures, the atoms continue to oscillate, but with less intensity. If a hotter block of stone is put in contact with a cooler block, the intensely oscillating atoms at the edge of the hotter block gives off its kinetic energy to the less oscillating atoms at the edge of the cool block. In this case there is energy transfer between these two blocks and heat flows from the hotter to the cooler block by this random vibrations. The modern view is to ascribe the energy transfer to lattice waves induced by atomic motion. In an electrical insulators, the energy transfer is exclusively via these lattice waves. In a conductor, it is also due to the translational motion of the free electrons.

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Convection

In general, convection is either the mass transfer or the heat transfer due to bulk movement of molecules within fluids such as gases and liquids. Although liquids and gases are generally not very good conductors of heat, they can transfer heat quite rapidly by convection.

Convection takes place through advection, diffusion or both. Convection cannot take place in most solids because neither significant diffusion of matter nor bulk current flows can take place. Diffusion of heat takes place in rigid solids, but that is called thermal conduction.

The process of heat transfer between a surface and a fluid flowing in contact with it is called convective heat transfer. In engineering, convective heat transfer is one of the major mechanisms of heat transfer. When heat is to be transferred from one fluid to another through a barrier, convection is involved on both sides of the barrier. In most cases the main resistance to heat flow is by convection. Convective heat transfer take place both by thermal diffusion (the random motion of fluid molecules) and by advection, in which matter or heat is transported by the larger-scale motion of currents in the fluid.

Radiation Heat Transfer

In preceding chapters, we have discussed convection and conduction, which require the presence of matter as a medium to carry the heat from the hotter to the colder region. But a third type of heat transfer, radiation heat transfer, occurs without any medium at all. In general, the radiation heat transfer from one surface to another is the radiation leaving the first surface for the other minus that arriving from the second surface. Radiation heat transfer is mediated by electromagnetic radiation, known as thermal radiation, that arises due to the temperature of a body. Any material that has a temperature above absolute zero gives off some radiant energy. Most energy of this type is in the infra-red region of the electromagnetic spectrum although some of it is in the visible region. One of most important examples of radiation heat transfer is the Earth’s absorption of solar radiation, followed by its outgoing thermal radiation. These processes determine the temperature and climate of the Earth.

See also:

Heat Transfer

Stefan–Boltzmann Law

Radiation heat transfer rate, q [W/m²], from a body (e.g. a black body) to its surroundings is proportional to the fourth power of the absolute temperature and can be expressed by the following equation:

q =  εσT⁴

where σ is a fundamental physical constant called the Stefan–Boltzmann constant, which is equal to 5.6697×10^-8 W/m²K⁴. The Stefan–Boltzmann constant is named after Josef Stefan (who discovered the Stefa-Boltzman law experimentally in 1879) and Ludwig Boltzmann (who derived it theoretically soon after). As can be seen, radiation heat transfer is important at very high temperatures and in a vacuum.

Emissivity

The emissivity, ε, of the surface of a material is its effectiveness in emitting energy as thermal radiation and varies between 0.0 and 1.0.

By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies. To take into account the fact that real objects are gray bodies, the Stefan-Boltzmann law must include emissivity. Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Absorptivity

Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, value of absorptivity is in the range 0 < α < 1.

From its definition, a blackbody, which is an idealized physical body, absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect absorber. Since for real objects the absorptivity is less than unity, a real object can not absorb all incident light. The incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

Q = εσA_1-2(T⁴₁ −T⁴₂)  [J/s]

q =  εσ(T⁴₁ −T⁴₂) [J/m²s]

The area factor A_1-2, is the area viewed by body 2 of body 1, and can become fairly difficult to calculate.

Blackbody Radiation

It is known that the amount of radiation energy emitted from a surface at a given wavelength depends on the material of the body and the condition of its surface as well as the surface temperature. Therefore, various materials emit different amounts of radiant energy even whhen they are at the same temperature. A body that emits the maximum amount of heat for its absolute temperature is called a blackbody.

A blackbody is an idealized physical body, that has specific properties. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Since the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, a blackbody is also a perfect absorber of electromagnetic radiation.

Kirchhoff’s Law of thermal radiation:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

A blackbody absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Its absorptivity is therefore equal to unity, which is also the highest possible value. That is, a blackbody is a perfect absorber (and a perfect emitter).

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

See also:

Radiation

Absorptivity – Absorptance

Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, value of absorptivity is in the range 0 < α < 1.

