VALIDATION OF A TRANSIENT THERMAL-FLUID SYSTEMS … · 1 validation of a transient thermal-fluid systems cfd model for a packed bed high temperature gas-cooled nuclear reactor pg

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VALIDATION OF A TRANSIENT THERMAL-FLUID SYSTEMS CFD MODEL FOR A

PACKED BED HIGH TEMPERATURE GAS-COOLED NUCLEAR REACTOR

PG Rousseau*, CG du Toit* and WA Landman** *Energy Systems Research, Faculty of Engineering, North-West University,

Private Bag X6001, Potchefstroom 2520, South Africa **M-Tech Industrial (Pty) Ltd, 5 Luke Street, Potchefstroom, South Africa

Corresponding author - PG Rousseau: Fax: +27 18 299 1320 Email: [email protected]

ABSTRACT

This paper provides an overview of the theoretical basis for a new thermal-fluid systems CFD simulation model for high temperature gas-cooled reactors, contained in the Flownex software code. Flownex provides for detailed steady state and transient thermal-fluid simulations of the complete power plant, fully integrated with core neutronics and controller algorithms. The reactor model is founded on a fundamental approach for the conservation of mass, momentum and energy for the compressible fluid flowing through a fixed bed, as well as the heat transfer in the pebbles and core structures. The time-wise integration of the resulting differential equations is based on an implicit pressure correction algorithm. This allows for the use of rather large time steps making it very suitable for simulating the slow transients that can be expected to follow incidents like reactor shutdowns. The paper also compares the Flownex results for four transient tests with the measured results from the SANA test facility as well as to the results of simulations with the Thermix/DIREKT code that were done at the Research Centre Jülich. The Flownex results compare well with the Thermix/DIREKT results for all the cases presented here. Good comparison was also obtained between the simulated and measured results, except at two points within the pebble bed near the inner wall. The fact that quick computer simulation times were obtained indicates that the new model indeed achieves a fine balance between accuracy and simplicity. However, the discrepancies obtained at the two points near the inner wall, together with the fact that additional uncertainty was introduced in the original SANA test set-up by not being able to control the temperature of the outer wall, highlight the need for additional systematic tests to be performed in order to better validate the new model.

INTRODUCTION

The Pebble Bed Modular Reactor (PBMR) power plant is currently being developed by PBMR (Pty) Ltd in South Africa in association with ESKOM and other industrial partners. This high temperature gas cooled reactor (HTGR) plant is based on a Brayton gas turbine cycle with helium gas as coolant. The complexity associated with the thermal-fluid design of the cycle requires the use of a variety of analysis techniques and simulation tools. These range from simple one-dimensional models that do not capture all the significant physical phenomena to large-scale three-dimensional CFD codes that, for practical reasons, can not simulate the entire plant as a single integrated model.

One of the most prominent codes that provide a suitable compromise is the thermal-fluid systems CFD or network simulation code Flownex [1]. Flownex allows detailed steady state and transient thermal-fluid simulations of the complete power plant, fully integrated with core neutronics and controller algorithms. The reactor model contained in the code is based on the fundamental equations for the conservation of mass, momentum and energy for the compressible fluid flowing through a fixed bed, as well as the equations for the conservation of energy for the pebbles and core structures.

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Detailed comparisons of results obtained with the reactor model and measured results obtained from the 1996 SANA test facility [2],[3], have recently been presented [4]. These included various steady-state tests, covering a range of temperatures as well as two different fluids and different heating configurations. The good comparison obtained between the simulated and measured results showed that the systems CFD approach sufficiently accounts for all of the important phenomena encountered in the quasi-steady natural convection driven flows that will prevail after critical events in a reactor. However, results have not yet been presented for transient events. The purpose of this paper is to provide an overview of the theoretical basis and present the simulated results for transient tests that were also carried out on the SANA test facility. The Flownex results are once again compared to the measured results as well as to the results of simulations carried out at Forschungszentrum Jülich as part of the SANA project.

