NHÀ MÁY ĐIỆN HẠT NHÂN

Attila Aszódi, BME NTI 13 – Power, power density – Fuel and cladding temperature – Cladding integrity and durability – Maximizing fuel burnup to enhance fuel economy – Maximal fuel tempe

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HUVINETT 2012/1

Thermal hydraulics

of nuclear reactors

Thermal hydraulics Prof Dr Attila Aszódi, BME NTI 1

Prof Dr Attila ASZÓDI

Director

Budapest University of Technology and Economics

Institute of Nuclear Techniques (BME NTI)

General considerations

Basic considerations of nuclear safety

• Nuclear Power Plant in normal operation: very low

emission, practically only thermal load to the environment.

• But high risk: Large amount of highly dangerous

radioactive material generated and accumulated in the

reactor core

• Safety objective: protect the environment from this highly

dangerous radioactive material

– Safety objective is ranked higher than electricity production!

• Specialties of nuclear fuel and NPPs:

– No real environmental risk of fresh (non-irradiated) fuel

– High risk of irradiated fuel

– High thermal density

– Reactor power possible in a very wide range (even over nominal power, in a

short pulse hundreds of nominal power – see Chernobyl)

– Cooling of irradiated fuel needed after shut-down (removal of remanent (decay)

heat in the first ~5 years in water feasible, later in gas atmosphere over hundreds

of years possible)

Design of NPPs

Design necessary not only for normal operation, but also for anticipated operational transients and for wide range of so called Design Basis Accidents.

– Beside operational systems separated safety systems required

– Due to single failure criteria multiple independent safety systems needed for the same function

Construction of an NPP is much more expensive than of a fossil power plant, where safety is not that

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Example: Bulgaria, Kozloduy NPP, VVER-1000

Example: Bulgaria, Kozloduy NPP, VVER-1000 Example: Bulgaria, Kozloduy NPP, VVER-1000

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Independent and redundant water source to avoid loss of ultimate heat sink

Basic safety functions

– Efficient control of the chain reaction and hence the

power produced.

– Fuel cooling assured under thermal hydraulic conditions

designed to maintain fuel clad integrity, thus constituting

an initial containment system.

– Containment of radioactive products in the fuel but

also in the primary coolant, in the reactor building constituting the containment or in other parts of the plant unit.

unit.

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Fuel design

• Limits in design: thermal hydraulics

(cooling) is more limiting than reactor

physics

Na, Pb), pressure, temperature

– Choice of flow velocity and other flow

parameters (turbulence, mixing, boiling)

• Many parameters have to be optimized:

– Enrichment

– Burn-up

– Power, power density

– Fuel and cladding temperature

– Cladding integrity and durability

– Maximizing fuel burnup to enhance fuel economy

– Maximal fuel temperature (to avoid fuel melting), – Maximal cladding temperature (to avoid cladding

– Maximal cladding temperature (to avoid cladding oxidation, decrease in strength)

– Maximal coolant temperature (to avoid boiling or boiling crisis)

– Other thermal limits (to keep reactivity feedbacks between reactor physical limits)

Fundamentals of heat

transport

Fundamentals of heat transport

• Heat: energy transport due to temperature difference.

• Conduction: In heat transfer, conduction (or heat conduction) is the transfer of

thermal energy between neighboring molecules in a substance due to a temperature gradient It always takes place from a region of higher temperature to a region of lower temperature, and acts to equalize temperature differences Conduction needs matter and does not require any bulk motion of matter

lower temperature, and acts to equalize temperature differences Conduction needs matter and does not require any bulk motion of matter

• Convection in the most general terms refers to the movement of molecules within

fluids (i.e liquids, gases) Convection is one of the major modes of heat transfer and mass transfer In fluids, convective heat and mass transfer take place through both diffusion – the random Brownian motion of individual particles in the fluid – and by advection, in which matter or heat is transported by the larger-scale motion of currents in the fluid In the context of heat and mass transfer, the term "convection"

is used to refer to the sum of advective and diffusive transfer.

• Thermal radiation is electromagnetic radiation emitted from the surface of an

object which is due to the object's temperature Infrared radiation from a common household radiator or electric heater is an example of thermal radiation, as is the light emitted by a glowing incandescent light bulb Thermal radiation is generated when heat from the movement of charged particles within atoms is converted to electromagnetic radiation.

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The definition of heat transfer

• Heat transfer: across the surfaces of two different

materials it is a complex physical process which combines

the three fundamental heat transport methods (conduction,

convection, radiation).

