Thermal Engineering

What is Grashof Number

The Grashof number is a dimensionless number, named after  Franz Grashof. The Grashof number is defined as the ratio of the buoyant to viscous force acting on a fluid in the velocity boundary layer. Its role in natural convection is much the same as that of the Reynolds number in forced convection.

Natural convection is used if this motion and mixing is caused by density variations resulting from temperature differences within the fluid. Usually the density decreases due to an increase in temperature and causes the fluid to rise. This motion is caused by the buoyant force. The major force that resists the motion is the viscous force. The Grashof number is a way to quantify the opposing forces.

The Grashof number is defined as:

where:

g is acceleration due to Earth’s gravity

β is the coefficient of thermal expansion

T_wall is the wall temperature

T_∞ is the bulk temperature

L is the vertical length

ν is the kinematic viscosity.

For gases β = 1/T where the temperature is in K. For liquids β can be calculated if variation of density with temperature at constant pressure is known. For a vertical flat plate, the flow turns turbulent for value of Gr.Pr > 10⁹. As in forced convection the microspic nature of flow and convection correlations are distinctly different in the laminar and turbulent regions.

The Grashof number is closely related to Rayleigh number, which is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity.

Example: Grashof Number

A vertical plate is maintained at 50°C in 20°C air. Determine the height at which the boundary layer will turn turbulent if turbulence sets in at Gr.Pr = 10⁹.

Solution:

The property values required for this example are:

ν = 1.48 x 10^-5 m²/s

ρ = 1.17 kg/m³

Pr = 0.700

β = 1/ (273 + 20) = 1/293

We know the natural circulation becomes turbulent at Gr.Pr > 10⁹, which is fulfilled at the following height:

See also:

Characteristic Numbers

What is Rayleigh Number

The Rayleigh number is a dimensionless number, named after  Lord Rayleigh. The Rayleigh number is closely related to Grashof number and both numbers are used to decribe natural convection (Gr) and heat transfer by natural convection (Ra). The Rayleigh number is simply defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity.

Ra_x = Gr_x . Pr

The Grashof number is defined as the ratio of the buoyant to viscous force acting on a fluid in the velocity boundary layer. Its role in natural convection is much the same as that of the Reynolds number in forced convection. Natural convection occurs if this motion and mixing is caused by density variations resulting from temperature differences within the fluid. Usually the density decreases due to an increase in temperature and causes the fluid to rise. This motion is caused by the buoyant force. The major force that resists the motion is the viscous force. The Grashof number is a way to quantify the opposing forces.

The Rayleigh number is used to express heat transfer in natural convection. The magnitude of the Rayleigh number is a good indication as to whether the natural convection boundary layer is laminar or turbulent. The simple empirical correlations for the average Nusselt number, Nu, in natural convection are of the form:

Nu_x = C. Ra_xⁿ

The values of the constants C and n depend on the geometry of the surface and the flow regime, which is characterized by the range of the Rayleigh number. The value of n is usually n = 1/4 for laminar flow and n = 1/3 for turbulent flow.

The Rayleigh number is defined as:

where:

g is acceleration due to Earth’s gravity

β is the coefficient of thermal expansion

T_wall is the wall temperature

T_∞ is the bulk temperature

L is the vertical length

α is thermal diffusivity

ν is the kinematic viscosity.

For gases β = 1/T where the temperature is in K. For liquids β can be calculated if variation of density with temperature at constant pressure is known. For a vertical flat plate, the flow turns turbulent for value of:

Ra_x = Gr_x . Pr > 10⁹ 

As in forced convection the microspic nature of flow and convection correlations are distinctly different in the laminar and turbulent regions.

Example: Rayleigh Number

A vertical plate is maintained at 50°C in 20°C air. Determine the height at which the boundary layer will turn turbulent if turbulence sets in at Ra = Gr.Pr = 10⁹.

