the analysis of the pebble-bed modular reactor thermal ... · the analysis of the pebble-bed...

THE ANALYSIS OF THE PEBBLE-BED MODULAR REACTOR THERMAL-FLUID CYCLE USING A NETWORK CODE.

G.P. Greyvenstein*, P.G. Rousseau** and C.G. du Toit****

School for Mechanical and Materials Engineering

Potchefstroom University for CHE Private Bag X6001, Potchefstroom 2520, South Africa

Abstract

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 three-shaft Brayton cycle with helium gas as coolant. This is a complex thermal-fluid system consisting of many interacting components such as pipes, valves, heat exchangers, compressors, turbines, pumps, blowers and the nuclear reactor itself. Engineers are faced with two major challenges when carrying out the thermal-fluid design of the PBMR power plant. The first challenge is to predict the performance of all the thermal-fluid components of the plant such as pipes, valves, heat exchangers, turbo machines and the reactor. The second challenge is to predict the performance of the integrated plant consisting of all its sub-systems.

System performance predictions must be done for both steady-state and transient conditions. Steady-state analyses are required to study normal operating conditions and to generate initial values for transient analyses. Transient analyses, in contrast, are required for control studies and for studying operating procedures such as start-up, load rejection and load following, as well as when analysing accident events. 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.

Various approaches have been developed to model complete thermal-fluid systems. One approach is to build a custom computer model for a specific system layout or to use commercially available modelling tools such as EES or Aspen Custom Modeller (ACM). An approach that has gained wide acceptance is known as the network approach. With this approach models of standard components are developed that can be interconnected in any arbitrary way. The name “network” is derived from the graphical representation of thermal-fluid systems as networks of interconnected components. With the network approach the system layout can easily be reconfigured without having to change any computer code. This paper gives an overview of one of the most prominent codes that provide a suitable compromise, namely the thermal-fluid network simulation code Flownex. Flownex allows detailed steady state and transient thermal-fluid simulations of the complete power plant, fully integrated with core neutronics and controller algorithms. This is illustrated through several examples, including the PMBR power plant.

* Dean, Faculty of Engineering ** Director, Research *** Director, School for Mechanical and Materials Engineering

Potchefstroomse UniversiteitvIr Christelike Hoër Onderwys

Mechanical and MaterialsEngineering

THE ANALYSIS OF THE PEBBLE-BED MODULAR REACTOR

THERMAL-FLUID CYCLE USING A NETWORK CODE

GP Greyvenstein, PG Rousseau and CG du Toit

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SCOPE

• Pebble-bed Modular Reactor Main Power System • Flow Network or Integrated Systems CFD Approach• Implicit Pressure Correction Method• Reactor Model• Validation• PBMR Load Rejection• Conclusions

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PBMR MAIN POWER SYSTEM

Reactor

HP Turbo Unit

LP Turbo Unit

Power Turbine and Generator

RecuperatorPre-coolerInter-cooler

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PBMR

RXHPC

PBR

HPT LPT

PCIC

PT

LPC

SBS

G

GBPLPBHPB

SIV

SBSIV

PV

CWP

NIV

SBSOV

NEV

R

SCHEMATIC LAYOUT OF MPS

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PBMR: ISSUES

• Complex thermal-fluid system consisting of many interacting components.

• Performance of all the thermal-fluid components must be predicted.

• The performance of the integrated plant must be predicted.

• System performance predictions must be done for both steady-state and transient conditions.

• The system performance predictions can be done using a network or integrated systems CFD approach.

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Pump

PipeOrifice Compressor

Turbine

Reservoir

Heat exchanger Diffuser

Node NodeElement

NETWORK APPROACH

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1Reactor_Core (R)

3

Reactor_Outlet_Connection_to_Core_Outlet_Pi

pe (FP)

7 Core_Outlet_Pipe (FP)

9CM

11

HPT_Diffuser (FP)

13

HPT_Outlet_Pipe (FP)

17

Low_Pressure_Turbine (T)

CM19

LPT_Diffuser (FP)

21LPT_Outlet_Pipe (FP)

27PT_Intake (FP)

29

Power_Turbine (T)

31PT_Diffuser (FP)

35

PT_Outlet_Contraction (FP)