• In the technical practise: investigation of the heat transfer

process between solid wall and liquid, solid wall and

steam/gas, for example:

– Cooling the fuel rods;

– Heat transfer through a surface between the primary and secondary

side of a steam generator

• Thermal hydraulics: coupled thermal- and hydrodynamics

analysis of the reactors

Energy production and heat transfer

• Volumetric heat power rate:

Q & / ≡ ′′′

• Core volumetric power density: Q / V ≡ Q ′′′

Energy production in fuel

• Energy release in fission: 200 MeV/fission

– 92-94% discharged in fuel

– 2,5% discharged in moderator

• Calculation of volumetric heat power generation

from the thermal neutron flux

i i

Energy release in one fission of an „i” type atom [J/fission]

Energy production and heat transfer

• Thermal hydraulical design of the fuel:

∫∫∫

V A

dV r q dA n A

dV r q dz z

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Newton's law of cooling

• A related principle, Newton's law of cooling, states that the rate of heat loss of a

body is proportional to the temperature difference between the body and its

surroundings, or environment The law is

the thermal power from the solid surface to the fluid, W;

Q&

• In the technical practise the average heat transfer coefficient is often used In this case temperature difference is an averaged value

along the F surface.

• In some cases the heat transfer coefficient strongly alters along the

) ( T w − T∞

surface due to the changing influential properties This is the local

heat transfer coefficient: at the x place on the dF surface the thermal

power is the following:

where:

dF T

T Q

d & x = α x ⋅ ( w , x − ∞ ) ⋅

The aim of investigation on thermal processes is to describe the heat

transfer coefficient.

Local heat transfer coefficient – example:

The distribution of the average heat

w CFX

T T

Spacer Grid 1

ave w w GEN

T T

q

−

= & ''

α

Stockholm, 10.10.2006 S Tóth, A Aszódi, BME NTI 18

• The average heat transfer coefficient calculated according to the general definition

agrees well with the value calculated by Dittus-Boelter correlation

• The T ∞ temperature is equal to that temperature which can

be measured far from the heated surface in case of flow in half unlimited space

• In closed channel flow, if the mass flow rate is , the

density is ρ, the velocity is w and the enthalpy is h = h(T) : m

&

density is ρ, the velocity is w and the enthalpy is h = h(T) :

If the specific enthalpy is h = c p ·T and c p = constant, than

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The equilibrium equations - Boussinesq-approach

• The flow velocity field: , in Descartes coordinate

system :

• The temperature field:

• The pressure field:

( x y z )

w

w x = x , , w y = w y ( x , y , z ) w z = w z ( x , y , z )

( ) r w

( ) T ( x y z )

T

• The pressure field:

• The general differential equation of heat

( λ ), isobaric heat capacity ( c ), density ( ρ ) are constant,

( λ ), isobaric heat capacity ( c p ), density ( ρ ) are constant,

steady-state condition):

T a

T = ⋅ ∇ 2

∇

p c

a

⋅

= ρ

λ (thermal diffusivity)

The equilibrium equations - Boussinesq-approach Navier-Stokes equations (steady-state, kinematic viscosity is const.):

Conditions for the above equations:

( w ⋅ ∇ ) ⋅ w = ⋅ ∇ w − ∇ p + g − β ⋅ ∆ T ⋅ g

ρ

Conditions for the above equations:

– The gravity has only a component in z direction, whereas

– The density dependency on the temperature is only considered at the gravity term of the above equation to consider the temperature difference ( ∆T ) which causes a buoyancy force due to the density difference (β is the volumetric heat expansion or

k

g = − g ⋅ , where: k is the unit vector in z direction;

the above equation to consider the temperature difference ( ∆T ) which causes a buoyancy force due to the density difference (β is the volumetric heat expansion or

compressibility )

– ∆T is the temperature difference, T is the temperature and T ∞ is the average

temperature difference.

The hydraulic boundary layer

be divided into two regions at the vicinity of solid wall:

– first: inside the boundary layer, where viscosity is

dominant and the majority of the drag experienced by a

body immersed in a fluid is created,

– second: outside the boundary layer where viscosity can

be neglected without significant effects on the solution.

• The thickness of the hydraulic boundary layer

( δ ): if the velocity deviate from the main flow velocity

( w ∞ ) more than 1% then it is inside the hydraulic boundary

The bubbles have very slow lifting very slow lifting movement so they are suitable to visualize the flow.

