Heat Generation in Fuel Elements and Temperature Distributions

Lecture - 2

Contents of Lecture 2 -1

• How fuel is arranged in reactors? • What is heat generation shape in the core radially & axially? • What is heat generation rate in a fuel assembly? • Components of a fuel pin • What is heat generation rate with in a fuel pin? • Why do we need to know temperature distribution with in a

fuel pin? • Why 1-D conduction is generally assumed in a fuel pin? • Concepts of thermal resistance for estimating temperature

distribution • How to estimate temperature distribution with in bare fuel,

clad & gap conductance?

Contents of Lecture 2 -2 • Differential equation for conduction and its solution

• Definitions such as Q’’’, Q’’, Q’, LHGR, Kdo, power density, specific power, burn up etc.

• Variation of thermo physical properties of UO2 with temperature & burn up

• Estimation of temperature distribution with temperature dependent properties

• How to make use of concept of kdo

• Effect on temperature distribution of the assumption of Uniform vs depressed power generation

• Application of differential equation in 2-D & 3-D and corresponding boundary conditions

• Boundary conditions for surface heat generation and change of material

• Is uniform volumetric heat generation in a fuel pin conservative

Fuel arrangement in a reactor core • Fuel in a water reactor is arranged in form of cylindrical rods.

These rods are arranged in form of clusters. One cluster is called a fuel assembly. The core consists of several fuel assemblies. These assemblies are arranged in different geometrical forms in a core. These forms could be rectangular or hexagonal or triangular. In some cores each fuel assembly is enclosed in a shroud. The coolant flows through the shroud over fuel rods. In the process it collects heat from the fuel rods and gets heated up. The coolant not flowing through shrouds (may be flowing in between the shrouds) and thus not participating in the heat removal from the fuel is known as bypass flow. This is about 2 to 3 percent of the total flow. Heat is generated in the fuel pins/rods due to a nuclear chain reaction being maintained in the core

What is heat generation shape in the core radially?

• Fission of an atom produces heat at a very short distance from fission location. For heat generation estimation it is assumed at the same location where fission is taking place. Rate of fission depends on the neutron flux population present. Consequently it is assumed that rate of heat production is proportional to the flux causing fission. In a thermal reactor the most of the fissions are caused by thermal neutrons of energy about 0.023 ev. Thus the shape of heat production rate is proportional to thermal neutron flux present.

• The unit of thermal flux is neutrons/ .s (neutron density * velocity)

2m

Heat generation rate in a fuel rod/pin

• The heat generation in a fuel rod is proportional to the thermal neutron flux. It is a cosine function axially and follows a Bessel function radially.

• In a fuel pin thermal neutrons are absorbed and fast neutrons are generated. Consequently thermal neutron distribution takes a dip at the center due to absorption and fast neutrons peaks at the center for a symmetrical case

• For a fuel assembly the distribution of neutron flux is similar due to similar reasons

Shape of radial & axial thermal flux in reactor core

• Radial Shape in a finite cylinder

• Axial Flux Shape

)/405.2(00 RrJ

)/cos(0 Lz

Fuel Pin Bare fuel: consists of fuel pellets where heat is generated

Fuel clad: zircaloy clad to keep the fuel in coolable geometry

Gap between fuel & Clad

Fuel Pin

Heat generation rate in fuel, clad, gap in a thermal reactor

Why do we need to know temperature distribution with in a fuel pin?

• We need to know that fuel does not melt for safety and other considerations. For this we need to know the temperature distributions in the fuel pin. The melting point of , the commonly used fuel in thermal reactors is around 2800 C. To know this temperature we need to solve 1-D differential equation for heat conduction in radial direction with convective heat removal by coolant as one boundary condition.

2UO

Why 1-D conduction equation?

• The temperature gradients in the radial direction are of the order 1000 C in about 0.6 cm as compared the gradients in axial direction which are of the order of 50 to 70 C in about 6 m. Consequently heat conduction in axial direction can be neglected as compared to radial direction. If the heat generation is symmetric the heat flow in the third direction can also be neglected.