From its definition, a blackbody, which is an idealized physical body, absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect absorber. Since for real objects the absorptivity is less than unity, a real object can not absorb all incident light. The incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

See also:

Radiation

Emissivity

The emissivity, ε, of the surface of a material is its effectiveness in emitting energy as thermal radiation and varies between 0.0 and 1.0.

By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies. To take into account the fact that real objects are gray bodies, the Stefan-Boltzmann law must include emissivity. Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, value of absorptivity is in the range 0 < α < 1.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

See also:

Radiation

Blackbody Radiation

It is known that the amount of radiation energy emitted from a surface at a given wavelength depends on the material of the body and the condition of its surface as well as the surface temperature. Therefore, various materials emit different amounts of radiant energy even whhen they are at the same temperature. A body that emits the maximum amount of heat for its absolute temperature is called a blackbody.

A blackbody is an idealized physical body, that has specific properties. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m²). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Since the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, a blackbody is also a perfect absorber of electromagnetic radiation.

Kirchhoff’s Law of thermal radiation:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

A blackbody absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Its absorptivity is therefore equal to unity, which is also the highest possible value. That is, a blackbody is a perfect absorber (and a perfect emitter).

Note that visible radiation occupies a very narrow band of the spectrum from 400 to 760 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

Blackbody Emissive Power

The blackbody emissive power, E_b [W/m²], from a blackbody to its surroundings is proportional to the fourth power of the absolute temperature and can be expressed by the following equation:

E_b =  σT⁴

where σ is a fundamental physical constant called the Stefan–Boltzmann constant, which is equal to 5.6697×10^-8 W/m²K⁴ and T is the absolute temperature of the surface in K.

The term blackbody was introduced by a German physicist Gustav Kirchhoff in 1860. Blackbody radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation. Three following laws are associated with blackbody radiation:

• Kirchhoff’s law. This law gives the relationship between the emissivity and absorptivity of an object.
• Planck’s law. This law describes the spectrum of blackbody radiation, which depends only on the object’s temperature.
• Wien’s displacement law. This law determines the most likely frequency of the emitted radiation.
• Stefan–Boltzmann law. This law gives the radiant intensity.

All bodies above absolute zero temperature radiate some heat. The sun and earth both radiate heat toward each other. This seems to violate the Second Law of Thermodynamics, which states that heat cannot spontaneously flow from cold system to hot system without external work being performed on the system. The paradox is resolved by the fact that each body must be in direct line of sight of the other to receive radiation from it. Therefore, whenever the cool body is radiating heat to the hot body, the hot body must also be radiating heat to the cool body. Moreover, the hot body will radiate more energy than cold body. The case of different emissivities is solved by the Kirchhoff’s Law of thermal radiation, which states that object with low emissivity have also low absorptivity. As a result, heat cannot spontaneously flow from cold system to hot system and the second law is still satisfied.

Spectrum – Blackbody Radiation

The Stefan–Boltzmann law determines the total blackbody emissive power, E_b, which is the sum of the radiation emitted over all wavelengths. Planck’s law describes the spectrum of blackbody radiation, which depends only on the object’s temperature and relates the spectral blackbody emissive power, E_bλ. This law is named after a German theoretical physicist Max Planck, who proposed it in 1900. Planck’s law is a pioneering result of modern physics and quantum theory. Planck’s hypothesis that energy is radiated and absorbed in discrete “quanta” (or energy packets) precisely matched the observed patterns of blackbody radiation and resolved the ultraviolet catastrophe.

Using this hypothesis, Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by:

where

• B_ν(v,T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T
• h is the Planck constant
• c is the speed of light in a vacuum
• k_B is the Boltzmann constant
• ν is the frequency of the electromagnetic radiation
• T is the absolute temperature of the body

The Planck’s law has the following important features:

• The emitted radiation varies continuously with wavelength.
• At any wavelength the magnitude of the emitted radiation increases with increasing temperature.
• The spectral region in which the radiation is concentrated depends on temperature, with comparatively more radiation appearing at shorter wavelengths as the temperature increases (Wien’s Displacement Law).

See also:

Radiation

Flow Boiling – Forced Convection Boiling

In flow boiling (or forced convection boiling), fluid flow is forced over a surface by external means such as a pump, as well as by buoyancy effects. Therefore, flow boiling is always accompanied by other convection effects. Conditions depend strongly on geometry, which may involve external flow over heated plates and cylinders or internal (duct) flow. In nuclear reactors, most of boiling regimes are just forced convection boiling. The flow boiling is also classified as either external and internal flow boiling depending on whether the fluid is forced to flow over a heated surface or inside a heated channel.

