Thermal Engineering

Drag Force in Nuclear Engineering

Analysis of the hydraulic lift force is one of the most important analysis in designing a fuel assembly and analyzing the hydraulic compatibility of mixed cores. The vertical forces are induced by upward high-velocity flow through the reactor core. The flow path for the reactor coolant through the reactor vessel would be:

• 

  The coolant enters the reactor vessel at the inlet nozzle and hits against the core barrel.

• The core barrel forces the water to flow downward in the space between the reactor vessel wall and the core barrel, this space is usually known as the downcomer.
• From the bottom of the pressure vessel, the flow is reversed up through the core in order to pass through the fuel assemblies, where the coolant temperature increases as it passes through the fuel rods.
• Finally the hotter reactor coolant enters the upper internals region, where it is routed out the outlet nozzle into the hot legs of primary circuit and goes on to the steam generators.

Fuel assemblies are held by the upper guide structure assembly, which defines the top of the core. This assembly is made of stainless steel and has many purposes. The upper guide structure assembly exerts an axial force on fuel assemblies (through springs in the top nozzle), thus defines the exact position of the fuel assembly in the core. The upper guide structure assembly flange is held in place and preloaded by the RPV closure head flange. The upper guide structure assembly also guides and protects control rod assemblies and in-core instrumentation.

Required downforce of the upper guide structure assembly on fuel assemblies must very carefully calculated. Insufficient downforce can result in the lift of the fuel assembly, on the other hand, an excessive downforce can result in bowing of fuel assembly, which is also unacceptable.

What is Drag in Physics

In fluid dynamics, drag is a force acting opposite to the relative motion of any moving object. The force a flowing fluid exerts on a body in the flow direction. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag forces depend on velocity. Drag force is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. Drag is generally caused by two phenomena:

• 

  Skin Friction.  In general, when a fluid flows over a stationary surface, e.g. the flat plate, the bed of a river, or the wall of a pipe, the fluid touching the surface is brought to rest by the shear stress to at the wall. The region in which flow adjusts from zero velocity at the wall to a maximum in the main stream of the flow is termed the boundary layer. Therefore, a moving fluid exerts tangential shear forces on the surface because of the no-slip condition caused by viscous effects.  This type of drag force, depends especially on the geometry, the roughness of the solid surface and on the type of fluid flow.

• Form Drag. Form drag known also as pressure drag arises because of the shape and size of the object. This type of drag force is an interesting consequence the Bernoulli’s effect. According to Bernoulli’s principle, faster moving air exerts less pressure. This causes, that there can be a pressure difference between surfaces of the object. The general size and shape of the body are the most important factors in form drag. In general, bodies with a larger presented geometric cross-section will have a higher drag than thinner bodies.

Both of these forces, in general, have components in the direction of flow, and thus the resulting drag force is due to the combined effects of pressure and skin friction forces in the flow direction.

When the friction and pressure drag coefficients are available, the total drag coefficient is determined by simply adding them:

At low Reynolds numbers, most drag is due to friction drag. This is especially the case for highly streamlined bodies such as airfoils. On the other hand, at high Reynolds number, the pressure drop is significant, which increases form drag.

The components of the pressure and skin friction forces in the normal direction to flow tend to move the body in that direction, and their sum is called lift.

In aeronautics, the lift is an upward-acting force on an aircraft wing or airfoil. Bernoulli’s principle requires airfoil to be of an asymmetrical shape.

Drag Force – Drag Equation

The drag force, F_D,depends on the density of the fluid, the upstream velocity, and the size, shape, and orientation of the body, among other things. One way to express this is by means of the drag equation. The drag equation is a formula used to calculate the drag force experienced by an object due to movement through a fluid.

The reference area, A, is defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion. For hollow objects, the reference area may be significantly larger than the cross sectional area, but for non-hollow objects, it is exactly the same as a cross sectional area.

Example: Drag Force – Drag Coefficient – Fuel Bundle

Calculate the friction drag of a single fuel rod inside a reactor core at normal operation (design flow rate). Assume that this fuel rod is part of a fuel bundle with the rectangular fuel lattice and this fuel bundle does not contain spacing grids. Its height is h = 4m and the core flow velocity is constant and equal to V_core = 5 m/s.