THEORETICAL OVERVIEW

Reactor geometry. Figure 1 shows a simplified section through a part of what can typically be expected in the reactor geometry. The fuel pebbles are located in the annular volume formed between the central reflector and the core structures. The gas is fed into a ring-shaped inlet manifold. From there it flows up through vertical riser channels situated at discrete intervals around the circumference. The riser channels are intersected at the top by horizontal inlet slots that feeds the gas inward and into the pebble bed at the top. Besides the riser channels and inlet slots, the core structures can also contain vertical control rod channels. The control rod channels are also situated at discrete intervals around the circumference, but alternating with the riser channels. Note that the gas flowing in the riser channels does not mix with the gas contained in the control rod channels. The main core structures are typically surrounded by an annular gas-filled cavity contained within the core barrel. The core barrel in turn is contained within the reactor pressure vessel (RPV).

Figure 1 Section through part of a typical reactor geometry.

Horizontal

inlet slots

Control rod

channels

Central

reflector

Pebble bed

Vertical

riserchannels

Core barrel

annulus

Core barrel

Gas inlet

manifold

Core

structures

3

Simulation methodology. The simulation philosophy on which the model is based is to achieve a fine balance between accuracy and simplicity and to guard against simply developing another detailed CFD code that does not allow quick integrated plant simulations. This means that the simplest possible model was derived that can sufficiently account for all of the important phenomena.

The first simplification is that the model is based on a two-dimensional axi-symmetric coordinate system rather than a full three-dimensional cylindrical coordinate system. This implies that all variations in geometry or material properties around the perimeter of the reactor will be spread evenly around the circumference to form a material with constant properties at each given height and radius. For instance, the inlet slots situated at discrete intervals around the perimeter is represented by a material with representative unidirectional radial flow permeability. Similarly the discrete vertical control rod channels are represented by a material with representative unidirectional axial flow permeability. The pebble bed in turn is represented by a porous medium with multidirectional porosity rather than a unidirectional permeability. Note however, that the magnitude of the permeability or porosity may vary between control volumes in both the radial and axial directions. By employing this two-dimensional approach with representative permeability or porosity, all of the desired phenomena may be simulated, although in less detail than would be the case in a traditional CFD code. However, the reduction in the size of the required computational grid is substantial thus resulting is much faster simulation times.

Figure 2 Integrated network representation of the pebble-bed and core structure solids and flow paths

Governing differential equations. The reactor model is based on the fundamental equations for the conservation of mass, momentum and energy for the compressible fluid flow, as well as the equations for the conservation of energy for a pebble, energy transfer between the surfaces of adjacent pebbles, and the solid materials comprising the core structure. Through a rigorous analysis [4] the equations can be reduced and recast in a form that is suitable for incorporation in a network

Pebble bed

void volumes

Pebble bedflow resistance

elements

Inlet slot flowresistance

elements

Riser channelflow resistance

elements

Inlet manifoldPebble surface

temperature

Pebble internalsolid mass

temperatures

Core structure

Solid masstemperatures

Conduction

elements

Effective

conductionand radiation in

pebble bed

Control rod

channel flowresistance

elements

Convective heat transfer elements

between solid and gas

4

code. This formulation of the equations results in a collection of one-dimensional nodes (control volumes) and elements (models) that can be used to construct a comprehensive multi-dimensional model of the reactor. The elements account for the pressure drop through the reactor; the convective heat transport by the gas; the convection heat transfer between the gas and the solids; the radiative, contact and convection heat transfer between the pebbles and the heat conduction in the pebble and core structure materials. A greatly simplified network presentation is shown schematically in Figure 2.