39

Recuperator_LP (H)RX

41

Pre-cooler_outer_bundle (H)CX

43

Pre-cooler_inner_bundle (H)CX

47

PC_Outlet_Contraction (FP)

51

LPC_Intake (FP)

53CM

55

LPC_Diffuser (FP)

57

LPC_Outlet_Pipe (FP)

61

Intercooler_outer_bundle (H)CX

63

Intercooler_inner_bundle (H)CX

67

IC_Outlet_ducting(FP)

71

HPC_Intake (FP)

73CM

75

HPC_Diffuser (FP)

77HPCe_Holes_to_Manifold (FP)

79

Maintanence_Valve (V)RD

83 RX(HP)_Inlet_Pipes (FP)

85RX

87 RX(HP)_Outlet_Pipes(a) (FP)

89

Core_Inlet_Pipes(e) (FP)

91

Reactor_Annular_Pipes (FP)

93

Reactor_Gas_Inlet_Plenum_Holes (FP)

95Control_Rod_Cooling_flow_connection (FP)

102CX

104CX

109CX

111CX

78P

113

Manifold_to_HPTe_Leakage (L)RD

78P

116RD

119HPC_Bypass_Valve_(HPB) (V)

120

LPC_Bypass_Valve_1_(LPB1) (V)

121

Gas_Cycle_Bypass_Valve_1_(GBP1) (V)

78P

133

Manifold_to_HPTi_Leakage (L)RD

78P

135

RX_bypass_valve_(1 TO 4)_(RBP1) (V)

136Reactor_Plenum_Reduction_Area (FP)

137 Reactor_Outlet_Slots (FP)

138

Control_Rod_Bypass(filled) (FP)

140

Control_Rod_Bypass(empty) (FP)

142

Leakflow_from_Control_Rod_to_Outlet_Manifold (FP)

RD

144

Inside_Cavity_Slots_to_Core (FP)RD

146Reactor_Outside_

Cavity (FP)

148 Reactor_Leak_into_Manifold (FP)

RL

150HPT_Intake (FP)

152

LPT_Intake (FP)

158

RX(LP)_to_PC_connection (FP)

160

Annular_flow_path_to_IC (FP)

162

Annular_flow_path_. . ._to_IC_core (FP)

163

Recuperator_Outlet_Pipe (FP)

165

Core_Inlet_Pipes(a) (FP)

167Core_Inlet_Pipes(b) (FP)

169

Core_Inlet_Pipes(c) (FP)

171

Core_Inlet_Pipes(d) (FP)

174

175

LPC_Bypass_Valve_Pipe2 (FP)

178


180Bypass_Valve_Pipe (FP)

181 HPCe_to_HPCi_Leakage (L)

RD

78P

182

Manifold_to_LPTi_Leakage (L)

RD

183

LPCe_to_LPCi_Leakage (L)RD

184 Casing_cooling_flow (L)RD

56185

LPCe_to_PT_casing_flow (FP)

186

PTi_to_Basket_leak (L)RD

188

188 25%_casing_cooling_flow (L)

RD190


RD

192

Basket_to_PTe_leak (L)RD

194

PT_internal_fp2 (FP)

RD

196

196 Buffer_to_gaspath_leak (L)

197

197 ECLS_upper (L)

198

ECLS_lower (L)

300

ECLS_to_PToutlet (L)RD

301

dummy

303

HPS_supply_line (FP)RD

305

PTG_Shaft_Labyrinth_Seal(a) (L)

307

307 PTG_Shaft_Labyrinth_Seal(b) (L)

309

PT_top_volume_to_PTe

(FP)

78P

310

Shaft_valve (V)RD

153312

Pipe_RXe_to_SBSi_2 (FP)

314

SBS_Isolation_valve_2(SBSV2) (V)

316

SBS_Blower2 (CM)

M

CM327 318

Pipe_SBSe2_to_RX_manifold (FP)

153319


321


323

SBS_Blower1 (CM)

M

CM327 325


326

SBS_Inline_valve_1_(SIV1) (V)

78P

329

HP_coolant_valve_1_(HCV1) (V)78

P330

LP_coolant_valve_1_(LCV1) (V)

332SBS_Labyrinth_Seal2 (L)


336

RX(HP)_Outlet_Pipes

(b) (FP)

338 RX_inlet_header_pipe (FP)

340

PC_Impingement_plate(FP)