Assume that:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm
• the pitch of fuel pins is: p = 13 mm
• the relative roughness is ε/D = 5×10^-4
• the fluid density is: ρ = 714 kg/m³
• the core flow velocity is constant and equal to V_core = 5 m/s
• the average temperature of reactor coolant is: T_bulk = 296°C

Calculation of the Reynolds number

To calculate the Reynolds number, we have to know:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm (to calculate the hydraulic diameter)
• the pitch of fuel pins is: p = 13 mm  (to calculate the hydraulic diameter)
• the dynamic viscosity of saturated water at 300°C is: μ = 0.0000859 N.s/m²
• the fluid density is: ρ = 714 kg/m³

The hydraulic diameter, D_h, is a commonly used term when handling flow in non-circular tubes and channels. The hydraulic diameter of the fuel channel, D_h, is equal to 13,85 mm.

See also: Hydraulic Diameter

The Reynolds number inside the fuel channel is then equal to:

This fully satisfies the turbulent conditions.

Calculation of the Skin Friction Coefficient

The friction factor for turbulent flow depends strongly on the relative roughness. It is determined by the Colebrook equation or can be determined using the Moody chart. The Moody chart for Re = 575 600 and ε/D = 5 x 10^-4 returns following values:

• the Darcy friction factor is equal to f_D = 0.017
• the Fanning friction factor is equal to f_F = f_D/4 = 0.00425

Therefore the skin friction coefficient is equal to:

Calculation of the Drag Force

To calculate the drag force, we have to know:

• the skin friction coefficient, which is: C_D,friction = 0.00425
• the area of pin surface, which is: A = π.d.h = 0.1169 m²
• the fluid density, which is: ρ = 714 kg/m³
• the core flow velocity, which is constant and equal to V_core = 5 m/s

From the skin friction coefficient, which is equal to the Fanning friction factor we can calculate the frictional component of the drag force. The drag force is given by:

Assuming that a fuel assembly can have, for example, 289 fuel pins (17×17 fuel assembly), the frictional component of the drag force is then of the order of kilonewtons. Moreover, this drag force originates purely from the skin friction on fuel bundle. But typical PWR fuel assembly contains another components, which influence the fuel assembly hydraulics:

• Fuel rods. Fuel rods contain the fuel and burnable poisons.
• Top nozzle. Provides the mechanical support for the fuel assembly structure.
• Bottom nozzle. Provides the mechanical support for the fuel assembly structure.
• Spacing grid. Ensures an exact guiding of the fuel rods.
• Guide thimble tube. Vacant tube for control rods or in-core instrumentation.

As was written, the second component of the drag force is the form drag. Form drag known also as pressure drag arises because of the shape and size of the object. The pressure drag is proportional to the difference between the pressures acting on the front and back of the immersed body, and the frontal area.

Pressure Drop – Fuel Assembly

In general, total fuel assembly pressure drop is formed by fuel bundle frictional drop (dependent on relative roughness of fuel rods, reynolds number, hydraulic diameter etc.) and other pressure drops of structural elements (top and bottom nozzle, spacing grids or mixing grids).

In general, it is not so simple to calculate pressure drops in fuel assemblies (especially the spacing grids) and it belongs to key know-how of certain fuel manufacturer. Mostly, pressure drops are measured in experimental hydraulic loops, rather than calculated.

Engineers use the pressure loss coefficient, PLC. It is noted K or ξ  (pronounced “xi”). This coefficient characterizes pressure loss of a certain hydraulic system or of a part of a hydraulic system. It can be easily measured in hydraulic loops. The pressure loss coefficient can be defined or measured for both straight pipes and especially for local (minor) losses.

Using data from above mentioned example, the pressure loss coefficient (only frictional from straight pipe) is equal to ξ = f_DL/D_H = 4.9. But the overall pressure loss coefficient (including spacing grids, top and bottom nozzles etc.) is usually about three times higher. This PLC (ξ = 4.9) causes that the pressure drop is of the order of (using the previous inputs) Δp_friction =  4.9 x 714 x 5²/ 2 = 43.7 kPa (without spacing grids, top and bottom nozzles). About three times higher real PLC means about three times higher Δp_fuel will be.