Reference: Lajos Tamás: Az áramlástan alapjai (Fundamentals of fluid

mechanics – in Hungarian, CD), Műegyetemi Kiadó, Budapest, 2004.

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The hydraulic boundary layer in the flow

near a solid plane

The bubbles have very slow lifting

Reference: Lajos Tamás: Az áramlástan alapjai (Fundamentals of fluid

mechanics – in Hungarian, CD), Műegyetemi Kiadó, Budapest, 2004.

very slow lifting movement so they are suitable to visualize the flow.

The hydraulic boundary layer in pipe flow

Physical model:

• The boundary layer on the pipe wall abuts after a certain

distance from the entrance of the pipe.

• The constant velocity field before the entrance modify and a

characteristic velocity profile builds up after the suitable

distance from the entrance After this certain length this is

the so called fully developed flow.

The hydraulic boundary layer in pipe flow

Physical model:

The flow area is A , the mass flow rate is , the density is ρ.

A

m w

⋅

= ρ

The flow can be laminar along the whole pipe length, if the diameter:

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The thermal boundary layer

the hydraulic boundary layer If the wall temperature ( T w ) is

not equal to the bulk temperature ( T ∞ ), than:

– the fluid temperature is equal to the wall temperature on the solid

wall;

– approaching to the bulk from the wall the temperature approximate

the bulk temperature.

• The thermal boundary layer ( δ

t ): if the temperature deviate from the main flow temperature ( T ∞ ) more than 1% then it is inside

the thermal boundary layer.

The heat transfer NUSSELT’s equation:

• The mechanism of heat transfer: if the heat radiation is negligible,

then the energy from the wall (which has T w temperature) to the fluid

(which has T ∞ temperature) transported by heat conduction through the boundary layer at x position where the thickness of boundary layer

is δ x The δ x alters along the heated length but the viscose sub layer exists everywhere near the solid wall x x

exists everywhere near the solid wall.

• At a certain place (x) the surface heat flux can be defined by two

ways , if the local heat transfer coefficient is α x :

• The heat transfer NUSSELT’s equation:

λ is the heat conductivity of the fluid,

• The heat transfer NUSSELT’s equation is not equal to the 3rd kind boundary condition.

Methods to calculate the heat transfer coef.

• Solving the steady state equations by using certain

boundary conditions the velocity and temperature

field can be determined Using the temperature field

and the NUSSELT equation the heat transfer

coefficient can be calculated

• Measuring the heat transfer coefficient and

the generalization of the experimental data,

using them for similar flows (Similarity

theory) is possible

Similarity of heat transfer processes 1.

• Two physical processes are similar, if

– The differential equations are the same;

– The geometries are similar;

– The initial and boundary conditions of the differential equations can be transformed in the same values using proper ratios.

• The similarity theory enables to investigate an unmeasured heat transfer process using a geometrically similar and previously measured example.

• To characterize the similarity many similarity number are

• To characterize the similarity many similarity number are used in practice.

• In the followings the practically important similarity numbers are defined.

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• To define the similarity numbers let’s use the

Boussinesq-approach of the steady state equations:

• The z component of Navier-Stokes equation:

T g g p w

w w w

w

w ⋅ ∂ z + ⋅ ∂ z + ⋅ ∂ z = ν ⋅   ∂2 z + ∂2 z + ∂2 z  − 1 ⋅ ∂ − − ⋅ β ⋅ ∆

• The energy equation:

• The continuity equation:

T g g z

p z

w y

w x

w z

w w y

w w

x

w

z z y z

∂

∂ +

∂

⋅ +

∂

∂ +

∂

⋅ +

T x

T a z

T w y

T w

• Let’s transform the steady state equations into

dimensionless form using normalization The base value of

the normalization must be well measurable L is the characteristic geometrical parameter, w ∞ is the characteristic

characteristic geometrical parameter, w ∞ is the characteristic

velocity, T w is the surface temperature, T ∞ is the flow characteristic temperature!