Internal flow boiling is much more complicated in nature than external flow boiling because there is no free surface for the vapor to escape, and thus both the liquid and the vapor are forced to flow together. The two-phase flow in a tube exhibits different flow boiling regimes, depending on the relative amounts of the liquid and the vapor phases. Therefore internal forced convection boiling is commonly referred to as two-phase flow.

• In the case of steam and liquid water the density of the two phases differs by a factor of about 1000. Therefore the influence of gravitational body force on multiphase flows is of much greater importance than in the case of single-phase flows.
• The sound speed changes dramatically for materials undergoing phase change, and can be orders of magnitude different. This significantly influences a flow through an orifice.
• The relative concentration of different phases is usually a dependent parameter of great importance in multiphase flows, while it is a parameter of no consequence in single-phase flows.
• The change of phase means flow-induced pressure drops can cause further phase-change (e.g. water can evaporate through an orifice) increasing the relative volume of the gaseous, compressible medium and increasing efflux velocities, unlike single-phase incompressible flow where decreasing of an orifice would decrease efflux velocities.
• The spatial distribution of the various phases in the flow channel strongly affects the flow behavior.
• There are many types of instabilities in multiphase flow.

Flow Boiling – Vertical Channel

In this chapter, we will study flow boiling in a vertical channel of a boiling water reactor. The regimes of boiling and the heat flux curve are similar to the ones in pool boiling. The process occurs also in modern high pressure forced circulation boilers.

In BWRs there is a phenomenon, that is of the highest importance in reactor safety. This phenomenon is known as the “dryout” and it is directly associated with changes in flow pattern during evaporation. At normal the fuel surface is effectively cooled by boiling coolant. However when the heat flux exceeds a critical value (CHF – critical heat flux) the flow pattern may reach the dryout conditions (thin film of liquid disappears). The heat transfer from the fuel surface into the coolant is deteriorated, with the result of a drastically increased fuel surface temperature. In the high-quality region, the crisis occurs at a lower heat flux. Since the flow velocity in the vapor core is high, post-CHF heat transfer is much better than for low-quality critical flux (i.e. for PWRs temperature rises are higher and more rapid).

Typical flow boiling modes in a vertical channel are depicted in the figure. This figure shows the typical order of the flow regimes that are encountered from inlet to outlet of a heated channel. At the inlet, the liquid enters subcooled (at the lower temperature than saturation). In this region the flow is single-phase. As the liquid heats up, the wall temperature correspondingly rises. As the wall temperature exceeds the saturation temperature (e.g. 285°C at 6.8 MPa), subcooled nucleate boiling begins. Bubbles nucleate in the superheated thermal boundary layer on the heated wall but tend to condense in the subcooled bulk.

Further increase in liquid temperature causes, that the liquid bulk reaches its saturation temperature and the convective boiling process passes through the bubbly flow into the slug flow. Increasing void fraction causes that the structure of the flow becomes unstable. The boiling process passes through the slug and churn flow into the annular flow regime with its characteristic annular film of the liquid. At given combinations of flow rate through a channel, pressure, flow quality, and linear heat rate, the wall liquid film may exhaust and the wall may be dried out. At the dryout point the wall temperature significantly rises in order to dissipate the applied heat flux. The post-dryout flow (mist or drop flow) in the heated channel is undesirable, because the presence of such flow regime is accompanied with significantly higher wall temperatures and high fluctuation of wall temperatures. Correlations used to determine heat transfer coefficients in two phase flow are described below.

Special Reference: Tong, L. S., Weisman, Joel. Thermal Analysis of Pressurized Water Reactors. Amer Nuclear Society, 3rd edition, 5/1996. ISBN-13: 978-0894480386.

Single-phase Forced Convection – Heat Transfer Correlation

For fully developed (hydrodynamically and thermally) turbulent flow in a smooth circular tube, the local Nusselt number may be obtained from the well-known Dittus-Boelter equation. The Dittus–Boelter equation is easy to solve but is less accurate when there is a large temperature difference across the fluid and is less accurate for rough tubes (many commercial applications), since it is tailored to smooth tubes.

The Dittus-Boelter correlation may be used for small to moderate temperature differences, T_wall – T_avg, with all properties evaluated at an averaged temperature T_avg.