The overall reactor pressure loss, Δp_reactor, must include:

• downcomer and reactor bottom
• lower support plate
• fuel assembly including spacing grids, top and bottom nozzles and other structural components – Δp_fuel
• upper guide structure assembly

In result the overall reactor pressure loss – Δp_reactor is usually of the order of hundreds kPa (let say 300 – 400 kPa) for design parameters.

See also:

Nuclear Fuel

Pressure Drop – Fuel Assembly

In general, total fuel assembly pressure drop is formed by fuel bundle frictional drop (dependent on relative roughness of fuel rods, reynolds number, hydraulic diameter etc.) and other pressure drops of structural elements (top and bottom nozzle, spacing grids or mixing grids).

See also: Fluid Acceleration – Pressure Loss

In general, it is not so simple to calculate pressure drops in fuel assemblies (especially the spacing grids) and it belongs to key know-how of certain fuel manufacturer. Mostly, pressure drops are measured in experimental hydraulic loops, rather than calculated.

Engineers use the pressure loss coefficient, PLC. It is noted K or ξ  (pronounced “xi”). This coefficient characterizes pressure loss of a certain hydraulic system or of a part of a hydraulic system. It can be easily measured in hydraulic loops. The pressure loss coefficient can be defined or measured for both straight pipes and especially for local (minor) losses.

Using data from below mentioned example, the pressure loss coefficient (only frictional from straight pipe) is equal to ξ = f_DL/D_H = 4.9. But the overall pressure loss coefficient (including spacing grids, top and bottom nozzles etc.) is usually about three times higher. This PLC (ξ = 4.9) causes that the pressure drop is of the order of (using the previous inputs) Δp_friction =  4.9 x 714 x 5²/ 2 = 43.7 kPa (without spacing grids, top and bottom nozzles). About three times higher real PLC means about three times higher Δp_fuel will be.

The overall reactor pressure loss, Δp_reactor, must include:

• downcomer and reactor bottom
• lower support plate
• fuel assembly including spacing grids, top and bottom nozzles and other structural components – Δp_fuel
• upper guide structure assembly

In result the overall reactor pressure loss – Δp_reactor is usually of the order of hundreds kPa (let say 300 – 400 kPa) for design parameters.

It is an illustrative example, previous data do not correspond to any reactor design.

Example: Frictional Pressure Loss – Fuel Bundle

Calculate the frictional pressure loss of a single fuel rod inside a reactor core at normal operation (design flow rate). Assume that this fuel rod is part of a fuel bundle with the rectangular fuel lattice and this fuel bundle does not contain spacing grids. Its height is h = 4m and the core flow velocity is constant and equal to V_core = 5 m/s.

Assume that:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm
• the pitch of fuel pins is: p = 13 mm
• the relative roughness is ε/D = 5×10^-4
• the fluid density is: ρ = 714 kg/m³
• the core flow velocity is constant and equal to V_core = 5 m/s
• the average temperature of reactor coolant is: T_bulk = 296°C

Calculation of the Reynolds number

To calculate the Reynolds number, we have to know:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm (to calculate the hydraulic diameter)
• the pitch of fuel pins is: p = 13 mm  (to calculate the hydraulic diameter)
• the dynamic viscosity of saturated water at 300°C is: μ = 0.0000859 N.s/m²
• the fluid density is: ρ = 714 kg/m³

The hydraulic diameter, D_h, is a commonly used term when handling flow in non-circular tubes and channels. The hydraulic diameter of the fuel channel, D_h, is equal to 13,85 mm.

See also: Hydraulic Diameter

The Reynolds number inside the fuel channel is then equal to:

This fully satisfies the turbulent conditions.

Calculation of the Darcy Friction Factor

The friction factor for turbulent flow depends strongly on the relative roughness. It is determined by the Colebrook equation or can be determined using the Moody chart. The Moody chart for Re = 575 600 and ε/D = 5 x 10^-4 returns following values:

• the Darcy friction factor is equal to f_D = 0.017
• the Fanning friction factor is equal to f_F = f_D/4 = 0.00425

Therefore, the pressure loss coefficient (only frictional from straight pipe) is equal to ξ = f_DL/D_H = 4.9.