Solution:

The property values required for this example are:

ν = 1.48 x 10^-5 m²/s

ρ = 1.17 kg/m³

Pr = 0.700

β = 1/ (273 + 20) = 1/293

We know the natural circulation becomes turbulent at approximately Ra = Gr.Pr > 10⁹, which is fulfilled at the following height:

See also:

Characteristic Numbers

What is Péclet Number

The Péclet number is a dimensionless number, named after the French physicist Jean Claude Eugène Péclet. The Péclet number is defined as the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion (matter or heat) of the same quantity driven by an appropriate gradient. Therefore we must distinguish between Peclet number for mass transfer and heat transfer.

Peclet Number in Mass Transfer

In mass transfer problems, the Péclet number is the product of the Reynolds number, which describes the flow regime, and the Schmidt number, which is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes.

where:

• u is the flow velocity,
• L is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter etc.)
• D is the mass diffusivity [m²/s]

Peclet Number in Heat Transfer

In heat transfer problems, the Péclet number is the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid.  The Péclet number is simply defined as the product of the Reynolds number, which describes the flow regime, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. If Pe is small, conduction is important.

where:

• u is the flow velocity,
• L is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter etc.)
• α is thermal diffusivity [m²/s]

Thermal diffusivity represents how fast heat diffuses through a material and has units m²/s. In other words, it is the measure of thermal inertia of given material. Thermal diffusivity is usually denoted α and is given by:

As can be seen it measures the ability of a material to conduct thermal energy(represented by factor k) relative to its ability to store thermal energy (represented by factor ρ.c_p). Materials of large α will respond quickly to changes in their thermal environment, whereas materials of small α will respond more slowly (heat is mostly absorbed), taking longer to reach a new equilibrium condition.

Mass Diffusivity and Fick’s law

Diffusivity is encountered in Fick’s law, which states:

If the concentration of a solute in one region is greater than in another of a solution, the solute diffuses from the region of higher concentration to the region of lower concentration, with a magnitude that is proportional to the concentration gradient.

In one (spatial) dimension, the law is:

where:

• J is the diffusion flux,
• D is the diffusion coefficient,
• φ (for ideal mixtures) is the concentration.

The use of this law in nuclear reactor theory leads to the diffusion approximation.

See also:

Characteristic Numbers

Critical Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent.

The critical Reynolds number is associated with the laminar-turbulent transition, in which a laminar flow becomes turbulent. This is an extraordinarily complicated process, which at present is not fully understood.

Critical Reynolds Number for Flow in a Pipe

For flow in a pipe of diameter D, experimental observations show that for “fully developed” flow, the critical Reynolds number is about Re_d,crit = 2300.

• Laminar flow. For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. The accepted transition Reynolds number for flow in a circular pipe is Re_d,crit = 2300.
• Transitional flow. At Reynolds numbers between about 2000 and 4000 the flow is unstable as a result of the onset of turbulence. These flows are sometimes referred to as transitional flows.
• Turbulent flow. If the Reynolds number is greater than 3500, the flow is turbulent.

Note that, the critical Reynolds number is different for every geometry.

Critical Reynolds Number in boundary layer flow over a flat plate.

In determining whether the boundary layer is laminar or turbulent, it is frequently reasonable to assume that transition begins at some location x_crit, as shown in figure. This location is determined by the critical Reynolds number, Re_x,crit. For flow over a flat plate,

transition from laminar to turbulent boundary layer occurs when Reynolds number at x exceeds Re_x,crit ~ 500,000. Transition may occur earlier, but it is dependent especially on the surface roughness. The turbulent boundary layer thickens more rapidly than the laminar boundary layer as a result of increased shear stress at the body surface.

Example: Critical Reynolds Number

A long thin flat plate is placed parallel to a 1 m/s stream of water at 20°C. Assume that kinematic viscosity of water at 20°C is equal to 1×10^-6 m²/s.

At what distance x from the leading edge will be the transition from laminar to turbulent boundary layer (i.e. find Re_x ~ 500,000).