Thermal Resistances

• For a cylindrical hollow slab

• For a rectangular slab

)2/()/ln( 12 lkrrQ

T

)/(/ kAlQT

These resistances can be combined in series or parallel just as in electrical network

Convective Heat Resistance

)/(.1)/( hAQT

Differential heat conduction equation in cylindrical coordinates

0)/(

/1

Q

r

rTkrr

Boundary Conditions: 1. At the center

0

r

T

2. At the surface r

TKQ

Bare fuel surface temperature is determined from coolant side

Initial (steady conditions)Conditions (BCs?)

0/)/(/1 QrrTkrr

Clad surface temperature Outer

Clad surface temperature

Fuel Surface Temperature

Fuel C/L Temperature

Significance of Gap Conductance?

• Gap Conductance increases significantly not only the fuel center line temperature but also the fuel average temperature. In other words it increase the fuel stored heat above the clad temperature

Solution of differential equation (Constant thermal conductivity)

0/)/(/1 QrrTkrrAssuming K constant and integrating we get

02

1

2

CQr

rTKr

Integrating again

At r=0 temperature derivative is zero hence 01 C

04

2

2

CrQ

KT

2C is equal to 4/)( 2aQkTs

)4/()(22 KraQTT isi

How tp find fuel surface temperature?

• Fuel surface temperature =

Coolant temperature +Temperature drop in the clad + temperature drop in fuel –clad gap

lrrQTT iocoolants 2/()/ln(

Some important definitions -1

• Linear Heat Generation Rate (LHGR): is the total amount of heat generated in fuel per unit length of fuel.

• is the volumetric source of heat generation in the fuel.

• Surface heat flux

Q

2* rQLHGR

LHGRQ

)/( DQQ

Some important definitions - 2

• Power density is the amount of heat generated in a reactor per unit volume of the reactor

• Specific power is the amount of heat generated per unit weight of the fuel

• Burn up is the amount of energy extracted per unit weight of the fuel

• For the same linear heat generation rate does the center line temperature depend on the fuel radius for the same fuel surface temperature?

Thermal Conductivity of UO 2

Solution of differential equation (Variable thermal conductivity)

0/)/(/1 QrrTkrr

0)/(/1)/)(/()/( 22 rTKrrTrKrTK

r

T

T

K

r

K

For solving the differential equation by finite difference methods the above mentioned equation should be used. Derivative of thermal conductivity wrt temperature should be calculated and fed into the program

Concept

0/)/(/1 QrrTkrr

02/2

rQ

r

TKr

4//)4/( 2 LHGRrQTKcenter

surface

T

T

KdT

Term on the left can be obtained from the K graph shown earlier by plotting area under the graph against temperature. The graph so obtained can be used for estimating temperature distribution in the fuel pin

Steps involved in estimating center line temperature of fuel from surface

temperature and graph of KdT

1. Get onto x axis of the graph to the point where t=surface temperature

2. Go vertically up and find the value of KdT corresponding to surface temperature (say ONE)

3. Calculate from LHGR the value of Kdt change in the fuel pin

4. Add the value to ONE (say TWO) 5. Find the value of temperature corresponding to TWO 6. This will be equal to center line temperature Can we find the temperature distribution in the fuel

pin using this concept?

Effect on temperature distribution of the assumption of Uniform vs depressed power generation

• A computer code has to be run to find out which case is conservative. However, a qualitative argument can prove that assumption of uniform volumetric heat generation is always conservative assumption

2 2 2'''

2 2 2 2

1 10

T T T Tk q

r r r r z

2 2 2

2 2 2

''' 0T T T

k qx y z

Differential equation for heat conduction in rectangular, cylindrical & spherical coordinates

(i) Dirichlet Boundary Condition

The temperature at the boundaries is maintained

constant. This condition is normally applicable if a phase

change (boiling or condensation) is taking place which

occurs at constant temperature. This is represented by

= Constant

. (ii) Neumann Boundary Condition The heat flux at the boundary is maintained constant. This is normally the case for a steady constant heat flux process such as taking place in a nuclear fuel pin surface. This is represented by

surfaceT

'' - .dT

q k consts dx surface

(iii) Cauchy’s Boundary Conditions

This is convective surface boundary condition

where heat flux at the surface of a solid cooled by

a liquid is proportional to the temperature

difference between surface temperature and liquid

temperature.

)( TThdx

dTkq s

s

s