See also:

Nuclear Fuel

What is Liquid Metal

In physics, liquid metal consists of alloy with very low melting points which form a eutectic that is liquid at room temperature. In reactor engineering, liquid metals are alloys with low melting point allowing for reactor coolant to be liquid in operating range of temperatures (usually above the room temperature).

Liquid metals can be used as a reactor coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. The use of liquid metal coolants made it possible to provide high rate of heat transfer in power plants as well as the temperatures of working surfaces of their constructions close to coolant temperature.

Moreover, liquid metals used in reactor engineering are very weak absorbers of neutrons allowing for liquid metal reactors to operate with fast neutron spectrum. A liquid metal fast reactor is a high power density reactor, which does not need neutron moderator.

The main differences between thermal and fast reactors are, of course, in neutron cross-sections, that exhibit significant energy dependency. It can be characterized by capture-to-fission ratio, which is lower in fast reactors. There is also a difference in the number of neutrons produced per one fission, which is higher in fast reactors than in thermal reactors. These very important differences are caused primarily by differences in neutron fluxes. Therefore it is very important to know detailed neutron energy distribution in a reactor core.

The disadvantage of many liquid metals is also their high chemical activity at interaction with oxygen, water and structural materials, which may cause heat transfer deterioration in the plant under certain conditions.

Sodium and NaK – Reactor Coolant

Sodium-cooled fast reactors (SFRs) are the most common fast reactor design. They use molten sodium or a eutectic sodium-potassium alloy (NaK) as the reactor coolant. Melting and boiling points of sodium and NaK are:

• 

  sodium

  • melting point – 97.72°C
  • boiling point – 883°C
• NaK – eutectic mixture
  • melting point – (-12°C)
  • boiling point – 785°C

NaK containing 40% to 90% potassium by weight is liquid at room temperature. The eutectic mixture consists of 77% potassium and 23% sodium. Sodium and NaK do not corrode steel to any significant degree and are compatible with many nuclear fuels, allowing for a wide choice of structural materials. Because sodium reacts violently with water, however, SFRs require the placement of an intermediate heat exchanger between the reactor core and the steam generator. This hi-tech technology requires a lot of experience, therefore only few countries have developed their own fast reactor design.

Lead and Lead-bismuth Eutectic – Reactor Coolant

Lead, lead-bismuth eutectic, and other metals have also been proposed and occasionally used. The lead-cooled fast reactor is a nuclear reactor design that features a fast neutron spectrum and molten lead or lead-bismuth eutectic coolant. Lead-Bismuth Eutectic or LBE is a eutectic alloy of lead (44.5%) and bismuth (55.5%). Molten lead or lead-bismuth eutectic can be used as the primary coolant because lead and bismuth have low neutron absorption and relatively low melting points.

Melting and boiling points of lead and lead-bismuth eutectic mixture are:

• lead
  • melting point – 327.5°C
  • boiling point – 1749°C
• lead-bismuth – eutectic mixture
  • melting point – 123.5°C
  • boiling point – 1670°C

As compared to sodium-based liquid metal coolants such as liquid sodium or NaK, lead-based coolants have significantly higher boiling points, meaning a reactor can be operated without risk of coolant boiling at much higher temperatures. Lead and LBE also do not react readily with water or air, in contrast to sodium and NaK which ignite spontaneously in air and react explosively with water. Due to its denseness and high atomic number, lead and bismuth are also an excellent gamma radiation shield, while simultaneously being virtually transparent to neutrons.

On the other hand, lead and LBE coolant are more corrosive to steel than sodium or NaK eutectic alloy. This and the very high density of lead puts an upper limit on the velocity of coolant flow through the reactor due to safety considerations. Furthermore, the higher melting points of lead and LBE (327 °C and 123.5 °C respectively) may mean that solidification of the coolant may be a greater problem when the reactor is operated at lower temperatures.

Nusselt Number for Liquid Metal Reactors

A liquid metal cooled reactor is an advanced type of nuclear reactor where the primary coolant is a liquid metal.  Liquid metals can be used as a coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. This very high thermal conductivity together with low viscosity causes, that typical heat transfer correlations (e.g. Dittus-Boelter) can not be used.