• The dimensionless parameters:

T T

L

y

= η

L

z

= ζ

• Replacing the dimensionless parameters into the Navier-Stokes

equations and reassembling:

• Replacing the dimensionless parameters into the energy equation

β

ζ π ζ

ω η

ω ξ ω ν

ζ

ω η

ω ω

∂

∂ +

L

z z y

z

22222

• Replacing the dimensionless parameters into the energy equation

∂

∂ +

∂

⋅ +

ζ ϑ

η ϑ

ξ ϑ ζ

ϑ ω η

ϑ ω ξ

L grad ϑ

λ

α

ϑ = − ⋅ ⋅

• The dimensionless similarity parameters in the above mentioned

equations are the following:

– The Péclet number (this number shows a relationship between the

velocity field and temperature field): w L

Pe = ∞⋅

velocity field and temperature field):

– The Reynolds number (this number shows the relationship between

the inertia and drag forces):

– The Nusselt number (the criteria of the similarity of the heat

transfer):

ν

L w

– The Froude number (this number shows the relationship between

the gravity and inertia forces):

– The Archimedes number (this number shows the relationship

between the buoyancy and inertia force): ( )

Ar β W

g L

w Fr

⋅

= ∞

λ

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• Further practically used similarity numbers

– The Prandtl number defines the ratio of the thickness of the

hydraulic and thermal boundary layer:

Pe ν

– The Grasshoff number:

– The Rayleigh number:

– The Stanton number:

2

32

– The Stanton number:

• Hereafter the fluid fully fill the whole domain for all of the discussed

heat transfer cases (there is no free surface) or the solid body is fully

bounded by the fluid Therefore the Froude number do not has an

Nu St

p

ρ α

• The Nusselt number (which consist of the α ) depends on the

Reynolds-, Grasshoff- and Prandtl number, furthermore the geometry

and boundary conditions:

Nu = f (Re, Gr, Pr, Geometry, Boundary

• But: the similarity numbers are defined using the

Boussinesq-approach which has the constant material properties (ρ, λ, ν, c p ) versus the temperature The temperature dependency of the material

properties influences the velocity and temperature fields, so finally the heat transfer coefficient too.

Nu = f (Re, Gr, Pr, Geometry, Boundary

conditions)

heat transfer coefficient too.

• This effect can be considered by a Φ T correction factor:

Nu = f (Re, Gr, Pr, Geometry, Boundary conditions)· Φ T

Practical determination of αααα 1.

• The similarity theory enable to generalize the experimental

results of heat transfer If the similarity numbers are used

for the experimental results then the Nusselt number can be

expressed by the similarity numbers and their powers:

• Forced convection (the flow created by external force)

often the buoyancy effect is negligible, therefore the Nusselt

number is independent from the Grasshoff number.

T p n m

Φ Pr Gr

• Natural convection (due to inhomogeneous temperature

field the density field is inhomogeneous, which induces the

flow) here the Nusselt number is independent from the

Reynolds number

Nu = f (Gr, Pr, Geometry, Boundary conditions)· Φ T

Practical determination of αααα 2.

• The Nusselt number correlations are generally semi-empirical

equations using experimentally determined coefficient sets.

• The correlations are valid in their definition range.

• The error range of calculated α is typically ± 20…40%, at complex phenomenon that could be higher (for example boiling, condensation)!

• The values of the similarity numbers must be calculated at

reference temperature.

• All parameters must be used with correct units.

• Near the critical point of the fluid the following correlations and equations are not valid.

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Heat transfer coefficient at forced convection

• Single phase heat transfer in horizontal tubes. The

characteristic of α strongly depends on the flow mode (laminar or

turbulent)

– The averaged velocity in the Reynolds number is a value calculated

by the fluid mass flow rate ( ), the flow area ( A ) and the fluid

density (ρ):

– The characteristic length is the diameter ( D ), standard temperature is

the flow characteristic temperature ( T ∞ ), a needed additional size is

pipe length ( L ).

m &

A

m w

⋅

= ρ

&

pipe length ( L ).

• Single phase heat transfer in horizontal tubes.

• The α for laminar, hydraulically developed flow [1]:

Valid, if Re < 2300 and 0,1< Re·Pr·D/L < 10 4

(constant wall temperature)

.

T

L

D Pr Re

3/2

1 1

7 , 12 1

Pr 1000 8

Valid, if 3000 < Re < 5·10 6 and 0,1 < D/L and 0,5 < Pr < 2000

• Correlation for temperature:

where Pr W is the Prandtl number at T W wall temperature.

, ,

8 1 7

, 12

14,0

W T

12,0

T T

• Single phase heat transfer in horizontal tubes.