Du Toit et al. [4] have shown that the equations for the conservation of mass, momentum and energy (total specific enthalpy) for the fluid can be expressed in axi-symmetric cylindrical coordinates as

( ) ( ) ( )1

0r zru ut r r z

ερ ερ ερ∂ ∂ ∂

+ + =∂ ∂ ∂

(1)

2

2o or

r

o o

T pu V p yg u

t T r p r r

ρερ ε ε ερ ρεβ

∂ ∂∂ ∂= − − − −

∂ ∂ ∂ ∂ (2)

2

2o oz

z

o o

T pu V p yg u

t T z p z z

ρερ ε ε ερ ρεβ

∂ ∂∂ ∂= − − − −

∂ ∂ ∂ ∂ (3)

( ) ( ) ( ) ( )

( )

1 1o o r o z

r r z z sf

T Th r h u h u p rk k

t r r z t r r r z z

g u g u q

ερ ε ρ ερ ε ε ε

ερ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

′′′+ + +

(4)

Three energy equations can be distinguished in the case of the solids, i.e., the conduction in the core structures, conduction in the pebbles and the heat transfer between the surfaces of the pebbles due to contact, radiation and convection. Du Toit et al. [4] have shown that the respective energy conservation equations can be expressed in axi-cylindrical coordinates as

( ) ( ) ( ) ( )( )11 1 1 1s s s fs s

T Te r k k q q

t r r r z zε ρ ε ε ε

∂ ∂ ∂ ∂ ∂ ′′′ ′′′ − = − + − + − + ∂ ∂ ∂ ∂ ∂

(5)

( ) 2

2

1v p p

Tc T k r q

t r r rρ

∂ ∂ ∂ ′′′= + ∂ ∂ ∂

(6)

1

0 eff eff

T Trk k

r r r z z

∂ ∂ ∂ ∂ = +

∂ ∂ ∂ ∂ (7)

The variation in the porosity in the radial direction due to the influence of the walls is taken into account by the correlation of Hunt and Tien [5], whilst the resistance coefficient β due to the

pebbles is determined using the KTA Ergun equation [6]. The modified Zehner-Schlünder correlation [4] is employed to determine the effective conductivity effk , whilst the heat transfer

between the pebbles and the fluid is modelled with the aid of the effectiveness-NTU method [4].

Integrated equations. Integrating equations (1) to (4) for the fluid over a control volume and over a discrete time step leads to the following algebraic equations.

Integration of the equation for the conservation of mass gives

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )1 0

o

p p p

p r e r w z n z se w n s

o

r e r w z n z se w n s

V u A u A u A u At

u A u A u A u A

ε ρ ρα ερ ερ ερ ερ

α ερ ερ ερ ερ

− + − + − + ∆

− − + − =

(8)

5

Integrating the equations for the conservation of momentum leads to

( ) ( )

( )( ) ( )

( )( ) ( )

[ ] ( ) } ( )( )

( ) ( )

( )( ) ( ) [ ] ( )

2

2

2

12

0

o

r rp p p p p

p p r p r o o p r o oe w e w

o op p

p p

p p r e w p p r r p r o op e w

o p

o

p

p r o o p p r e w p p r re w p

o p

u u V pA r A T T A p p

t T p

Vg A y y u A r A T T

T

pA p p g A y y u A r

p

ρε ρ α ε ε

ρε ρ ε ρ β α ε

ε ε ρ ε ρ β

− ∆ + − + − ∆

+ − + ∆ + − −

+ − + − + ∆ =

(9)

( ) ( )

( )( ) ( ) ( ) ( )

[ ] ( ) } ( )( )

( ) ( )

( ) ( ) [ ] ( )

2

2

2

12

0

o

z zp p p p

p p z p z o o p z o on s n s

o op p

p p

p p z n s p p z p z p z o on s

o p

o

p z o o p p z n s p p z p zn s

o p

u u V pA z A T T A p p

t T p

Vg A y y u A z A T T

T

pA p p g A y y u A z

p

ρε ρ α ε ε

ρε ρ ε ρ β α ε

ε ε ρ ε ρ β

− ∆ + − + − ∆

+ − + ∆ + − −

+ − + − + ∆ =

(10)

The integration of the equation for the conservation of energy for the fluid gives

( ) ( ) ( )( ) ( ){

( ) ( ) ( ) ( )