Solution:

In order to locate the transition from laminar to turbulent boundary layer, we have to find x at which Re_x ~ 500,000.

x = 500 000 x 1×10^-6 [m²/s] / 1 [m/s] = 0.5 m

See also:

Reynolds Number

What is 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.

Mechanism on Convection

In thermal conduction, energy is transferred as heat either due to the migration of free electrons or lattice vibrational waves (phonons). There is no movement of mass in the direction of energy flow. Heat transfer by conduction is dependent upon the driving “force” of temperature difference. Conduction and convection are similar in that both mechanisms require the presence of a material medium (in comparison to thermal radiation). On the other hand they are different in that convection requires the presence of fluid motion.

It must be emphasized, at the surface, energy flow occurs purely by conduction, even in conduction. It is due to the fact, there is always a thin stagnant fluid film layer on the heat transfer surface. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. Higher the rate of mass movement, thinner the stagnant fluid film layer will be and higher will be the heat flow rate.

It must be noted, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

As was written, heat transfer through a fluid is by convection in the presence of mass movement and by conduction in the absence of it. Therefore, thermal conduction in a fluid can be viewed as the limiting case of convection, corresponding to the case of quiescent fluid.

Convection as a Conduction with Fluid Motion

Some experts do not consider convection to be a fundamental mechanism of heat transfer since it is essentially heat conduction in the presence of fluid motion. They consider it to be a special case of thermal conduction, known as “conduction with fluid motion”. On the other hand, it is practical to recognize convection as a separate heat transfer mechanism despite the valid arguments to the contrary.

In practice, analysis of heat transfer by convection is treated empirically (by direct experimental observation). Most of problems can be solved using so called characteristic numbers (e.g. Nusselt number). Characteristic numbers are dimensionless numbers used to describe a character of heat transfer and can be used to compare a real situation (e.g. heat transfer in a pipe) with a small-scale model. Experience shows that convection heat transfer strongly depends on the fluid properties dynamic viscosity, thermal conductivity, density, and specific heat, as well as the fluid velocity. It also depends on the geometry and the roughness of the solid surface, in addition to the type of fluid flow. All these conditions affect especially the stagnant film thickness.

Convection involves the transfer of heat between a surface at a given temperature (T_wall) and fluid at a bulk temperature (T_b). The exact definition of the bulk temperature (T_b) varies depending on the details of the situation.

• For flow adjacent to a hot or cold surface, T_b is the temperature of the fluid “far” from the surface.
• For boiling or condensation, T_b is the saturation temperature of the fluid.
• For flow in a pipe, T_b is the average temperature measured at a particular cross-section of the pipe.

Newton’s Law of Cooling

Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton’s law of cooling, which states that:

The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings provided the temperature difference is small and the nature of radiating surface remains same.

Note that, ΔT is given by the surface or wall temperature, T_wall and the bulk temperature, T_∞, which is the temperature of the fluid sufficiently far from the surface.

Convective Heat Transfer Coefficient

As can be seen, the constant of proportionality will be crucial in calculations and it is known as the convective heat transfer coefficient, h. The convective heat transfer coefficient, h, can be defined as:

The rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference.

The convective heat transfer coefficient is dependent upon the physical properties of the fluid and the physical situation. The convective heat transfer coefficient is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables influencing convection such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity.

Typically, the convective heat transfer coefficient for laminar flow is relatively low compared to the convective heat transfer coefficient for turbulent flow. This is due to turbulent flow having a thinner stagnant fluid film layer on the heat transfer surface.

It must be noted, this stagnant fluid film layer plays crucial role for the convective heat transfer coefficient. It is observed, that the fluid comes to a complete stop at the surface and assumes a zero velocity relative to the surface. This phenomenon is known as the no-slip condition and therefore, at the surface, energy flow occurs purely by conduction. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. As was written, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

A similar phenomenon occurs for the temperature. It is observed, that the fluid’s temperature at the surface and the surface will have the same temperature at the point of contact. This phenomenon is known as the no-temperature-jump condition and it is very important for theory of nucleate boiling.