• The α can be calculated for turbulent flows by Dittus-Boelter formula

[3]:

Valid, if Re > 10 5 and 0,6 < Pr < 160 and L/D > 10

, where

T n

T ha 4 , 0

W

T

T n

Valid, if Re > 10 5 and 0,6 < Pr < 160 and L/D > 10

• If the tube cross section is not circular then the so called equivalent

diameter ( D e ) must be used:

• For curved pipe the heat transfer coefficient ( α R ) is higher compared

= 1 1,77

1st example – Heat transfer at the primary side of a steam generator

The diameter of heat exchanger pipe in a PWR steam generator is

10 mm The pressure of the primary side is 124 bar and the velocity of the primary water is 2 m/s The characteristic temperature of the primary water is 282 °C and the wall temperature at the primary side

is 260 °C The length of the pipe is far higher than the diameter of the pipe

Define the heat transfer coefficient between the primary water and wall!

The material properties of the water at 124 bar:

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Solution.

Heat transfer in straight tube:

Characteristic temperature:

Characteristic dimension : the inner diameter of the tube,

Material properties at temperature with linear interpolation :

C

=

D

C

PrW =

The temperature correlation:

so the flow is turbulent.

2300 10

595 , 1 10 254 , 1

01 , 0 2

003 , 1 8159 , 0 8331 , 0 Pr

W T

Solution (continuation):

For turbulent flow the following formulas must be used:

The Nusselt number:

) 64 , 1 10 595 , 1 log 82 , 1 (

1 )

64 , 1 log 82 , 1 (

1

25

102

1 , 289 003 , 1 01632 , 0 1 8331 , 0 7 , 12 1

8331 , 0 1000 10 595 , 1 8 01632 , 0

1 7

, 12 1

Pr 1000 8

325

ξ ξ

L D Pr

⋅

−

⋅ +

3/2

1

8 1 7

, 12 1

Pr 1000 8

ξ ξ

The heat transfer coefficient:

8 01632 , 0 1 8331 , 0 7 , 12 1 8

1 7

, 12

1 + ⋅   Pr3 −   ⋅ ξ + ⋅   3 −   ⋅

K D

, 0 5881 , 0 1 , 289 λ α

2nd example – Cooling a fuel rod

On the below figure a simplified sub-channel of a fuel rod

can be seen The mass flow rate of the coolant (water) in the

core is 8800 kg/s at 124 bar The number of fuel assemblies

is 349 Each fuel assembly has 126 fuel rods

The inlet water temperature is 267 °C

The outlet temperature is 297 °C

between the wall and water temperature!) The material properties of the water at 124 bar:

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Solution.

Using these signs:

the flow area ( F ) and the equivalent diameter ( D e ):

mm.

1 , 6 2

12,2 m mm

222

2

m 10 5282 , 6 mm 282 , 65 9

61 , 6 6 6

The characteristic temperature

The material properties at 124 bar , T m = 282 °C with linear

C 282 2

297 267 2

T

25

mm 282 , 65 4

9 3

61 , 6 6 4 3 6

235 , 9 9

282 , 65 4 4

co e

D

F K

F

D

out in

The material properties at 124 bar , T m = 282 °C with linear

n n

m w

kaz p

055 , 4 10 5282 , 6 99 , 755 349 126

The Reynolds number:

The Nusselt number for turbulent flow by the Dittus-Boelter formula,

if ΦT = 1 :

s F

n

np⋅ kaz ⋅ ρ ⋅ 126 ⋅ 349 ⋅ 755 , 99 ⋅ 6 , 5282 ⋅ 10

( 2,986 10 ) ( 0 , 8331 ) 513 , 0 023

, 0 Pr Re 023 ,

2,986 10

254 , 1 0,009235 055

, 4

The heat transfer coefficient:

( 2,986 10 ) ( 0 , 8331 ) 513 , 0 023

, 0 Pr Re 023 ,

0 ⋅ 0.8⋅ 0.4 = ⋅ ⋅ 5 0.8⋅ 0.4 =

=

Nu

K m

W Nu

De ⋅ = ⋅ = ⋅

= 513 , 0 32669 , 0 2

009235 , 0 5881 , 0 λ

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Temperature dependency of λ Ι.

3 11

) 273 T ( 10 1256 6 T 4 402

3824 )

compared to the metals.

The Lyon formula:

) 273 T ( 10 1256 6 T 4 402 ) T

+

= λ

UO 2 hõvezetési tényezõje a Lyon-összefüggés alapján

The integral of Lyon formula:

0

− +

⋅ +

T

λ

Lyon összefüggés integrálja

300040005000600070008000

The integral of the Lyon formula

0100020003000

The influence of Porosity (density) I.

sintering.

• 90% or more can be reached compared to the theoretical density.