For flows characterized by large property variations, the corrections (e.g. a viscosity correction factor μ/μ_wall) must be taken into account, for example, as Sieder and Tate recommend.

Nucleate Boiling Correlations – Flow Boiling

McAdams Correlation

In fully developed nucleate boiling with saturated coolant, the wall temperature is determined by local heat flux and pressure and is only slightly dependent on the Reynolds number. For subcooled water at absolute pressures between 0.1 – 0.6 MPa, McAdams correlation gives:

Thom Correlation

The Thom correlation is for the flow boiling (subcooled or saturated at pressures up to about 20 MPa) under conditions where the nucleate boiling contribution predominates over forced convection. This correlation is useful for rough estimation of expected temperature difference given the heat flux:

Chen’s Correlation

In 1963, Chen proposed the first flow boiling correlation for evaporation in vertical tubes to attain widespread use. Chen’s correlation includes both the heat transfer coefficients due to nucleate boiling as well as forced convective mechanisms. It must be noted, at higher vapor fractions, the heat transfer coefficient varies strongly with flow rate. The flow velocity in a core can be very high causing very high turbulences. This heat transfer mechanism has been referred to as “forced convection evaporation”. No adequate criteria has been established to determine the transition from nucleate boiling to forced convection vaporization. However, a single correlation that is valid for both nucleate boiling and forced convection vaporization has been developed by Chen for saturated boiling conditions and extended to include subcooled boiling by others. Chen proposed a correlation where the heat transfer coefficient is the sum of a forced convection component and a nucleate boiling component. It must be noted, the nucleate pool boiling correlation of Forster and Zuber (1955) is used to calculate the nucleate boiling heat transfer coefficient, h_FZ and the turbulent flow correlation of Dittus-Boelter (1930) is used to calculate the liquid-phase convective heat transfer coefficient, h_l.

The nucleate boiling suppression factor, S, is the ratio of the effective superheat to wall superheat. It accounts for decreased boiling heat transfer because the effective superheat across the boundary layer is less than the superheat based on wall temperature. The two-phase multiplier, F, is a function of the Martinelli parameter χ_tt.

Boiling Crisis – Critical Heat Flux

As was written, in nuclear reactors, limitations of the local heat flux is of the highest importance for reactor safety. For pressurized water reactors and also for boiling water reactors, there are thermal-hydraulic phenomena, which cause a sudden decrease in the efficiency of heat transfer (more precisely in the heat transfer coefficient). These phenomena occur at certain value of heat flux, known as the “critical heat flux”. The phenomena, that cause the deterioration of heat transfer are different for PWRs and for BWRs.

In both types of reactors, the problem is more or less associated with departure from nucleate boiling. The nucleate boiling heat flux cannot be increased indefinitely. At some value, we call it the “critical heat flux” (CHF), the steam produced can form an insulating layer over the surface, which in turn deteriorates the heat transfer coefficient. Immediately after the critical heat flux has been reached, boiling become unstable and film boiling occurs. The transition from nucleate boiling to film boiling is known as the “boiling crisis”. As was written, the phenomena, that cause the deterioration of heat transfer are different for PWRs and for BWRs.

Departure From Nucleate Boiling – DNB

In case of PWRs, the critical safety issue is named DNB (departure from nucleate boiling), which causes the formation of a local vapor layer, causing a dramatic reduction in heat transfer capability. This phenomenon occurs in the subcooled or low-quality region. The behaviour of the boiling crisis depends on many flow conditions (pressure, temperature, flow rate), but the boiling crisis occurs at a relatively high heat fluxes and appears to be associated with the cloud of bubbles, adjacent to the surface. These bubbles or film of vapor reduces the amount of incoming water. Since this phenomenon deteriorates the heat transfer coefficient and the heat flux remains, heat then accumulates in the fuel rod causing dramatic rise of cladding and fuel temperature. Simply, a very high temperature difference is required to transfer the critical heat flux being produced from the surface of the fuel rod to the reactor coolant (through vapor layer).

In case of PWRs, the critical flow is inverted annular flow, while in BWRs, the critical flow is usually annular flow. The difference in flow regime between post-dryout flow and post-DNB flow is depicted in the figure. In PWRs at normal operation the flow is considered to be single-phase. But a great deal of study has been performed on the nature of two-phase flow in case of transients and accidents (such as the loss-of-coolant accident – LOCA or trip of RCPs), which are of importance in reactor safety and in must be proved and declared in the Safety Analysis Report (SAR).