Values of the heat transfer coefficient, h, have been measured and tabulated for the commonly encountered fluids and flow situations occurring during heat transfer by convection.

Nusselt Number

The Nusselt number is a dimensionless number, named after a German engineer Wilhelm Nusselt. The Nusselt number is closely related to Péclet number and both numbers are used to describe the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid. Nusselt number is equal to the dimensionless temperature gradient at the surface, and it provides a measure of the convection heat transfer occurring at the surface. The conductive component is measured under the same conditions as the heat convection but with a stagnant fluid. The Nusselt number is to the thermal boundary layer what the friction coefficient is to the velocity boundary layer. Thus, the Nusselt number is defined as:

where:

k_f is thermal conductivity of the fluid [W/m.K]

L is the characteristic length

h is the convective heat transfer coefficient [W/m².K]

For illustration, consider a fluid layer of thickness L and temperature difference ΔT. Heat transfer through the fluid layer will be by convection when the fluid involves some motion and by conduction when the fluid layer is motionless.

In case of conduction, the heat flux can be calculated using Fourier’s law of conduction. In case of convection, the heat flux can be calculated using Newton’s law of cooling. Taking their ratio gives:

The preceding equation defines the Nusselt number. Therefore, the Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer.  A Nusselt number of Nu=1 for a fluid layer represents heat transfer across the layer by pure conduction. The larger the Nusselt number, the more effective the convection. A larger Nusselt number corresponds to more effective convection, with turbulent flow typically in the 100–1000 range. For turbulent flow, the Nusselt number is usually a function of the Reynolds number and the Prandtl number.

Example – Convective Heat Transfer – Cladding Surface Temperature

Cladding is the outer layer of the fuel rods, standing between the reactor coolant and the nuclear fuel (i.e. fuel pellets). It is made of a corrosion-resistant material with low absorption cross section for thermal neutrons, usually zirconium alloy. Cladding prevents radioactive fission products from escaping the fuel matrix into the reactor coolant and contaminating it. Cladding constitute one of barriers in ‘defence-in-depth‘ approach, therefore its coolability is one of key safety aspects.

Consider the fuel cladding of inner radius r_Zr,2 = 0.408 cm and outer radius r_Zr,1 = 0.465 cm. In comparison to fuel pellet, there is almost no heat generation in the fuel cladding (cladding is slightly heated by radiation). All heat generated in the fuel must be transferred via conduction through the cladding and therefore the inner surface is hotter than the outer surface.

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 thermal conductivity of saturated water at 300°C is: k_H2O = 0.545 W/m.K
• 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 specific heat is: c_p = 5.65 kJ/kg.K
• the core flow velocity is constant and equal to V_core = 5 m/s
• the temperature of reactor coolant at this axial coordinate is: T_bulk = 296°C
• the linear heat rate of the fuel is q_L = 300 W/cm (F_Q ≈ 2.0) and thus the volumetric heat rate is q_V = 597 x 10⁶ W/m³

Calculate the Prandtl, Reynolds and Nusselt number for this flow regime (internal forced turbulent flow) inside the rectangular fuel lattice (fuel channel), then calculate the heat transfer coefficient and finally the cladding surface temperature, T_Zr,1.

To calculate the cladding surface temperature, we have to calculate the Prandtl, Reynolds and Nusselt number, because the heat transfer for this flow regime can be described by the Dittus-Boelter equation, which is:

Calculation of the Prandtl number

To calculate the Prandtl number, we have to know:

• the thermal conductivity of saturated water at 300°C is: k_H2O = 0.545 W/m.K
• the dynamic viscosity of saturated water at 300°C is: μ = 0.0000859 N.s/m²
• the specific heat is: c_p = 5.65 kJ/kg.K

Note that, all these parameters significantly differs for water at 300°C from those at 20°C.  Prandtl number for water at 20°C is around 6.91. Prandtl number for reactor coolant at 300°C is then:

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 Nusselt number using Dittus-Boelter equation

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.