341HPC_Bypass_Control_Valve_(HPBC) (V)

342

LPC_Bypass_Control_Valve[LPBC] (V)

343

SBS_Bypass_valve_(SBP) (V)

344

SBS_Bypass_Control_Valve_(SBPC) (V)

345


346


347

348

349

350

351

352

353 Gas_Cycle_Bypass_Valve_8_(GBP8)(V)

78P

354

HP_coolant_valve_2_(HCV2) (V)

78P

355


356


357


78P

361

78P

362

78P

363

364

HPC_Bypass_Control_Valve_Pipe (FP)

365LPC_Bypass_Control_Valve

_Pipe2 (FP)

366

LPC_Bypass_Control_Valve_Pipe1 (FP)

368

PT_1st_stg_stator (FP)

370

370 PT_1st_stg_rotor_inlet (FP)

371

371 Inlet_1st_stg_throat (FP)

373

373 25%_rim_cooling_flow (L)

RD

375


RD

376

Rim_cooling_flow (L)RD

377

OD_u_ECLS_backup_lab (L)

380

380 ID_u_ECLS_backup_lab (L)56 381

PT_internals_cooling flow

(FP)

383


RD

387

Inlet_to_SBS_Blower1 (FP)

388

SBS_Blower1_Outlet_Diffuser (FP)

390

Outlet_From_SBS_Blower1 (FP)

393


394


396


78P

1702

Basket_ventilation_flow (L)RD

1703Reactor_Outlet_Pipe_to_CCS (FP)

1705

CCS_Splitting_Restrictor (FP)RD

1707

CCS_Outlet_Pipe_to_Reactor (FP)

2Reactor_Core_Outlet

H

4

6

Reactor_Outlet_Manifold

8 10 12

HPT_Diffuser_Volume

16 18 20

LPT_Diffuser_Volume

26

PT_Inlet_Volume

28 30

34

PT_Diffuser_Volume

38RX(LP)_Inner_Vess

el_Volume40PC_Inlet_Volume

42

46

PC_Outlet_Volume

5052545658

IC_Inlet_Volume

62

66

IC_Outlet_Volume

707274

76HPC_Diffuser_Volume

78

Manifold

P82

RX(HP)_Inlet_Plenum

84RX(HP)_Core_Inlet

86

88 RX(HP)_Outlet_Plenum

90

Reactor_Inlet_Manifold

92

Reactor_Gas_Inlet_Plenum_Volume1

94Reactor_Gas_Inlet_Plenum_Volume2

96

98

Void_above_the_Reactor_Core

H

99IC_Waterside_Inlet

PT

103

105

IC_Waterside_OutletM

106

PC_Waterside_Inlet

PT110 112

PC_Waterside_Outlet

M

139141

143

Reactor_Inside_Cavity

T

145

Reactor_Upper_Volume

T

147

Reactor_Lower_Volume

T

149HPT_Intake_Volume

151

LPT_Intake_Volume

153

RX_Outer_Volume

159161

164

166

168

170172

173

176 177

179

187

187 Turbine_casing_temp

T

189PT_casing_volume

191

PT_basket_volume

195

PT_Inner_Volume

199

302

HPS_outlet_purified

MT304

Generator_Volume

306

308

308 Volume_on_top_of_PT

311

HPS Extraction

M

313315

Inlet_Volume_To_SBS_Blower2

317

320322


324

327RX_Outer_Annulus_Volume

328

333SBS_Motor_Casing_Volume2

335

SBS_Motor_Casing_Volume1

337Inlet to RX(HP)_Outlet_Pipes(b)

339

367

369

372 374

374 Turbine_rim_temp

T

378

378 u_ECLS_OD_volume

379

379 u_ECLS_ID_volume382 384

386389

SBS_Blower1_Dump_Volume

391


392395


397


398

1700 1701

1704CCS_Volume1

1706

Reactor

LP Turbo Unit

Power Turbine

Pre-cooler

Recuperator

Start-UP Blower System

HP turbo Unit

Intercooler

MPS NETWORK MODEL

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SHELL-AND-TUBE SUBNETWORK

Discretization of a section of a shell-and-tube heat exchanger

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NETWORK SOLUTION METHODS

• Methods can be broadly classified as explicit and implicit.