( )

( ) ( ) ( )1

oo o

p p o p p o p p pp p

p p o r e o r we w

o r e o r w o z n o z s ee w n se

w n s p p r r z z p

w n s p

o r e o re w

h h p pV V h u A h u A

t t

Th u A h u A h u A h u A k A

r

T T Tk A k A k A g u g u V

r z z

h u A h u

ε ρ ε ρ εα ερ ερ

ερ ερ ερ ερ ε

ε ε ε ε ρ

α ερ ερ

− −− + −

∆ ∆

∂ + − + − −

∂

∂ ∂ ∂ + − + − +

∂ ∂ ∂

+ − − ( ) ( ) ( ){

( )

( ) }

w o r e o r w o z ne w n

o z s e w n sse w n s

o

p p r r z z sf pp

A h u A h u A h u A

T T T Th u A k A k A k A k A

r r z z

g u g u V q V

ερ ερ ερ

ερ ε ε ε ε

ε ρ

+ − +

∂ ∂ ∂ ∂ − − + − +

∂ ∂ ∂ ∂

′′′− + =

(11)

Terms with the superscript o refer to the previous time step, whilst terms without a superscript refer to the current time step. α is weighting factor between the previous and the current time steps and can vary between 0 and 1. When 0α = the scheme becomes fully explicit, with 1α = it becomes fully implicit and when 0.5α = the time integration is equivalent to that of the so-called Crank-Nicholson method. For 1α = the method is first-order accurate in time, whilst for 0.5α = it is second-order accurate. For -valuesα close to 0.5 the scheme sometimes becomes unstable and an

-valueα of 0.6 offers a good compromise between accuracy and stability [7]. Integrating equations (5) to (7) for the solids and the pebbles also leads to the following algebraic equations.

6

The equation for the conservation of energy for the solids gives

( )[ ] [ ]

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 1 1

1 1 1 1

1 1

o

p p

p s e s w

e w

s n s s s

n s e

s w s

w

e e T TV k A k A

t r r

T T Tk A k A k A

z z r

T Tk A k

r z

ρ ρε α ε ε

ε ε α ε

ε ε

− ∂ ∂ − + − − + − ∆ ∂ ∂

∂ ∂ ∂ − − + − + − − − ∂ ∂ ∂

∂ ∂ + − − − ∂ ∂

( )

( )( )

1

1

o

n s s

n s

p fs s p

TA k A

z

q q V

ε

ε

∂ + − ∂

′′′ ′′′= − +

(12)

The integration of the equation for the conservation of energy for a pebble results in

( )1

o

p pp p

p p e p w

e w

o

p e p w p p

e w

e e T TV k A k A

t r r

T Tk A k A q V

r r

ρ ρα

α

− ∂ ∂ + − + ∆ ∂ ∂

∂ ∂ ′′′+ − − + = ∂ ∂

(13)

and finally the equation for the transfer of heat between adjacent pebbles becomes

0eff e eff w eff n eff s

e w n s

T T T Tk A k A k A k A

r r z z

∂ ∂ ∂ ∂ − + − = ∂ ∂ ∂ ∂

(14)

The numerical formulation of the equations is based on a staggered grid approach with the pressures, densities and temperatures defined at cell (control volume) centers (nodes) and the velocities are defined at cell boundaries (elements). A computationally effective segregated implicit pressure correction method [7] is employed to solve the resulting equations.

SANA TEST FACILITY AND COMPUTATIONAL MODEL

Having developed the theoretical basis for the new systems CFD model, it is important to prove its validity by comparing the numerical results with that of experiments. Of particular importance is the prediction of the pebble temperatures obtained in the case of power transients. Although limited, such data does exist in the form of the experimental results obtained from the SANA test facility [2],[3]. The SANA test facility was installed at the Research Centre Jülich in Germany specifically to investigate the heat transport mechanisms inside the core of a high temperature gas cooled reactor (HTGR). Besides the physical tests, simulations were also conducted at Jülich with two software models namely TINTE and Thermix/DIREKT, both of which allow both steady state and transient simulations. Results of these simulations compared well with measurements.