To calculate the Nusselt number, we have to know:

• the Reynolds number, which is Re_Dh = 575600
• the Prandtl number, which is Pr = 0.89

The Nusselt number for the forced convection inside the fuel channel is then equal to:

Calculation of the heat transfer coefficient and the cladding surface temperature, T_Zr,1

Detailed knowledge of geometry, fluid parameters, outer radius of cladding, linear heat rate, convective heat transfer coefficient allows us to calculate the temperature difference ∆T between the coolant (T_bulk) and the cladding surface (T_Zr,1).

To calculate the the cladding surface temperature, we have to know:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm
• the Nusselt number, which is Nu_Dh = 890
• the hydraulic diameter of the fuel channel is: D_h = 13,85 mm
• the thermal conductivity of reactor coolant (300°C) is: k_H2O = 0.545 W/m.K
• the bulk temperature of reactor coolant at this axial coordinate is: T_bulk = 296°C
• the linear heat rate of the fuel is: q_L = 300 W/cm (F_Q ≈ 2.0)

The convective heat transfer coefficient, h, is given directly by the definition of Nusselt number:

Finally, we can calculate the cladding surface temperature (T_Zr,1) simply using the Newton’s Law of Cooling:

For PWRs at normal operation, there is a compressed liquid water inside the reactor core, loops and steam generators.  The pressure is maintained at approximately 16MPa. At this pressure water boils at approximately 350°C(662°F). As can be seen, the surface temperature T_Zr,1 = 325°C ensures, that even subcooled boiling does not occur. Note that, subcooled boiling requires T_Zr,1 = T_sat. Since the inlet temperatures of the water are usually about 290°C (554°F), it is obvious this example corresponds to the lower part of the core. At higher elevations of the core the bulk temperature may reach up to 330°C. The temperature difference of 29°C causes the subcooled boiling may occur (330°C + 29°C > 350°C). On the other hand, nucleate boiling at the surface effectively disrupts the stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid. As a result, the convective heat transfer coefficient significantly increases and therefore at higher elevations, the temperature difference (T_Zr,1 – T_bulk) significantly decreases.

See also:

Heat Transfer

What is Convection

What is 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.

Mechanism on Convection

In thermal conduction, energy is transferred as heat either due to the migration of free electrons or lattice vibrational waves (phonons). There is no movement of mass in the direction of energy flow. Heat transfer by conduction is dependent upon the driving “force” of temperature difference. Conduction and convection are similar in that both mechanisms require the presence of a material medium (in comparison to thermal radiation). On the other hand they are different in that convection requires the presence of fluid motion.

It must be emphasized, at the surface, energy flow occurs purely by conduction, even in conduction. It is due to the fact, there is always a thin stagnant fluid film layer on the heat transfer surface. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. Higher the rate of mass movement, thinner the stagnant fluid film layer will be and higher will be the heat flow rate.

It must be noted, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

As was written, heat transfer through a fluid is by convection in the presence of mass movement and by conduction in the absence of it. Therefore, thermal conduction in a fluid can be viewed as the limiting case of convection, corresponding to the case of quiescent fluid.

Newton’s Law of Cooling

Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton’s law of cooling, which states that:

The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings provided the temperature difference is small and the nature of radiating surface remains same.

Note that, ΔT is given by the surface or wall temperature, T_wall and the bulk temperature, T_∞, which is the temperature of the fluid sufficiently far from the surface.

See also:

Convection

Mechanism on Convection

Mechanism on Convection

 

In thermal conduction, energy is transferred as heat either due to the migration of free electrons or lattice vibrational waves (phonons). There is no movement of mass in the direction of energy flow. Heat transfer by conduction is dependent upon the driving “force” of temperature difference. Conduction and convection are similar in that both mechanisms require the presence of a material medium (in comparison to thermal radiation). On the other hand they are different in that convection requires the presence of fluid motion.