In pressurized water reactors, one of key safety requirements is that a departure from nucleate boiling (DNB) will not occur during steady state operation, normal operational transients, and anticipated operational occurrences (AOOs).  Fuel cladding integrity will be maintained if the minimum DNBR remains above the 95/95 DNBR limit for PWRs ( a 95% probability at a 95% confidence level). DNB criterion is one of acceptance criteria in safety analyses as well as it constitutes one of safety limits in technical specifications.

Critical Heat Flux for DNB – Correlations

As was written, the boiling crisis can be classified as dryout (will be described below DNB) in the high-quality region and departure from nucleate boiling (DNB) in the subcooled or low-quality region (approximate quality range: from –5% to +5%). But the critical heat flux is used for both regimes.

DNB – W-3 Correlation

One of the most well known design correlations for predicting departure from nucleate boiling is the W-3 correlation developed at the Westinghouse Atomic Power Division by Tong. It is applicable for subcooled and low to moderate quality flows.The W-3 correlation is a function of coolant enthalpy (saturated and inlet), pressure, quality and coolant mass flux:

The correlation W-3 is for critical heat flux in uniformly heated channels. To account for non-uniform heat fluxes, Tong introduced the correction factor, F.

Special Reference: Tong, L. S., Weisman, Joel. Thermal Analysis of Pressurized Water Reactors. Amer Nuclear Society, 3rd edition, 5/1996. ISBN-13: 978-0894480386.

Cold Wall Factor – CWF

Tong, L. S. and Weisman, Joel also introduces a new factor known as the “cold wall factor”, which corrects CHF in a channel containing an unheated wall (e.g. channel adjacent to control rod guide tube). In these channels, liquid film builds up along the cold wall and this fluid is not effective in cooling the heated surface and the fluid cooling the heated surface is at higher enthalpy than calculated without assumption of cold wall. Note that, there is an assumption that cold wall deteriorates heat transfer compared to channel with all sides heated at the same bulk exit enthalpy.

CHF Look-up Tables

CHF look-up tables are used widely for the prediction of the critical heat flux (CHF). The CHF look-up table is basically a normalized data bank for a vertical 8 mm water-cooled tube. The 2006 CHF look-up table is based on a database containing more than 30,000 data points and they cover the ranges of 0.1–21 Mpa pressure, 0–8000 kg.m^–2.s^-1 (zero flow refers to pool-boiling conditions) mass flux and –0.5 to 1 vapour quality (negative qualities refer to subcooled conditions).

Special Reference: GROENEVELD, D.C. et al., The 2006 look-up table, Nuclear Engineering and Design 237 (2007), 1909–1922.

Departure from Nucleate Boiling Ratio – DNBR

As was written, in case of PWRs, the critical safety issue is named DNB (departure from nucleate boiling), which causes the formation of a local vapor layer, causing a dramatic reduction in heat transfer capability. Note that, even for BWRs, which have a significantly bottom-peaked axial power profile, the DNB-risk have to be taken into account.

DNB occurs, when the local heat flux reaches value of critical heat flux. This phenomenon occurs in the subcooled or low-quality region (approximate quality range: from –5% to +5%). The behaviour of this type of boiling crisis depends on many flow conditions (pressure, temperature, flow rate), since the critical heat flux is generally a function  of coolant enthalpy (saturated and inlet), pressure, quality and coolant mass flux:

This type of boiling crisis occurs at a relatively high heat fluxes and appears to be associated with the cloud of bubbles, adjacent to the surface. These bubbles or film of vapor reduces the amount of incoming water. Since this phenomenon deteriorates the heat transfer coefficient and the heat flux remains, heat then accumulates in the fuel rod causing dramatic rise of cladding and fuel temperature. Simply, a very high temperature difference is required to transfer the critical heat flux being produced from the surface of the fuel rod to the reactor coolant (through vapor layer). In case of PWRs, the critical flow is inverted annular flow, while in BWRs, the critical flow is usually annular flow.

In pressurized water reactors, one of key safety requirements is that a departure from nucleate boiling (DNB) will not occur during steady state operation, normal operational transients, and anticipated operational occurrences (AOOs).  Fuel cladding integrity will be maintained if the minimum DNBR remains above the 95/95 DNBR limit for PWRs ( a 95% probability at a 95% confidence level). DNB criterion is one of acceptance criteria in safety analyses as well as it constitutes one of safety limits in technical specifications. Needless to say, the establishment of a minimum DNB ratio provides a major limitation on the design of water cooled reactors. This phenomenon limits the maximal thermal power of each PWR.