SANA test facility. The test facility consisted of a heated pebble bed inside a furnace to simulate the thermal conditions of such a HTGR-core. Different heater configurations were possible but Figure 3 shows a schematic of the test facility with a single central heating element. The diameter of the pebble bed is 1.5 m and the height is 1.0 m. The overall height of the facility is 3.2 m and the maximum heating capacity of the single central heating element is 35 kW. The top and bottom of the facility was well-insulated while the outside of the furnace was open to atmosphere. More than 50 steady-state as well as some transient tests were carried out on the facility. In these experiments all the main parameters of a pebble bed were varied, such as pebble material, pebble diameter, gas type, heating power and heating geometry.

7

Figure 3. Schematic of SANA test facility taken from [2].

For the tests conducted with the 60 mm diameter graphite pebbles, measurements were taken of the pebble temperatures at different radial positions close to the bottom of the pebble bed (height 90 mm) as well as at the center (height 500 mm) and top (height 910 mm) as shown in Figure 4.

Figure 4. Schematic of the temperature measuring points on the SANA test facility taken from [2].

8

Figure 5. Schematic of the discretization scheme used in all of the Flownex simulations.

Computational grid. In all of the Flownex simulations the pebble bed was discretised into control volumes with equal heights of 100 mm and equal radial widths of 29.6 mm. Half control volumes were employed at all the boundaries as shown in Figure 5. The bottom and top boundaries were assumed to be adiabatic since in practice it was very well insulated. The heat input to the heating element was distributed uniformly at the inner boundary and no allowance was made for conductive heat transfer within the material of the heating element sheath. At the outer boundary a 5 mm thick stainless steel wall with heat conduction and heat capacity was specified, connected to a fixed ambient temperature through convective heat transfer elements. Due to the lack of detailed information from the test reports, the ambient temperature was simply fixed at 26 °C for all cases while values for the convective heat transfer coefficient were fixed between 20 and 25 W/m2K. These values were estimated based on inspection of the steady-state test results. This highlighted a shortcoming in the original test set-up namely that the temperature of the outer wall could not be controlled and therefore additional uncertainty is introduced in all of the measured results.

In all of the cases the simulations were based on 60-second time-steps, which provided a time-step independent solution. The computer simulation time for obtaining the steady-state initial values were all in the region of 4.5 seconds and the transient simulations for a sixty-hour period took in the order of 450 seconds on a notebook computer with a 1.6 GHz Centrino processor and 512 Megabytes of memory.

RESULTS

This section will present comparisons between results produced with Flownex and measured results obtained with SANA facility as well as comparisons between the results of the Flownex and Thermix/DIREKT computational models for four transient cases. In all four cases the furnace contained graphite pebbles with 60 mm diameter, which is naturally of particular interest to the PBMR application. Heating was done within the central electrode along the full height of the

Adiabatic boundary

Adiabatic boundary

Fix

ed

am

bie

nt

tem

pera

ture

bo

un

dary

Fix

ed

heat

inp

ut

bo

un

dary

Top

Centre

Bottom

9

pebble bed. The first two cases involve a linear ramp-down of the power input, one with Helium as coolant and the other with Nitrogen. The other two cases involve an instantaneous step-up of the power input, again one with Helium as coolant and the other with Nitrogen. In all cases the calculated or measured temperature distribution at different radial positions at the center along the height of the pebble bed (i.e. height 500 mm) is compared.

Ramp-down of power input: The transient starts with a steady-state condition at a nominal power input of 30 kW. The power input is linearly ramped down to 10 kW over a period of 50 hours, i.e. at 0.4 kW/h, after which it is kept constant to eventually again reach steady-state conditions.