Heat transfer coefficients for sodium flow through fuel channel are based on the Prandtl number and Péclet number. Pitch-to-diameter (P/D) also enters many calculations of heat transfer in liquid metal reactors. Convective heat transfer correlations are usually presented in terms of Nusselt number versus Péclet number. Typical Péclet number for normal operation are from 150 to 300 in the fuel bundles. As for another flow regimes,  the Nusselt number and a given correlation can be used to determine the convective heat transfer coefficient.

Graber-Rieger Correlation

FFTF Correlation

• Sodium-cooled fast reactor
• Lead-cooled fast reactor

One of the main challenges in numerical simulation is the reliable modeling of heat transfer in liquid-metal cooled reactors by Computational Fluid Dynamics (CFD). Heat transfer applications with low-Prandtl number fluids are often in the transition range between conduction and convection dominated regimes.

See also:

Materials

Properties of Liquid Metals

Properties of Liquid Metals

In physics, liquid metal consists of alloy with very low melting points which form a eutectic that is liquid at room temperature. In reactor engineering, liquid metals are alloys with low melting point allowing for reactor coolant to be liquid in operating range of temperatures (usually above the room temperature).

Liquid metals can be used as a reactor coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. The use of liquid metal coolants made it possible to provide high rate of heat transfer in power plants as well as the temperatures of working surfaces of their constructions close to coolant temperature.

Moreover, liquid metals used in reactor engineering are very weak absorbers of neutrons allowing for liquid metal reactors to operate with fast neutron spectrum. A liquid metal fast reactor is a high power density reactor, which does not need neutron moderator.

The main differences between thermal and fast reactors are, of course, in neutron cross-sections, that exhibit significant energy dependency. It can be characterized by capture-to-fission ratio, which is lower in fast reactors. There is also a difference in the number of neutrons produced per one fission, which is higher in fast reactors than in thermal reactors. These very important differences are caused primarily by differences in neutron fluxes. Therefore it is very important to know detailed neutron energy distribution in a reactor core.

The disadvantage of many liquid metals is also their high chemical activity at interaction with oxygen, water and structural materials, which may cause heat transfer deterioration in the plant under certain conditions.

Sodium and NaK – Reactor Coolant

Sodium-cooled fast reactors (SFRs) are the most common fast reactor design. They use molten sodium or a eutectic sodium-potassium alloy (NaK) as the reactor coolant. Melting and boiling points of sodium and NaK are:

• 

  sodium

  • melting point – 97.72°C
  • boiling point – 883°C
• NaK – eutectic mixture
  • melting point – (-12°C)
  • boiling point – 785°C

NaK containing 40% to 90% potassium by weight is liquid at room temperature. The eutectic mixture consists of 77% potassium and 23% sodium. Sodium and NaK do not corrode steel to any significant degree and are compatible with many nuclear fuels, allowing for a wide choice of structural materials. Because sodium reacts violently with water, however, SFRs require the placement of an intermediate heat exchanger between the reactor core and the steam generator. This hi-tech technology requires a lot of experience, therefore only few countries have developed their own fast reactor design.

Lead and Lead-bismuth Eutectic – Reactor Coolant

Lead, lead-bismuth eutectic, and other metals have also been proposed and occasionally used. The lead-cooled fast reactor is a nuclear reactor design that features a fast neutron spectrum and molten lead or lead-bismuth eutectic coolant. Lead-Bismuth Eutectic or LBE is a eutectic alloy of lead (44.5%) and bismuth (55.5%). Molten lead or lead-bismuth eutectic can be used as the primary coolant because lead and bismuth have low neutron absorption and relatively low melting points.

Melting and boiling points of lead and lead-bismuth eutectic mixture are:

• lead
  • melting point – 327.5°C
  • boiling point – 1749°C
• lead-bismuth – eutectic mixture
  • melting point – 123.5°C
  • boiling point – 1670°C

As compared to sodium-based liquid metal coolants such as liquid sodium or NaK, lead-based coolants have significantly higher boiling points, meaning a reactor can be operated without risk of coolant boiling at much higher temperatures. Lead and LBE also do not react readily with water or air, in contrast to sodium and NaK which ignite spontaneously in air and react explosively with water. Due to its denseness and high atomic number, lead and bismuth are also an excellent gamma radiation shield, while simultaneously being virtually transparent to neutrons.