It must be emphasized, at the surface, energy flow occurs purely by conduction, even in conduction. It is due to the fact, there is always a thin stagnant fluid film layer on the heat transfer surface. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. Higher the rate of mass movement, thinner the stagnant fluid film layer will be and higher will be the heat flow rate.

It must be noted, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

As was written, heat transfer through a fluid is by convection in the presence of mass movement and by conduction in the absence of it. Therefore, thermal conduction in a fluid can be viewed as the limiting case of convection, corresponding to the case of quiescent fluid.

Convection as a Conduction with Fluid Motion

Some experts do not consider convection to be a fundamental mechanism of heat transfer since it is essentially heat conduction in the presence of fluid motion. They consider it to be a special case of thermal conduction, known as “conduction with fluid motion”. On the other hand, it is practical to recognize convection as a separate heat transfer mechanism despite the valid arguments to the contrary.

In practice, analysis of heat transfer by convection is treated empirically (by direct experimental observation). Most of problems can be solved using so called characteristic numbers (e.g. Nusselt number). Characteristic numbers are dimensionless numbers used to describe a character of heat transfer and can be used to compare a real situation (e.g. heat transfer in a pipe) with a small-scale model. Experience shows that convection heat transfer strongly depends on the fluid properties dynamic viscosity, thermal conductivity, density, and specific heat, as well as the fluid velocity. It also depends on the geometry and the roughness of the solid surface, in addition to the type of fluid flow. All these conditions affect especially the stagnant film thickness.

Convection involves the transfer of heat between a surface at a given temperature (T_wall) and fluid at a bulk temperature (T_b). The exact definition of the bulk temperature (T_b) varies depending on the details of the situation.

• For flow adjacent to a hot or cold surface, T_b is the temperature of the fluid “far” from the surface.
• For boiling or condensation, T_b is the saturation temperature of the fluid.
• For flow in a pipe, T_b is the average temperature measured at a particular cross-section of the pipe.

See also:

Convection

Convection vs Conduction

Convection vs Conduction

In thermal conduction, energy is transferred as heat either due to the migration of free electrons or lattice vibrational waves (phonons). There is no movement of mass in the direction of energy flow. Heat transfer by conduction is dependent upon the driving “force” of temperature difference.

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.

Conduction and convection are similar in that both mechanisms require the presence of a material medium (in comparison to thermal radiation). On the other hand they are different in that convection requires the presence of fluid motion.

It must be emphasized, at the surface, energy flow occurs purely by conduction, even in conduction. It is due to the fact, there is always a thin stagnant fluid film layer on the heat transfer surface. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. Higher the rate of mass movement, thinner the stagnant fluid film layer will be and higher will be the heat flow rate.

It must be noted, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

As was written, heat transfer through a fluid is by convection in the presence of mass movement and by conduction in the absence of it. Therefore, thermal conduction in a fluid can be viewed as the limiting case of convection, corresponding to the case of quiescent fluid.

Convection as a Conduction with Fluid Motion

Some experts do not consider convection to be a fundamental mechanism of heat transfer since it is essentially heat conduction in the presence of fluid motion. They consider it to be a special case of thermal conduction, known as “conduction with fluid motion”. On the other hand, it is practical to recognize convection as a separate heat transfer mechanism despite the valid arguments to the contrary.

What is 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.

Fourier’s law of Thermal Conduction

Heat transfer processes can be quantified in terms of appropriate rate equations. The rate equation in this heat transfer mode is based on Fourier’s law of thermal conduction. This law states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows. Its differential form is:

What is 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.