Case 1. Ramp-down of power input from 30 kW to 10 kW in 50 hours with Helium as coolant. Figure 6 shows the comparison between the measured and simulated temperatures for different radial positions in the center along the height of the pebble bed while Figure 7 shows the comparison between the simulated Thermix/DIREKT and Flownex results.

From Figure 6 it can be seen that the Flownex results compare well with the measurements except for two points along the radius i.e. at radius = 10 cm and radius = 22 cm. Although the general trend over time is correct there seems to be an offset of approximately 60 °C throughout. Note that this offset is not only observed during transients but also at steady state conditions at the start of the transient. This difference therefore does not indicate a discrepancy in the thermal mass calculation or in the time-dependent numerical integration, but rather in the calculation of the effective thermal resistance in the radial coordinate direction.

However, the Flownex results compare very well with the Thermix/DIREKT results at all of the points, including the two points mentioned above. This raises the suspicion that the measuring points may perhaps have been positioned differently in the test facility, for instance one ball diameter to the inside of where it was supposed to be. Although it seems unlikely that such an error could have been made, there is unfortunately no way to verify it at this time.

Figure 6. Results of measured (SANA) and simulated (Flownex) pebble temperatures for Helium with a ramp-down of power input from 30 kW to 10 kW in 50 hours.

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Tem

pera

ture

(°C

)

Flownex (6.5) Sana (6.5 ) Flownex (22) Sana (22) Flownex (34) Sana (34 )

Flownex (46) Sana (46) Flownex (58) Sana (58) Flownex (70) Sana (70 )

Flownex (75) Sana (75) Flownex (10) Sana (10)

10

Figure 7. Results of pebble temperatures simulated with Thermix/DIREKT and Flownex respectively for Helium with a ramp-down of power input from 30 kW to 10 kW in 50 hours.

Case 2. Ramp-down of power input from 30 kW to 10 kW in 50 hours with Nitrogen as coolant. Figure 8 shows the comparison between the measured and simulated temperatures for different radial positions at the center along the height of the pebble bed while Figure 9 shows the comparison between the simulated Thermix/DIREKT and Flownex results.

Figure 8 shows that the initial temperatures are on average higher in the case of Nitrogen than in the case of Helium. Once again the Flownex results compare well with the measurements except at the two points mentioned earlier. In this case the Thermix/DIREKT results are much closer to the measured values at these two points. However, for all of the other points to the outside of radius = 34 cm, the Flownex results compare better with the measurements than the Thermix/DIREKT results.

Figure 8. Results of measured (SANA) and simulated (Flownex) pebble temperatures for Nitrogen with a ramp-down of power input from 30 kW to 10 kW in 50 hours.

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Te

mp

era

ture

(°C

)

Flownex (6.5) Thermix (6.5 ) Flownex (22) Thermix (22) Flownex (34) Thermix (34 )

Flownex (46) Thermix (46) Flownex (58) Thermix (58) Flownex (70) Thermix (70 )

Flownex (75) Thermix (75) Flownex (10) Thermix (10)

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Tem

pera

ture

(°C

)

Flownex (6.5) Sana (6.5 ) Flownex (22) Sana (22) Flownex (34) Sana (34 )

Flownex (46) Sana (46) Flownex (58) Sana (58) Flownex (70) Sana (70 )

Flownex (75) Sana (75) Flownex (10) Sana (10)

11

Figure 9. Results of pebble temperatures simulated with Thermix/DIREKT and Nitrogen respectively for Helium with a ramp-down of power input from 30 kW to 10 kW in 50 hours.

Step-up of power input: In the following cases the transient starts with a steady-state condition at a nominal power input of 10 kW, which is then instantaneously stepped up to 25 kW.

Case 3. Instantaneous step-up of power input from 10 kW to 25 kW with Helium as coolant. Figure 10 shows the comparison between the measured and simulated temperatures for different radial positions at the center along the height of the pebble bed while Figure 11 shows the comparison between the simulated Thermix/DIREKT and Flownex results.

Figure 10. Results of measured (SANA) and simulated (Flownex) pebble temperatures for Helium with an instantaneous step-up of power input from 10 kW to 25 kW.