On the other hand, lead and LBE coolant are more corrosive to steel than sodium or NaK eutectic alloy. This and the very high density of lead puts an upper limit on the velocity of coolant flow through the reactor due to safety considerations. Furthermore, the higher melting points of lead and LBE (327 °C and 123.5 °C respectively) may mean that solidification of the coolant may be a greater problem when the reactor is operated at lower temperatures.

Nusselt Number for Liquid Metal Reactors

A liquid metal cooled reactor is an advanced type of nuclear reactor where the primary coolant is a liquid metal.  Liquid metals can be used as a coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. This very high thermal conductivity together with low viscosity causes, that typical heat transfer correlations (e.g. Dittus-Boelter) can not be used.

Heat transfer coefficients for sodium flow through fuel channel are based on the Prandtl number and Péclet number. Pitch-to-diameter (P/D) also enters many calculations of heat transfer in liquid metal reactors. Convective heat transfer correlations are usually presented in terms of Nusselt number versus Péclet number. Typical Péclet number for normal operation are from 150 to 300 in the fuel bundles. As for another flow regimes,  the Nusselt number and a given correlation can be used to determine the convective heat transfer coefficient.

Graber-Rieger Correlation

FFTF Correlation

See also:

Liquid Metals

Sodium and NaK – Reactor Coolant

Sodium-cooled fast reactors (SFRs) are the most common fast reactor design. They use molten sodium or a eutectic sodium-potassium alloy (NaK) as the reactor coolant. Melting and boiling points of sodium and NaK are:

• 

  sodium

  • melting point – 97.72°C
  • boiling point – 883°C
• NaK – eutectic mixture
  • melting point – (-12°C)
  • boiling point – 785°C

NaK containing 40% to 90% potassium by weight is liquid at room temperature. The eutectic mixture consists of 77% potassium and 23% sodium. Sodium and NaK do not corrode steel to any significant degree and are compatible with many nuclear fuels, allowing for a wide choice of structural materials. Because sodium reacts violently with water, however, SFRs require the placement of an intermediate heat exchanger between the reactor core and the steam generator. This hi-tech technology requires a lot of experience, therefore only few countries have developed their own fast reactor design.

Properties of Liquid Metals

Properties of Liquid Metals

In physics, liquid metal consists of alloy with very low melting points which form a eutectic that is liquid at room temperature. In reactor engineering, liquid metals are alloys with low melting point allowing for reactor coolant to be liquid in operating range of temperatures (usually above the room temperature).

Liquid metals can be used as a reactor coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. The use of liquid metal coolants made it possible to provide high rate of heat transfer in power plants as well as the temperatures of working surfaces of their constructions close to coolant temperature.

Moreover, liquid metals used in reactor engineering are very weak absorbers of neutrons allowing for liquid metal reactors to operate with fast neutron spectrum. A liquid metal fast reactor is a high power density reactor, which does not need neutron moderator.

The main differences between thermal and fast reactors are, of course, in neutron cross-sections, that exhibit significant energy dependency. It can be characterized by capture-to-fission ratio, which is lower in fast reactors. There is also a difference in the number of neutrons produced per one fission, which is higher in fast reactors than in thermal reactors. These very important differences are caused primarily by differences in neutron fluxes. Therefore it is very important to know detailed neutron energy distribution in a reactor core.

The disadvantage of many liquid metals is also their high chemical activity at interaction with oxygen, water and structural materials, which may cause heat transfer deterioration in the plant under certain conditions.

Nusselt Number for Liquid Metal Reactors

A liquid metal cooled reactor is an advanced type of nuclear reactor where the primary coolant is a liquid metal.  Liquid metals can be used as a coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. This very high thermal conductivity together with low viscosity causes, that typical heat transfer correlations (e.g. Dittus-Boelter) can not be used.

Heat transfer coefficients for sodium flow through fuel channel are based on the Prandtl number and Péclet number. Pitch-to-diameter (P/D) also enters many calculations of heat transfer in liquid metal reactors. Convective heat transfer correlations are usually presented in terms of Nusselt number versus Péclet number. Typical Péclet number for normal operation are from 150 to 300 in the fuel bundles. As for another flow regimes,  the Nusselt number and a given correlation can be used to determine the convective heat transfer coefficient.