DNB ratio (DNBR – Departure from Nucleate Boiling Ratio) is the measure of the margin to critical heat flux. DNBR is defined as:

the critical heat flux at a specific location and specific coolant parameters divided by the operating local heat flux at that location.

The reactor core must be designed to keep the DNBR larger than the minimum allowable value (known as the correlation limit) during steady state operation, normal operational transients, and anticipated operational occurrences (AOOs). For predicting departure from nucleate boiling, CHF can be, for example, determined using the W-3 correlation developed at the Westinghouse Atomic Power Division. If these correlation were perfect (without uncertainties), the criterion would be simple:

Local heat flux must be lower than critical heat flux (i.e. DNBR must be higher than one).

But in reality, no correlation is perfect and uncertainties must be involved in this calculation. These uncertainty bands or error bounds establish a minimum acceptable value for the DNB Ratio, which may be significantly greater than one as indicated in the figure. Uncertainties may reach about 20% and therefore the DNBR must be larger than, for example, DNBR_lim = 1,2.

As can be seen from the figure, the CHF significantly decreases with increasing coolant enthalpy, therefore minimal value of DNBR is not necessarily in the center of the core. The Minimum DNB Ratio (MDNBR) occurs at the location where the critical heat flux and the operating heat flux are the closest and it is usually in the upper part of the core. Moreover, at the channel inlet where the coolant subcooling is the highest, we would expect the heat flux necessary to cause DNB at this location to be extremely high. On the other hand, at the channel exit where the coolant enthalpy is its highest, the heat flux necessary to cause DNB should be at its lowest.

Special Reference: Tong, L. S., Weisman, Joel. Thermal Analysis of Pressurized Water Reactors. Amer Nuclear Society, 3rd edition, 5/1996. ISBN-13: 978-0894480386.

Post-DNB Heat Transfer

The nucleate boiling heat flux cannot be increased indefinitely. At some value, we call it the “critical heat flux” (CHF), the steam produced can form an insulating layer over the surface, which in turn deteriorates the heat transfer coefficient. This is because a large fraction of the surface is covered by a vapor film, which acts as an thermal insulation due to the low thermal conductivity of the vapor relative to that of the liquid. Immediately after the critical heat flux has been reached, boiling become unstable and transition boiling occurs. The transition from nucleate boiling to film boiling is known as the “boiling crisis”. Since beyond the CHF point the heat transfer coefficient decreases, the transition to film boiling is usually inevitable.

A further increase in the heat flux is not necessary to maintain film boiling. A film of vapour fully covers the surface. This significantly reduces the convection coefficient, since the vapor layer has significantly lower heat transfer ability. As a result the excess temperature shoots up to a very high value. Beyond the Leidenfrost point, a continuous vapor film blankets the surface and there is no contact between the liquid phase and the surface. In this situation the heat transfer is both by radiation and by conduction to the vapour. Heated surface stabilizes stabilizes its temperature at point E (see figure). If the material is not strong enough for withstanding this temperature, the equipment will fail by damage to the material.

Critical Power Ratio – Dryout

In BWRs, similar phenomenon is known as “dryout” and it is directly associated with changes in flow pattern during evaporation in the high-quality region. At given combinations of flow rate through a channel, pressure, flow quality, and linear heat rate, the wall liquid film may exhaust and the wall may be dried out. At normal, the fuel surface is effectively cooled by boiling coolant. However when the heat flux exceeds a critical value (CHF – critical heat flux) the flow pattern may reach the dryout conditions (thin film of liquid disappears). The heat transfer from the fuel surface into the coolant is deteriorated, with the result of a drastically increased fuel surface temperature. In the high-quality region, the crisis occurs at a lower heat flux. Since the flow velocity in the vapor core is high, post-CHF heat transfer is much better than for low-quality critical flux (i.e. for PWRs temperature rises are higher and more rapid).

In this case, engineers define parameter known as the minimum critical power ratio (MCPR) instead of DNBR. The critical power ratio (CPR) is used for determining the thermal limits of boiling water reactors.

Definition of CPR :

The CPR is that power in the assembly that is calculated by application of the appropriate correlation(s) to cause some point in the assembly to experience boiling transition, divided by the actual assembly operating power.

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See also:

Boiling and Condensation