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FINITE DIFFERENCE ANALYSIS OF THE TRANSIENT TEMPERATURE

PROFILE WITHIN GHARR-1 FUEL ELEMENT

T. A. Annafi, A. A. Gyeabour I, E. H. K. Akaho, M. Annor-Nyarko and C. R. Quaye

Computational Science and Engineering, Department of Nuclear Engineering, School of

Nuclear and Allied Sciences, University of Ghana, P.O Box AE 1, Accra, Ghana.

ABSTRACT

Mathematical model of the transient heat distribution within Ghana Research Reactor -1

(GHARR-1) fuel element and related shutdown heat generation rates have been developed.

The shutdown heats considered were residual fission and fission product decay heat. A finite

difference scheme for the discretization by implicit method was used. Solution algorithms

were developed and MATLAB program implemented to determine the temperature

distributions within the fuel element after shutdown due to reactivity insertion accident. The

simulations showed a steady state temperature of about 341.3 K which deviated from that

reported in the GHARR-1 Safety Analysis report by 2 % error margin. The average

temperature obtained under transient condition was found to be approximately 444 K which

was lower than the melting point of 913 K for the Aluminium cladding.

Thus, the GHARR-1 fuel element was stable and there would be no release of radioactivity in

the coolant during accident conditions.

Key Words

Transient heat conduction, shutdown heat generation, residual fission power, Fission

Product decay power, finite difference method, fuel element, Cladding

1. INTRODUCTION

During the steady state operation of the reactor, temperature distribution of fuel element was

determined by thermal balance between heat generated and heat conducted to the coolant

(Akaho, Maaku, 2002). Provided there are adequate heat-removal systems, the temperature in

the core does not exceed specific safety limits, and damage to the fuel pins and other reactor

materials would be prevented (Yue-Wen, 1989).

An accurate description of the temperature distribution in the fuel element was essential for

the prediction of the behavior of the reactor component. Also, the impact of the fuel and

cladding temperature on the neutron reaction rates provide an incentive for accurate

modelling of the temperature behavior under transient as well as steady state operating

conditions (Todreas, Kazimi, 1993).

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Kreith et all (2011) showed that the numerical method is better than the analytical solution

for the estimation of the fuel temperatures because the analytical method often treats the

problem by simplifications of the reality (in cases of very complex geometry, boundary

conditions and temperature dependent thermal properties).

Transient temperature profile within the fuel pin of GHARR-1 has been solved analytically

using Bessel Functions by Adjei et all (2010); while the shutdown heat generation rates

(residual fission power and fission product decay power) after reactor shutdown due to

reactivity insertion accident has been estimated by Annafi et all (2013).

For the transient heat conduction problem considered, the residual fission power and fission

product decay power were the heat source. Also, the steady state temperature was used as the

initial temperature before shutdown.

The study presented in this paper was conducted to estimate the temperature variations

coupled with heat generation rates in GHARR-1 fuel rod using finite difference method.

2. THERMAL ANALYSIS FOR A CYLINDRICAL GHARR-1 FUEL ELEMENT

GHARR-1 is a tank-in-pool type, low power research reactor, which is under-moderated, and

with natural convection cooling. The reactor employs highly enriched uranium as fuel, light

water as moderator and coolant, and metallic beryllium as reflector. The rated power of the

non-uniform thermal neutron reactor is about 30 kW and the maximum thermal neutron flux

is 1.0 x 1012

n cm-2

s-1

with 10 irradiation sites (5 inside and 5 outside the beryllium annulus

reflector). The main components of GHARR-1 are the reactor vessel, reactor core (fuel

elements), beryllium reflectors, irradiation sites, control rod, and various detectors (Akaho et

all, 2006).

The methodology adopted in analysing transients in GHARR-1 involved solving the heat

conduction equation with heat generation at steady state by finite difference techniques. A

mathematical model was then developed for transient. The shutdown heat generation rates

considered were fission product decay and residual fission heating.

2.1 Mathematical Modelling

Fig. 1 shows the cross section of the GHARR-1 fuel element. There was no gap between the

fuel pellet and the surface of the cladding in GHARR-1 reactor and the heat generated by

nuclear fission was conducted through fuel meat to fuel cladding and to the coolant. A radial

heat conduction model was analyzed to determine the transient temperature distribution

within the fuel element immediately after shut down of the reactor. The heat distribution was

modelled based on the following assumptions that:

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a) Axial heat conduction was negligible.

b) Internal heat generations considered were residual fission heat and fission product

decay heat was uniformly distributed radially over the fuel pellet cross-section.

c) All fission product decay heat was deposited locally in the fuel meat.

d) Fuel element length to diameter ratio was large, hence axial temperature gradient was

small, and axial distribution of temperature remains constant.

e) One dimensional cylindrical model was adopted for calculating the temperature

distribution in the core.

f) Thermal conductivity of the fuel pellet and the cladding depends on the temperature.

g) Specific heat capacity, density of fuel pellet and coolant are temperature dependent.

h) The fuel meat was assumed to be homogeneous.