Heat transfer by convection is more difficult to analyze than heat transfer by conduction because no single property of the heat transfer medium, such as thermal conductivity, can be defined to describe the mechanism. Convective heat transfer is complicated by the fact that it involves fluid motion as well as heat conduction. Heat transfer by convection varies from situation to situation (upon the fluid flow conditions), and it is frequently coupled with the mode of fluid flow. In forced convection, the rate of heat transfer through a fluid is much higher by convection than it is by conduction.

Newton’s Law of Cooling

Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton’s law of cooling, which states that:

The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings provided the temperature difference is small and the nature of radiating surface remains same.

Note that, ΔT is given by the surface or wall temperature, T_wall and the bulk temperature, T_∞, which is the temperature of the fluid sufficiently far from the surface.

Example – Convection vs Conduction

See also:

Convection

Newton’s Law of Cooling

Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton’s law of cooling, which states that:

The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings provided the temperature difference is small and the nature of radiating surface remains same.

Note that, ΔT is given by the surface or wall temperature, T_wall and the bulk temperature, T_∞, which is the temperature of the fluid sufficiently far from the surface.

Convective Heat Transfer Coefficient

As can be seen, the constant of proportionality will be crucial in calculations and it is known as the convective heat transfer coefficient, h. The convective heat transfer coefficient, h, can be defined as:

The rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference.

The convective heat transfer coefficient is dependent upon the physical properties of the fluid and the physical situation. The convective heat transfer coefficient is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables influencing convection such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity.

Typically, the convective heat transfer coefficient for laminar flow is relatively low compared to the convective heat transfer coefficient for turbulent flow. This is due to turbulent flow having a thinner stagnant fluid film layer on the heat transfer surface.

It must be noted, this stagnant fluid film layer plays crucial role for the convective heat transfer coefficient. It is observed, that the fluid comes to a complete stop at the surface and assumes a zero velocity relative to the surface. This phenomenon is known as the no-slip condition and therefore, at the surface, energy flow occurs purely by conduction. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. As was written, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

A similar phenomenon occurs for the temperature. It is observed, that the fluid’s temperature at the surface and the surface will have the same temperature at the point of contact. This phenomenon is known as the no-temperature-jump condition and it is very important for theory of nucleate boiling.

Values of the heat transfer coefficient, h, have been measured and tabulated for the commonly encountered fluids and flow situations occurring during heat transfer by convection.

Example: Newton’s Law of Cooling

From: Example – Convective Heat Transfer

Detailed knowledge of geometry, fluid parameters, outer radius of cladding, linear heat rate, convective heat transfer coefficient allows us to calculate the temperature difference ∆T between the coolant (T_bulk) and the cladding surface (T_Zr,1).

To calculate the the cladding surface temperature, we have to know:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm
• the Nusselt number, which is Nu_Dh = 890
• the hydraulic diameter of the fuel channel is: D_h = 13,85 mm
• the thermal conductivity of reactor coolant (300°C) is: k_H2O = 0.545 W/m.K
• the bulk temperature of reactor coolant at this axial coordinate is: T_bulk = 296°C
• the linear heat rate of the fuel is: q_L = 300 W/cm (F_Q ≈ 2.0)

The convective heat transfer coefficient, h, is given directly by the definition of Nusselt number:

Finally, we can calculate the cladding surface temperature (T_Zr,1) simply using the Newton’s Law of Cooling:

For PWRs at normal operation, there is a compressed liquid water inside the reactor core, loops and steam generators.  The pressure is maintained at approximately 16MPa. At this pressure water boils at approximately 350°C(662°F). As can be seen, the surface temperature T_Zr,1 = 325°C ensures, that even subcooled boiling does not occur. Note that, subcooled boiling requires T_Zr,1 = T_sat. Since the inlet temperatures of the water are usually about 290°C (554°F), it is obvious this example corresponds to the lower part of the core. At higher elevations of the core the bulk temperature may reach up to 330°C. The temperature difference of 29°C causes the subcooled boiling may occur (330°C + 29°C > 350°C). On the other hand, nucleate boiling at the surface effectively disrupts the stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid. As a result, the convective heat transfer coefficient significantly increases and therefore at higher elevations, the temperature difference (T_Zr,1 – T_bulk) significantly decreases.