• Explicit methods are especially suited for solving fast transients, but stability governed by Courant number which dictates short time steps.

• Very slow when solving steady-state or slow transient problems.

• Implicit methods not governed by Courant number and are especially suited for steady-state and slow transient problems.

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NETWORK SOLUTION METHODS

• Generally much slower than explicit methods when solving fast transients.

• In the design of the PBMR plant the concern is mostly slow transients and steady-state solutions.

• The method used in the case of the PBMR is the Implicit Pressure Correction Method, which is suitable for both steady-state and transient analyses, slow and fast transients, gas and liquid flows, and for adiabatic and non-adiabatic flows.

• The method is embodied in the network or systems CFD code Flownex.

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GOVERNING EQUATIONS

• Mass conservation

( ) 0VAt A xρ ρ∂ ∂+ =

∂ ∂

• Momentum conservation

2( ) ( ) cos 02

f V VV V A p gt A x x D

ρρ ρ ρ θ∂ ∂ ∂+ + + + =

∂ ∂ ∂

• Energy conservation

0 0( ) ( ) 0h p VAh qt A x

ρ ρ∂ − ∂+ − =

∂ ∂

Fully conservative form of conservation equations:

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GOVERNING EQUATIONS

• Mass conservation

( ) 0V

t xρρ ∂∂

+ =∂ ∂

• Momentum conservation

2

cos 02 2

o o

o o

f V Vp TV p Vgt p x T x D

ρρρ ρ θ∂ ∂∂+ + + + =

∂ ∂ ∂

• Energy conservation

( ) ( ) 0o oh Vhp qt t xρ ρ∂ ∂∂

− + − =∂ ∂ ∂

For compressible gas flows:

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SOLUTION ALGORITHM

Conservation of mass:

Branch 1

1

2

j ik 1

2

J

Branch 2

Branch j Branch 1

1

2

j ik 1

2

J

Branch 2

Branch j

1

2

j ik 1

2

J

Branch 2

Branch j

Control surface

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1

2

j ik 1

2

J

Branch 2

Branch j

Control volume

Branch 1

1

2

j ik 1

2

J

Branch 2

Branch j

Control volume

Branch 1

SOLUTION ALGORITHM

Conservation of momentum:

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SOLUTION ALGORITHM

Conservation of energy:

1

2

j ik 1

2

J

Branch 2

Branch j

Control volume

Branch 1

1

2

j ik 1

2

J

Branch 2

Branch j

Control volume

Branch 1

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REACTOR MODEL

Horizontalinlet slots

Control rodchannels

Centralreflector

Pebble bed

Verticalriserchannels

Core barrelannulus

Core barrel

Gas inletmanifold

Corestructures

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REACTOR MODEL

• Gas flow channels and/or leak flow paths connected to inner and outer perimeters of core.

• Heat transfer and flow through core structures.• Fluid flow and heat transfer, including radiation and

convection in cavities between core structures.• Variable porosity.• Radial as well as axial power profiles.• Heat generation in core structures.

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Conservation of mass for gas:

{ } 0V A

d dV V n dAdt

ερ ε ρ+ =∫ ∫ i

Conservation of momentum for gas:

( ){ }V A V V

d V dV n V V n dA gdV B dVdt

ερ ε ρ ερ ε+ − = −∫ ∫ ∫ ∫σi

Conservation of energy for gas:

{ } ( )bfV A V

d E dV n EV V q dA g V q dVdt

ερ ε ρ ερ+ − + = +∫ ∫ ∫σi i

GOVERNING EQUATIONS

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Expressed in axi-symmetric cylindrical coordinates

Continuity:

( ) ( ) ( )1 0r zru ut r r zερ ερ ερ∂ ∂ ∂

+ + =∂ ∂ ∂

GOVERNING EQUATIONS

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GOVERNING EQUATIONSMomentum in radial direction:

( ) ( )

2

21

o or r zz

o o

rr zr r r

T pu u u V put z r T r p r

p r g Br r r z r

θθ

ρερ ερ ε

ετε ε τ ετ ερ ε

⎡ ⎤∂ ∂∂ ∂ ∂⎛ ⎞+ − = − +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦∂ ∂ ∂

− + + − + −∂ ∂ ∂

Momentum in axial direction:

( ) ( )

2

21

o oz z rr

o o

rz zz z z

T pu u u V put r z T z p z

p r g Bz r r z

ρερ ερ ε

ε ε τ ετ ερ ε

⎡ ⎤∂ ∂∂ ∂ ∂⎛ ⎞+ − = − +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦∂ ∂ ∂

− + + + −∂ ∂ ∂

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GOVERNING EQUATIONS

Energy:

( ) ( ) ( ) ( )

( )

[ ]( ) [ ]( )

1

1

1

o o r o z

r r z z bf

rr r rz z zr r zz z

h r h u h u pt r r z t

T Trk k g u g u qr r r z z

r u u u ur r z

ερ ε ρ ερ ε

ε ε ερ

ε τ τ ε τ τ

∂ ∂ ∂ ∂+ + =

∂ ∂ ∂ ∂∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ + + + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂

+ + + +∂ ∂

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GOVERNING EQUATIONS

For the pebble-bed:

1B L

V Vρ ⋅

Thus momentum in radial direction:2

2o or

ro o

T pu V p yg B pt T r p r r r

ρ εερ ε ε ερ ε∂ ∂∂ ∂ ∂

= − − − − −∂ ∂ ∂ ∂ ∂

Thus momentum in axial direction:2

2o oz

zo o

T pu V p yg B pt T z p z z z

ρ εερ ε ε ερ ε∂ ∂∂ ∂ ∂

= − − − − −∂ ∂ ∂ ∂ ∂

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GOVERNING EQUATIONS

Darcy-Weisbach pipe momentum:

Therefore momentum in radial direction:2

2 2r ro or

ro o

u uT pu V p y pgt T r p r r r

ρρ ερ ρ βε

∂ ∂∂ ∂ ∂= − − − − −

∂ ∂ ∂ ∂ ∂

2

2 2o o

o o

f V VT pV V p ygt T s p s s D

ρρρ ρ∂ ∂∂ ∂=− − − −

∂ ∂ ∂ ∂

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GOVERNING EQUATIONS

Energy:

( ) ( ) ( ) ( )

( )

1

1

o o r o z

r r z z bf

h r h u h u pt r r z t

T Trk k g u g u qr r r z z

ερ ε ρ ερ ε

ε ε ερ

∂ ∂ ∂ ∂+ + =

∂ ∂ ∂ ∂∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ + + + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

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GOVERNING EQUATIONS

Conservation of energy for pebble:

( ) 22

1 sinsinv s

Tc T kr qt r r rρ θ

θ∂ ∂ ∂⎡ ⎤⎛ ⎞= +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦

Convective, conductive and radiative heat transfer between pebbles:

10 eff eff fbT Trk k q

r r r z z∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

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GOVERNING EQUATIONS

Conservation of energy for reflector blocks:

( ) ( )

( ) ( )

11 1

1 1

v

fs

Tc T r kt r r r

Tk qz z

ε ρ ε

ε ε

∂ ∂ ∂⎛ ⎞⎡ ⎤− = −⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎢ ⎥∂ ∂ ∂⎣ ⎦⎝ ⎠∂ ∂⎛ ⎞⎡ ⎤+ − + −⎜ ⎟⎢ ⎥∂ ∂⎣ ⎦⎝ ⎠

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DICRETIZATION

Pebble bed Core structures

Pebble surfacetemperature

Pebble internalsolid mass

temperatures

Core structureSolid masstemperatures

Conductionelements

Effectiveconduction

and radiation inpebble bed

Core and core structures solid elements


Control rodchannels

Centralreflector

Pebble bed


Core barrelannulus

Core barrel

Gas inletmanifold

Corestructures

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DICRETIZATION


Control rodchannels

Centralreflector

Pebble bed


Core barrelannulus

Core barrel

Gas inletmanifold

Corestructures


Pebble bedvoid volumes

Pebble bedflow resistance

elements

Inlet slot flowresistanceelements

Riser channelflow resistanceelements

Inlet manifold

Riser channels, inlet slots and pebble bed flow paths

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DICRETIZATION

Control rod channel flow paths


Control rodchannel flowresistanceelements


Control rodchannels

Centralreflector

Pebble bed


Core barrelannulus

Core barrel

Gas inletmanifold

Corestructures

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DICRETIZATION

Integrated networkHorizontalinlet slots

Control rodchannels

Centralreflector

Pebble bed


Core barrelannulus

Core barrel

Gas inletmanifold

Corestructures

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T = 323 K

m = 5kg/s

Convection at 373 K

Radiation at 373 K

Composite layer reservoir

Node 1

Node 2

Node 3

VALIDATION

Heat transfer through a composite reservoir wall

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VALIDATION

322

324

326

328

330

332

334

336

338

340

342

0 50 100 150 200 250 300 350 400 450 500

Time [s]