Fig.1 The Cross section of a cylindrical fuel element with coolant region

The general form of the spatial distribution of temperature in a fuel element with internal heat

generation under transient condition is governed by the equation (Glasstone, Sesonske, 1980)

(1)

where is the thermal diffusivity of the material in m2s

-1,

, is the density of the

fuel in gm-3, is the thermal conductivity in W/mK, is the specific heat capacity of the

fuel in Jm-3

K-1

and is the rate of heat generation after shutdown in W.

But

where

is the residual fission power and is the fission product decay power estimated as

(2)

where Po is operating power in W and t is the time after shutdown in s

and

(3)

where Po is operating power, t is the time after shutdown; C and y are constants with values

provided in (Annafi et all, 2013)

For a cylinder, equation (1) becomes

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(4)

The convective heat transfer from the clad surface (at ) is governed by

(5)

where is the mean bulk temperature (K), is the clad outer surface temperature (K),

is the thermal conductivity of the clad (Wm-1

K-1

), is the outer clad radius (m) and h is the

convective heat transfer coefficient of the coolant (Wm-2

K-1

).

The boundary conditions employed in solving the above equation are

1.

(6)

2.

(7)

Where is the inner clad radius (m).

For the clad region, the heat conduction takes place according to

, (8)

where , represent the density (gm-3

), specific heat capacity (Jm-3

K-1

) and

thermal conductivity (Wm-1

K-1

) respectively for the clad region.

2.2 Numerical Analysis

Changing the equation into a more suitable form for finite difference solution by involving a

new variable such that

,

Then

(9)

and

(10)

By putting equations (9) and (10) into (4), the heat conduction within the fuel becomes

(11)

The system was treated as a set of grid points and the temperature represented as nodal

points. The dimension of each cell in the radial direction is , and the volume is where

is the cross-sectional area of the cell (Waqar et all, 2009).

The temperature at various nodes and time steps for the fuel region is

(

)

( )

(12)

where ,

, j means j-th time step.

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Thus,

( )

(13)

where F0 is a dimensionless number known as the Fourier number or modulus of the fuel.

Equation (13) can be written in the form

(14)

where

, ,

(15)

The implicit finite difference form of equation (8) which represents the temperature at each

node and time step within the clad also takes the form

( ) (16)

This implies

( ) (17)

where is the Fourier modulus of the clad

Equation (17) can be written as

(18)

where

, ,

Fig. 2 One dimensional nodal points in the fuel, cladding and water regions

Fig. 2 represents the one dimensional grid overlay of the GHARR-1 fuel element. The fuel

region was divided in to ten equal nodes having width Δ , while the clad region was divided

into three nodes with width Δ , in the radial direction.

At the fuel centreline (for i = 1), equation (4) revealed that the second term in the RHS would

become infinite which is unacceptable (i.e. the second term can be written as

)

From the first boundary condition, at ,

Therefore,

(19)

Applying L’Hopital’s rule to the centre condition,

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(20)

Putting equation (20) into equation (4),

(21)

In finite difference form, equation (21) becomes

( )

(22)

Using mirror-image technique at the centre of the fuel (r= 0), equation (22) becomes

(23)

Since

Equation (23) was re-written as

(24)

Similarly, putting into equations (13) and into (17) respectively and

applying the specified boundary conditions to obtain an expression in the form

(25)

At the fuel-clad interface ( , applying the derivative boundary condition gives

(26)

Using the implicit technique, equation (26) becomes

[

]

(27)

This can be written as

(28)

where [

],

and

At the clad-coolant interface (for

(29)

where is the bulk coolant temperature (K)

Using the conditions

i.e. (30)

where and are inlet and out outlet coolant temperatures respectively.

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(31)

where is the density of water (gm-3

), is the heat transfer coefficient of water (Wm-2

K-1

),

is the channel height through which water is flowing (m), is water velocity (ms-1

) and

is the width of fuel plate.

This implies,

(32)

Putting equation (30) into (32) and rearranging gives

(

)

(33)

Substituting equation (33) into equation (29) and simplifying, we obtain

*

+

(34)

Applying the implicit technique, we obtain

*

+

(35)

Equation (35) can be written as

(36)

where *

+,

and

The set of simultaneous difference equations (equations (24), (25), (27) and (35)) were

expressed in matrix notation as follows

[

]

[

]

[

]

(37)

where the blank spaces in the matrix represent zeros.