See also:

Convection

Newton’s Law of Cooling

Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton’s law of cooling, which states that:

The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings provided the temperature difference is small and the nature of radiating surface remains same.

Note that, ΔT is given by the surface or wall temperature, T_wall and the bulk temperature, T_∞, which is the temperature of the fluid sufficiently far from the surface.

Convective Heat Transfer Coefficient

As can be seen, the constant of proportionality will be crucial in calculations and it is known as the convective heat transfer coefficient, h. The convective heat transfer coefficient, h, can be defined as:

The rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference.

The convective heat transfer coefficient is dependent upon the physical properties of the fluid and the physical situation. The convective heat transfer coefficient is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables influencing convection such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity.

Typically, the convective heat transfer coefficient for laminar flow is relatively low compared to the convective heat transfer coefficient for turbulent flow. This is due to turbulent flow having a thinner stagnant fluid film layer on the heat transfer surface.

It must be noted, this stagnant fluid film layer plays crucial role for the convective heat transfer coefficient. It is observed, that the fluid comes to a complete stop at the surface and assumes a zero velocity relative to the surface. This phenomenon is known as the no-slip condition and therefore, at the surface, energy flow occurs purely by conduction. But in the next layers both conduction and diffusion-mass movement in the molecular level or macroscopic level occurs. Due to the mass movement the rate of energy transfer is higher. As was written, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid.

A similar phenomenon occurs for the temperature. It is observed, that the fluid’s temperature at the surface and the surface will have the same temperature at the point of contact. This phenomenon is known as the no-temperature-jump condition and it is very important for theory of nucleate boiling.

Values of the heat transfer coefficient, h, have been measured and tabulated for the commonly encountered fluids and flow situations occurring during heat transfer by convection.

Example: Convective Heat Transfer Coefficient

From: Example – Convective Heat Transfer

Detailed knowledge of geometry, fluid parameters, outer radius of cladding, linear heat rate, convective heat transfer coefficient allows us to calculate the temperature difference ∆T between the coolant (T_bulk) and the cladding surface (T_Zr,1).

To calculate the the cladding surface temperature, we have to know:

• the outer diameter of the cladding is: d = 2 x r_Zr,1 = 9,3 mm
• the Nusselt number, which is Nu_Dh = 890
• the hydraulic diameter of the fuel channel is: D_h = 13,85 mm
• the thermal conductivity of reactor coolant (300°C) is: k_H2O = 0.545 W/m.K
• the bulk temperature of reactor coolant at this axial coordinate is: T_bulk = 296°C
• the linear heat rate of the fuel is: q_L = 300 W/cm (F_Q ≈ 2.0)

The convective heat transfer coefficient, h, is given directly by the definition of Nusselt number:

Finally, we can calculate the cladding surface temperature (T_Zr,1) simply using the Newton’s Law of Cooling:

For PWRs at normal operation, there is a compressed liquid water inside the reactor core, loops and steam generators.  The pressure is maintained at approximately 16MPa. At this pressure water boils at approximately 350°C(662°F). As can be seen, the surface temperature T_Zr,1 = 325°C ensures, that even subcooled boiling does not occur. Note that, subcooled boiling requires T_Zr,1 = T_sat. Since the inlet temperatures of the water are usually about 290°C (554°F), it is obvious this example corresponds to the lower part of the core. At higher elevations of the core the bulk temperature may reach up to 330°C. The temperature difference of 29°C causes the subcooled boiling may occur (330°C + 29°C > 350°C). On the other hand, nucleate boiling at the surface effectively disrupts the stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid. As a result, the convective heat transfer coefficient significantly increases and therefore at higher elevations, the temperature difference (T_Zr,1 – T_bulk) significantly decreases.

See also:

Convection