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Te

mp

era

ture

(°C

)

Flownex (6.5) Thermix (6.5 ) Flownex (22) Thermix (22) Flownex (34) Thermix (34 )

Flownex (46) Thermix (46) Flownex (58) Thermix (58) Flownex (70) Thermix (70 )

Flownex (75) Thermix (75) Flownex (10) Thermix (10)

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Tem

pera

ture

(°C

)

Flownex (6.5) Sana (6.5 ) Flownex (22) Sana (22) Flownex (34) Sana (34 )

Flownex (46) Sana (46) Flownex (58) Sana (58) Flownex (70) Sana (70 )

Flownex (75) Sana (75) Flownex (10) Sana (10)

12

Figure 11. Results of pebble temperatures simulated with Thermix/DIREKT and Flownex respectively for Helium with an instantaneous step-up of power input from 10 kW to 25 kW.

In this case the results for the innermost points compares exceptionally well with the measured values. Discrepancies of as much as 100 °C are again observed at radius = 10 cm and radius = 22 cm. Fortunately, in almost all cases Flownex provides conservative results from the viewpoint of accident analyses, i.e. where the predicted pebble temperatures are higher than the measured values. The Flownex and Thermix/DIREKT results compare exceptionally well for all cases, except the inner wall temperature, where the Flownex results are much closer to those of the measured values.

Case 4. Instantaneous step-up of power input from 10 kW to 25 kW with Nitrogen as coolant. Figure 10 shows the comparison between the measured and simulated temperatures for different radial positions at the center along the height of the pebble bed while Figure 11 shows the comparison between the simulated Thermix/DIREKT and Flownex results.

These results show very much the same trends as those obtained for Helium with very good correlation between the two numerical models and Flownex providing conservative results compared to the measurements. However, in this case the Thermix/DIREKT results for the inner wall correlate better with the measured values.

For all of the cases presented above, there seem to be systematic errors between the simulated and measured values at two specific points along the radius. This either indicates that the effective thermal resistance is not modelled correctly in the region near the inner wall, which may be related to an incorrect prediction of the porosity distribution near the wall, or perhaps that the measuring probes may have been positioned incorrectly in the test facility. However, besides these two points, the Flownex results compare well with the measured values and generally provide conservative results from the point of view of safety analyses. The temperature gradients with respect to time are modelled correctly in all cases which indicates that the thermal storage effects are correctly accounted for and also provides confidence in the time-wise numerical integration scheme. In almost all cases the Flownex results compare well with that of the Thermix/DIREKT simulations.

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Te

mp

era

ture

(°C

)

Flownex (6.5) Thermix (6.5 ) Flownex (22) Thermix (22) Flownex (34) Thermix (34 )

Flownex (46) Thermix (46) Flownex (58) Thermix (58) Flownex (70) Thermix (70 )

Flownex (75) Thermix (75) Flownex (10) Thermix (10)

13

Figure 12. Results of measured (SANA) and simulated (Flownex) pebble temperatures for Nitrogen with an instantaneous step-up of power input from 10 kW to 25 kW.

Figure 13. Results of pebble temperatures simulated with Thermix/DIREKT and Flownex respectively for Nitrogen with an instantaneous step-up of power input from 10 kW to 25 kW.

CONCLUSIONS

An overview of the theoretical basis and conceptual formulation of a comprehensive reactor model to simulate the thermal-fluid phenomena of the PBMR reactor core and core structures was given. Through a rigorous analysis the fundamental equations were recast in a form that is suitable for incorporation in a systems CFD code. The formulation of the equations resulted in a collection of one-dimensional elements (models) that can be used to construct a comprehensive multi-dimensional network model of the reactor.