The matrix above can be written as

(38)

where = Ti,j+1 are the unknown transient temperatures (K) to be determined and d is a

column vector of known values that contains the heat source term (J).

For each time step, was found by solving the tridiagonal system since all the

coefficients , , and are known.

The algorithm for the numerical computation of transient temperature distribution with heat

generation rates was developed and a computer program implemented and simulated in

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Matlab version R2013a with Symbolic Math Toolbox. The Matlab code was implemented on

a 64 bit computer, with quad core processor, calibrated using published data and the output

verified and validated by manual calculations (Yue-Wen, 1989).

3. RESULTS AND DISCUSSION

The simulation data of the solution analysis conformed to that reported in the final safety

analysis report (Akaho et all, 2006). A uniform linear power of 380 Wm-1

, heat transfer

coefficient of 513 Wm-2

K-1

and inlet coolant temperature of 303 K were used for the accident

scenario considered.

Table 1 Simulation data for transient heat conduction within GHARR-1 fuel element

Fig. 3 represents the steady state temperature profile with a low linear power of 380 W/m for

GHARR-1. The initial inlet and outlet coolant temperatures assumed where 303 K and 332 K

respectively. The steady state temperature was found to be 342 K and the outer clad surface

temperature, 313 K. The steady state temperature was used as the initial radial temperature

during the transient analysis.

The temperature drop from the centre of the fuel to the surface of the clad was found to be

approximately 0.001 K. The small difference in temperature within the fuel element was

because there was no gas gap between the fuel rod and the cladding of GHARR-1.

Fig.3 Steady State Temperature Distribution within GHARR-1 fuel element

Fig. 4 represents the radial distribution up to 1500 s after reactor shutdown. The plots indicate

a parabolic thermal decay pattern from the center of the fuel to the clad-coolant surface. The

temperatures recorded both at the centerline and clad surface increased with time due to the

large value of the shutdown heat sources incorporated. The temperature range at the fuel

centre line (r = 0) was between 342 K and 448 K. At the clad-coolant interface, the

temperature recorded was between 341 K to 444 K. The increase in temperature was as a

result of heat being removed from the core at a slower rate compared to the rate at which heat

was produced.

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Furthermore, the residual fission and fission product decay heat rate generated after shutdown

accounted for the temperature rise.

Fig.4 Radial temperature distribution for up to 1500 s after shutdown

Fig.5 Radial Transient Temperature Profile between 1000 s and 2000 s

Fig. 5 shows radial temperature distribution between 1000 s and 2000 s. At t = 1500 s, the

temperature rose to a maximum of about 451 K at the centre and 447 K at the clad surface

after which decreased to approximately 450 K and 448 K at the centreline for simulated

times of t = 1750 and t = 2000 s respectively.

Fig.6 Plots of transient temperature distribution against shutdown time

Fig. 6 shows the transient temperature distribution with respect to shutdown time.

Immediately after shutdown, the centerline temperature, increased greatly up to temperature

of about 450 K at 800 s before decreasing steadily with time. Node 1 represents the fuel

centerline temperature; node 5, the temperature distribution within the fuel; node 11, the fuel-

clad interface temperature; node 12 shows the temperature distribution within the clad region

and node 14 represents the clad-coolant interface temperature profile.

4. CONCLUSIONS

Transient heat conduction models for GHARR-1 fuel element have been developed.

Discretization of the fuel pellet and cladding has been defined for ten and three nodes of

equal sizes, respectively. No gas gap was considered between the fuel rod and clad. Thermal

analysis of the cylindrical GHARR-1 fuel element was carried out using finite difference,

solution algorithm developed and Matlab code implemented.

The temperature profile after reactor shutdown showed parabolic decay pattern for every time

step simulated. Temperature increased at both the centreline and at the clad surface initially

because the rate of heat production was greater than heat removal from the system due to

transient. The temperatures decreased after some time (t = 1500 s) due to cooling by natural

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convection. Consequently, the heat generation rates considered decreased with time making

the temperatures recorded at clad surface decrease.

The computed results of peak centreline and cladding temperatures ranged between 332.003

K – 332.002 K and 411.600 K - 404.200 K respectively which conformed to that reported in

the final safety analysis of GHARR-1 (Akaho et all, 2006). The recorded fuel temperature of

approximately 411.6 K was below the melting point of 913 K of the U-Al alloy. Thus, for the

transient scenario considered, the GHARR-1 can be adjudged to be operating in safe mode to

forestall the incident of release of radioactive in to the coolant.

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