Tem

pera

ture

[K]

Bench Node 1 Bench Node 2 Bench Node 3FN Node 1 FN Node 3 FN Node 332

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VALIDATION

5000 m

0.5 mHelium

5000 kPa

107ºC

2000 m

Measuring point

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VALIDATION

0.00

1000.00

2000.00

3000.00

4000.00

5000.00

6000.00

7000.00

8000.00

9000.00

10000.00

0.00 10.00 20.00 30.00 40.00 50.00 60.00

Time (sec)

Tota

l Pre

ssur

e[kP

a]

Benchmark solution Flownet solution34

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VALIDATION

Pebble-bed micro model35

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VALIDATIONPBMM

RXHPC

HS

HPT LPT

PCIC

PT

LPC

CT

SBS

PTC

GBPLPBHPB

SIV

SBSOV

PTCV

PV

ELC

CWP

NIV

SBSIV

NEV

Schematic layout36

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VALIDATION

PBMM network model37

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VALIDATION

0

5

10

15

20

25

30

35

40

50 150 250 350 450 550 650

HPT inlet temperature [oC]

Pres

sure

incr

ease

ove

r SIV

[kPa

]

Measured Predicted

Pressure difference across system inline valve38

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1Reactor_Core (R)

3

Reactor_Outlet_Connection_to_Core_Outlet_Pi

pe (FP)

7 Core_Outlet_Pipe (FP)

9CM

11

HPT_Diffuser (FP)

13

HPT_Outlet_Pipe (FP)

17

Low_Pressure_Turbine (T)

CM19

LPT_Diffuser (FP)

21LPT_Outlet_Pipe (FP)

27PT_Intake (FP)

29

Power_Turbine (T)

31PT_Diffuser (FP)

35

PT_Outlet_Contraction (FP)

39

Recuperator_LP (H)RX

41

Pre-cooler_outer_bundle (H)CX

43

Pre-cooler_inner_bundle (H)CX

47

PC_Outlet_Contraction (FP)

51

LPC_Intake (FP)

53CM

55

LPC_Diffuser (FP)

57

LPC_Outlet_Pipe (FP)

61

Intercooler_outer_bundle (H)CX

63

Intercooler_inner_bundle (H)CX

67

IC_Outlet_ducting(FP)

71

HPC_Intake (FP)

73CM

75

HPC_Diffuser (FP)

77HPCe_Holes_to_Manifold (FP)

79

Maintanence_Valve (V)RD

83 RX(HP)_Inlet_Pipes (FP)

85RX

87 RX(HP)_Outlet_Pipes(a) (FP)

89

Core_Inlet_Pipes(e) (FP)

91

Reactor_Annular_Pipes (FP)

93

Reactor_Gas_Inlet_Plenum_Holes (FP)

95Control_Rod_Cooling_flow_connection (FP)

102CX

104CX

109CX

111CX

78P

113

Manifold_to_HPTe_Leakage (L)RD

78P

116RD

119HPC_Bypass_Valve_(HPB) (V)

120


121

Gas_Cycle_Bypass_Valve_1_(GBP1) (V)

78P

133

Manifold_to_HPTi_Leakage (L)RD

78P

135

RX_bypass_valve_(1 TO 4)_(RBP1) (V)

136Reactor_Plenum_Reduction_Area (FP)

137 Reactor_Outlet_Slots (FP)

138

Control_Rod_Bypass(filled) (FP)

140

Control_Rod_Bypass(empty) (FP)

142

Leakflow_from_Control_Rod_to_Outlet_Manifold (FP)

RD

144

Inside_Cavity_Slots_to_Core (FP)RD

146Reactor_Outside_

Cavity (FP)

148 Reactor_Leak_into_Manifold (FP)

RL

150HPT_Intake (FP)