The time-wise integration of the resulting differential equations is based on an implicit pressure correction algorithm that offers a good compromise between accuracy and stability. This allows for the use of rather large time steps in the case of the slow transients that can be expected to follow

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Tem

pera

ture

(°C

)

Flownex (6.5) Sana (6.5 ) Flownex (22) Sana (22) Flownex (34) Sana (34 )

Flownex (46) Sana (46) Flownex (58) Sana (58) Flownex (70) Sana (70 )

Flownex (75) Sana (75) Flownex (10) Sana (10)

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Time (h)

Te

mp

era

ture

(°C

)

Flownex (6.5) Thermix (6.5 ) Flownex (22) Thermix (22) Flownex (34) Thermix (34 )

Flownex (46) Thermix (46) Flownex (58) Thermix (58) Flownex (70) Thermix (70 )

Flownex (75) Thermix (75) Flownex (10) Thermix (10)

14

incidents like reactor shutdowns and other important transient events. The fact that computer simulation times of less than 10 minutes were achieved on a standard notebook computer for all of the simulations covering a 60-hour long transient also indicates that the new model indeed achieves a fine balance between accuracy and simplicity.

Good comparison was obtained between the simulated and measured results for all cases, except at two points within the bed near the inner wall. This indicates a systematic error but unfortunately it is not entirely possible to pinpoint the causes given the data available from the original SANA tests. The Flownex results compared well with the Thermix/DIREKT results for all the cases presented here. However, the discrepancy pointed out above, together with the fact that additional uncertainty was introduced in the original test set-up by not being able to control the temperature of the outer wall, highlight the need for additional systematic tests to be performed in order to better validate the new model.

ACKNOWLEDGEMENTS

The authors wish to thank PBMR (Pty) Ltd. whose financial support made this work possible as well as M-Tech Industrial (Pty) Ltd., developers of the Flownex software.

NOMENCLATURE

A area

vc specific heat capacity

ig gravitational acceleration component

oh specific total enthalpy

k thermal conductivity

L representative length

nr

outward pointing vector normal to surface

p static pressure

op total pressure

bfq heat flux from pebbles to coolant

fbq heat flux from coolant to pebbles

fsq heat flux from coolant to solid

r radial coordinate

T static temperature

oT total temperature

t time

iu velocity component

V velocity magnitude

V volume

z axial coordinate

15

Greek Letters

β resistance coefficient due to spheres

ε porosity of bed

ρ density of fluid

ijτ shear stress component

REFERENCES

[1] VAN DER MERWE, J. & VAN RAVENSWAAY, J.P., 2003. Flownex Version 6.4 User Manual, M-Tech Industrial, Potchefstroom, South Africa.

[2] NIESSEN, H-F. & STÖCKER, B., Data sets of SANA experiment: 1994-1996, 1997. JUEL-3409, Forschungszentrum Jülich.

[3] STÖCKER, B. 1998, Untersuchungen zur selbsttätigen Nachwärmeabfuhr bei Hochtemperaturreaktoren unter besonderer Berücksichtigung der Naturkonvektion, Jüll-3504, Forschungszentrum Jülich.

[4] DU TOIT, C.G. ROUSSEAU, P.G. GREYVENSTEIN, G.P. & LANDMAN, W.A. 2004. A systems CFD model of a packed bed high temperature gas-cooled nuclear reactor. (In De Vahl Davis, G. & Leonardi, E. eds. CHT’04 Advances in Computational Heat Transfer III : Proceedings of 3rd International Symposium on Advances in Computatonal Heat Transfer held on MS Midnatsol, Norway on 19–24 April 2004. New York : Begell House. [CD-ROM], Paper 157)

[5] HUNT, M.L. & TIEN, C.L. 1990. Non-Darcian flow, heat and mass transfer in catalytic packed-bed reactors. Chemical Engineering Science, 45, 55 – 63.

[6] KUGELER, K. & SCHULTEN, R. 1989. Hochtemperaturreaktortechnik, Heidelberg : Springer-Verlag.

[7] GREYVENSTEIN, G.P. 2002. An implicit method for the analysis of transient flows in pipe networks. Int. J. Num. Meths. Engrng, 53, 1127-1143.