Graber-Rieger Correlation

FFTF Correlation

See also:

Liquid Metals

Lead and Lead-bismuth Eutectic – Reactor Coolant

Lead, lead-bismuth eutectic, and other metals have also been proposed and occasionally used. The lead-cooled fast reactor is a nuclear reactor design that features a fast neutron spectrum and molten lead or lead-bismuth eutectic coolant. Lead-Bismuth Eutectic or LBE is a eutectic alloy of lead (44.5%) and bismuth (55.5%). Molten lead or lead-bismuth eutectic can be used as the primary coolant because lead and bismuth have low neutron absorption and relatively low melting points.

Melting and boiling points of lead and lead-bismuth eutectic mixture are:

• lead
  • melting point – 327.5°C
  • boiling point – 1749°C
• lead-bismuth – eutectic mixture
  • melting point – 123.5°C
  • boiling point – 1670°C

As compared to sodium-based liquid metal coolants such as liquid sodium or NaK, lead-based coolants have significantly higher boiling points, meaning a reactor can be operated without risk of coolant boiling at much higher temperatures. Lead and LBE also do not react readily with water or air, in contrast to sodium and NaK which ignite spontaneously in air and react explosively with water. Due to its denseness and high atomic number, lead and bismuth are also an excellent gamma radiation shield, while simultaneously being virtually transparent to neutrons.

On the other hand, lead and LBE coolant are more corrosive to steel than sodium or NaK eutectic alloy. This and the very high density of lead puts an upper limit on the velocity of coolant flow through the reactor due to safety considerations. Furthermore, the higher melting points of lead and LBE (327 °C and 123.5 °C respectively) may mean that solidification of the coolant may be a greater problem when the reactor is operated at lower temperatures.

Properties of Liquid Metals

Properties of Liquid Metals

In physics, liquid metal consists of alloy with very low melting points which form a eutectic that is liquid at room temperature. In reactor engineering, liquid metals are alloys with low melting point allowing for reactor coolant to be liquid in operating range of temperatures (usually above the room temperature).

Liquid metals can be used as a reactor coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. The use of liquid metal coolants made it possible to provide high rate of heat transfer in power plants as well as the temperatures of working surfaces of their constructions close to coolant temperature.

Moreover, liquid metals used in reactor engineering are very weak absorbers of neutrons allowing for liquid metal reactors to operate with fast neutron spectrum. A liquid metal fast reactor is a high power density reactor, which does not need neutron moderator.

The main differences between thermal and fast reactors are, of course, in neutron cross-sections, that exhibit significant energy dependency. It can be characterized by capture-to-fission ratio, which is lower in fast reactors. There is also a difference in the number of neutrons produced per one fission, which is higher in fast reactors than in thermal reactors. These very important differences are caused primarily by differences in neutron fluxes. Therefore it is very important to know detailed neutron energy distribution in a reactor core.

The disadvantage of many liquid metals is also their high chemical activity at interaction with oxygen, water and structural materials, which may cause heat transfer deterioration in the plant under certain conditions.

See also:

Liquid Metals

Nusselt Number for Liquid Metal Reactors

A liquid metal cooled reactor is an advanced type of nuclear reactor where the primary coolant is a liquid metal.  Liquid metals can be used as a coolant because they have excellent heat transfer properties and can be employed in low-pressure systems as is the case of sodium-cooled fast reactors (SFRs). The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically free electrons, giving them high electrical conductivity, high thermal conductivity. This very high thermal conductivity together with low viscosity causes, that typical heat transfer correlations (e.g. Dittus-Boelter) can not be used.

Heat transfer coefficients for sodium flow through fuel channel are based on the Prandtl number and Péclet number. Pitch-to-diameter (P/D) also enters many calculations of heat transfer in liquid metal reactors. Convective heat transfer correlations are usually presented in terms of Nusselt number versus Péclet number. Typical Péclet number for normal operation are from 150 to 300 in the fuel bundles. As for another flow regimes,  the Nusselt number and a given correlation can be used to determine the convective heat transfer coefficient.