152

LPT_Intake (FP)

158

RX(LP)_to_PC_connection (FP)

160

Annular_flow_path_to_IC (FP)

162

Annular_flow_path_. . ._to_IC_core (FP)

163

Recuperator_Outlet_Pipe (FP)

165

Core_Inlet_Pipes(a) (FP)

167Core_Inlet_Pipes(b) (FP)

169

Core_Inlet_Pipes(c) (FP)

171

Core_Inlet_Pipes(d) (FP)

174

175


178


180Bypass_Valve_Pipe (FP)

181 HPCe_to_HPCi_Leakage (L)

RD

78P

182

Manifold_to_LPTi_Leakage (L)

RD

183

LPCe_to_LPCi_Leakage (L)RD

184 Casing_cooling_flow (L)RD

56185

LPCe_to_PT_casing_flow (FP)

186

PTi_to_Basket_leak (L)RD

188


RD190


RD

192

Basket_to_PTe_leak (L)RD

194


RD

196

196 Buffer_to_gaspath_leak (L)

197

197 ECLS_upper (L)

198

ECLS_lower (L)

300

ECLS_to_PToutlet (L)RD

301

dummy

303

HPS_supply_line (FP)RD

305

PTG_Shaft_Labyrinth_Seal(a) (L)

307

307 PTG_Shaft_Labyrinth_Seal(b) (L)

309

PT_top_volume_to_PTe

(FP)

78P

310

Shaft_valve (V)RD

153312


314


316

SBS_Blower2 (CM)

M

CM327 318


153319


321


323

SBS_Blower1 (CM)

M

CM327 325


326


78P

329

HP_coolant_valve_1_(HCV1) (V)78

P330




336

RX(HP)_Outlet_Pipes

(b) (FP)

338 RX_inlet_header_pipe (FP)

340

PC_Impingement_plate(FP)

341HPC_Bypass_Control_Valve_(HPBC) (V)

342

LPC_Bypass_Control_Valve[LPBC] (V)

343

SBS_Bypass_valve_(SBP) (V)

344

SBS_Bypass_Control_Valve_(SBPC) (V)

345


346


347

348

349

350

351

352

353 Gas_Cycle_Bypass_Valve_8_(GBP8)(V)

78P

354

HP_coolant_valve_2_(HCV2) (V)

78P

355


356


357


78P

361

78P

362

78P

363

364

HPC_Bypass_Control_Valve_Pipe (FP)

365LPC_Bypass_Control_Valve

_Pipe2 (FP)

366

LPC_Bypass_Control_Valve_Pipe1 (FP)

368

PT_1st_stg_stator (FP)

370

370 PT_1st_stg_rotor_inlet (FP)

371

371 Inlet_1st_stg_throat (FP)

373


RD

375


RD

376

Rim_cooling_flow (L)RD

377

OD_u_ECLS_backup_lab (L)

380

380 ID_u_ECLS_backup_lab (L)56 381

PT_internals_cooling flow

(FP)

383


RD

387


388


390


393


394


396


78P

1702

Basket_ventilation_flow (L)RD

1703Reactor_Outlet_Pipe_to_CCS (FP)

1705

CCS_Splitting_Restrictor (FP)RD

1707

CCS_Outlet_Pipe_to_Reactor (FP)