• If the dimension of the pores is smaller then theoretically, the heat conductivity could be higher.

• But the gas phase fission products deforms the fuel pellet and causes over pressure in the fuel pellet The generated gas fills the pores, that is why there is

an optimal porosity for the manufacturing.

The porosity:

V

V V V

V V solid the of volume the

and V pores the of volume the

V pores the of volume the

) ( )

(

) (

V V V solid the of volume the

and V pores the of volume the

P

s p

=

) ( )

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The influence of Porosity (density) II.

λ

) 1 ( 1

⋅ +

−

=

5 , 0 1 1

The influence of oxygen/metal atomic ratio

• Both, the hyper- and hipostoichiometric ratio decreases the heat conductivity.

Pu)O 2 can be seen in

the function of the 2

the function of the

content of PuO 2 on

the right hand side.

stresses

↓

Enhancement of pellet cracking

decreases due to the increasing dimensions of cracks.

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The influence of burn up (power reactors)

structure; in the fuel porosity; material content of the fuel and in the

stoichiometric ratio of the fuel.

maximum 3%), for fast reactors this effect is higher (the 10% of

original U and Pu atoms could be burned in the nuclear fission

process).

heat conductivity.

effective heat conductivity of fuel pellets.

then the material structure of the oxide fuel modify which process

leads an increased density This increased density – which occurs in

the inner region of the fuel pellet – has an influence on the thermal

conductivity and the temperature field too.

Generation of gases (power reactors)

• Certain fission products in gas phase can leave the UO 2 fuel material at low temperature conditions.

• At high temperature when structural modification can

be occurred, significant amount of fission products in gas phase could leave the fuel material and mix with the filling gas of the fuel rod.

• The generated gas phase fission products causes

• The generated gas phase fission products causes increased pressure in the fuel rod which effect have to

be considered during the design of the fuel rods.

The melting point

The melting point of UO2 is 2840 °C.

The melting point for the oxide of uranium-plutonium

mixture in the function of plutonium content

The specific heat

The heat capacity of the fuel play an important role during the transient calculations.

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Thermal properties of different fuels

Up to melting point Average heat

Linear thermal expansion

coefficient [1/oC]

0,0000101 (400-1400 oC)

0,0000111 (20-1600 oC)

0,0000094 (1000 oC)

Structure of crystals

Below 655 oC:

a, orthorombic Above 770 oC: g, body- centered cubic

cubic centered

Thermal properties of different cladding

materials (power reactors)

Zircaloy 2 SS 316 Aluminium*

Density [kg/m 3 ] 6500 7800 2700 Melting point [ o C] 1850 1400 660 Heat conductivity at 400 o C [W/m o C] 13 23 237 (25°C) Heat capacity at 400 o C [J/kg o C] 330 580 910

Heat capacity at 400 o C [J/kg o C] 330 580 910 Linear thermal expansion coefficient[1/ o C] 5.9E-06 1.8E-05 2.31E-05

*the values can differ for different alloys

The temperature

distribution in the core

Subchannels

Hexagonal (for example VVER-440) and rectangular (for

example western PWRs) subchannels

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The temperature distribution of the fuel rod

General differential equation of transient heat conduction:

( ) ( ( ) T T ( ) r , t ) q ( ) r , t

t

t , r T

The temperature distribution into the cross

section of the fuel rod

Cladding Fuel pellet

Radial temperature distribution in the fuel rod

Gap Central bore

If L/D>10 then the axial temperature distribution can be considered as

0

0 )

( 1

2

= +

dT r

q dr

dT r dr

d r

&

λ λ

0 2

1 2 1

= +

′′′

+

= +

′′′

+

c

r q dr

dT r

c q

dr r

&

λ λ

In case of compact pellet (no central bore)

r dT

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In case of compact pellet (no central bore)

λ

r q dT

=

π λ

λ

4

4max

2 2

q R q

R q dT

r q dT

T

fo

fo T

T − fo = & ′

Using average heat transfer coefficient:

0

2

= +

r

R q c

dr

dT q

v v

2

0

2 1

′′′

=

−

= +

T

r c R r

q dT

r

c r q dr dT

ln 4

0 2

1 2 2 1

max

&

λ λ

Pellet with bore:

T

v

v v

T T

r R

R r

q dT

R

r R q R r q dT

ln 2

1

ln 2 4

2 2

2

2 2

2

maxmax

r r

R r

R R

fo v

fo

v fo

Pellet with bore:

T

fo v

fo vfo

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