Graber-Rieger Correlation

FFTF Correlation

See also:

Nusselt Number

Dynamic Pressure – Velocity Pressure

In general, pressure is a measure of the force exerted per unit area on the boundaries of a substance. The term dynamic pressure (sometimes called velocity pressure)  is associated with fluid flow and with the Bernoulli’s effect, which is described by the Bernoulli’s equation:

This effect causes the lowering of fluid pressure (static pressure) in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy (dynamic pressure – ½.ρ.v²) must increase at the expense of pressure energy (static pressure – p).

As can be seen, dynamic pressure is dynamic pressure is one of the terms of Bernoulli’s equation. In incompressible fluid dynamics, dynamic pressure is the quantity defined by:

The simplified form of Bernoulli’s equation can be summarized in the following memorable word equation:

static pressure + dynamic pressure = total pressure (stagnation pressure)

Total and dynamic pressure are not pressures in the usual sense – they cannot be measured using an aneroid, Bourdon tube or mercury column.

To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. The term static pressure is identical to the term pressure, and can be identified for every point in a fluid flow field. Dynamic pressure is the difference between stagnation pressure and static pressure.

Dynamic Pressure and Pressure Loss

Dynamic pressure is closely related to pressure losses. In the practical analysis of piping systems the quantity of most importance is the pressure loss due to viscous effects along the length of the system, as well as additional pressure losses arising from other technological equipments like, valves, elbows, piping entrances, fittings and tees. By observation, the  head loss is roughly proportional to the square of the flow rate in most engineering flows (fully developed, turbulent pipe flow). In fact, the  head loss is directly proportional to the dynamic pressure.

The constant of proportionality is the pressure loss coefficient. The pressure loss coefficient is noted K or ξ  (pronounced “xi”). This coefficient characterizes pressure loss of a certain hydraulic system or of a part of a hydraulic system. It can be easily measured in hydraulic loops. The pressure loss coefficient can be defined or measured for both straight pipes and especially for local (minor) losses. For example, the pressure loss coefficient ξ = 4.9  causes that the pressure drop will be about 4.9 times the dynamic pressure.

See also:

Pressure

Stagnation Pressure – Pitot Pressure

In general, pressure is a measure of the force exerted per unit area on the boundaries of a substance. In fluid dynamics and aerodynamics, stagnation pressure (or pitot pressure or total pressure) is the static pressure at a stagnation point in a fluid flow. At a stagnation point the fluid velocity is zero and all kinetic energy has been converted into pressure energy (isentropically). This effect is widely used in aerodynamics (velocity measurement or ram-air intake).

Stagnation pressure is equal to the sum of the free-stream dynamic pressure and free-stream static pressure.

Static pressure and dynamic pressure are terms of Bernoulli’s equation:

The Bernoulli’s effect causes the lowering of fluid pressure (static pressure – p) in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy (dynamic pressure – ½.ρ.v²) must increase at the expense of pressure energy (static pressure – p).

The simplified form of Bernoulli’s equation can be summarized in the following memorable word equation:

static pressure + dynamic pressure = total pressure (stagnation pressure)

Total and dynamic pressure are not pressures in the usual sense – they cannot be measured using an aneroid, Bourdon tube or mercury column.

Stagnation pressure is sometimes referred to as pitot pressure because it is measured using a pitot tube. A Pitot tube is a pressure measurement instrument used to measure fluid flow velocity. Velocity can be determined using the following formula:

where:

• u is flow velocity to be measured in m/s,
• p_s is stagnation or total pressure in Pa,
• p_t is static pressure in Pa,
• ρ is fluid density in kg/m³.

See also:

Pressure

Static Pressure

In general, pressure is a measure of the force exerted per unit area on the boundaries of a substance. In fluid dynamics, many authors use the term static pressure in preference to just pressure to avoid ambiguity. The term static pressure is identical to the term pressure, and can be identified for every point in a fluid flow field.

Static pressure is one of the terms of Bernoulli’s equation:

The Bernoulli’s effect causes the lowering of fluid pressure (static pressure – p) in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy (dynamic pressure – ½.ρ.v²) must increase at the expense of pressure energy (static pressure – p).

The simplified form of Bernoulli’s equation can be summarized in the following memorable word equation:

static pressure + dynamic pressure = total pressure (stagnation pressure)

Total and dynamic pressure are not pressures in the usual sense – they cannot be measured using an aneroid, Bourdon tube or mercury column.

See also:

Pressure