2Reactor_Core_Outlet

H

4

6

Reactor_Outlet_Manifold

8 10 12

HPT_Diffuser_Volume

16 18 20

LPT_Diffuser_Volume

26

PT_Inlet_Volume

28 30

34

PT_Diffuser_Volume

38RX(LP)_Inner_Vess

el_Volume40PC_Inlet_Volume

42

46

PC_Outlet_Volume

5052545658

IC_Inlet_Volume

62

66

IC_Outlet_Volume

707274

76HPC_Diffuser_Volume

78

Manifold

P82

RX(HP)_Inlet_Plenum

84RX(HP)_Core_Inlet

86

88 RX(HP)_Outlet_Plenum

90

Reactor_Inlet_Manifold

92

Reactor_Gas_Inlet_Plenum_Volume1

94Reactor_Gas_Inlet_Plenum_Volume2

96

98

Void_above_the_Reactor_Core

H

99IC_Waterside_Inlet

PT

103

105

IC_Waterside_OutletM

106

PC_Waterside_Inlet

PT110 112

PC_Waterside_Outlet

M

139141

143

Reactor_Inside_Cavity

T

145

Reactor_Upper_Volume

T

147

Reactor_Lower_Volume

T

149HPT_Intake_Volume

151

LPT_Intake_Volume

153

RX_Outer_Volume

159161

164

166

168

170172

173

176 177

179

187

187 Turbine_casing_temp

T

189PT_casing_volume

191

PT_basket_volume

195

PT_Inner_Volume

199

302

HPS_outlet_purified

MT304

Generator_Volume

306

308

308 Volume_on_top_of_PT

311

HPS Extraction

M

313315


317

320322


324

327RX_Outer_Annulus_Volume

328

333SBS_Motor_Casing_Volume2

335

SBS_Motor_Casing_Volume1

337Inlet to RX(HP)_Outlet_Pipes(b)

339

367

369

372 374

374 Turbine_rim_temp

T

378

378 u_ECLS_OD_volume

379

379 u_ECLS_ID_volume382 384

386389


391


392395


397


398

1700 1701

1704CCS_Volume1

1706

Reactor

LP Turbo Unit

Power Turbine

Pre-cooler

Recuperator

Start-UP Blower System

HP turbo Unit

Intercooler

MPS NETWORK MODEL

39

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LOAD REJECTION

• Initially the plant operates at maximum power and at time t=0.9 s the generator power is reduces by 50 percent.

• The generator speed is controlled by a PID controller that adjusts the bypass flow by opening and closing a control valve.

• Following figures shows the variation in turbine temperatures and the variation in compressor temperatures during the transient.

40

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LOAD REJECTION

500

550

600

650

700

750

800

850

900

950

0 2 4 6 8 10 12

Time (s)

Tem

pera

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(ºC

)

HPin LPin PTin PTout

Turbine temperatures

41

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LOAD REJECTION

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Time (s)

Tem

pera

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(ºC

)

LPin LPout HPin HPout

Compressor temperatures42

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Conclusions

• IPCM method as embodied in systems CFD code Flownex is capable of simulating complex thermal-fluid systems consisting of many interacting components.

• Flownex can predict performance of complex systems for both steady-state and transient conditions.

• Flownex can therefore be used to predict the performance of the integrated PBMR plant.

43

1

[1] TITLE: • My co-authors and colleagues are Proff Gideon Greyvenstein and Pieter

Rousseau [2] SCOPE: • Give a brief overview of PBMR main power system or power conversion unit

to sketch the context and put what will be said in context. • An overview of the essence of the network approach • A very brief introduction to the implicit pressure correction method which

forms the heart of the network code Flownex being used • Discussion of the reactor model as an example of the models incorporated in

the network code Flownex • Discuss a few validation cases to test / demonstrate the validity or fitness for

purpose of the code • An illustration of one scenario for the PBMR [3] PBMR MAIN POWER SYSTEM: • The slide shows a solid CAD representation of the main power system of the

proposed PBMR nuclear power plant currently under development in South Africa.

[4] SCHEMATIC LAYOUT OF MPS: • This slide shows a schematic layout of the main power system illustrating the

various components that from the system. [5] PBMR ISSUES: • High temperature gas reactors are complex thermal-fluid systems consisting

of many interacting components such as pipes, valves, heat exchangers, compressors, turbines, pumps, blowers and the nuclear reactor itself.

• Engineers are faced with two major challenges when carrying out the thermal-hydraulic design of a HTR power plant. The first challenge is to predict the performance of all the thermal-hydraulic components of the plant such as pipes, valves, heat exchangers, turbo machines and the reactor. The second challenge is to predict the performance of the integrated plant consisting of all its sub-systems

• System performance predictions must be done for both steady-state and transient conditions. Although it is obvious why steady-state analyses are required it is not so obvious why transient analyses are required. Transient analyses are required for control studies and for studying operating procedures such as start-up, load rejection and load following. It is also required when analyzing accident events.

2

• An approach that has gained wide acceptance is known as the network approach. With this approach models of standard components are developed that can be interconnected in any arbitrary way. The name “network” is derived from the graphical representation of thermal-fluid systems as networks of interconnected components. With the network approach one can easily reconfigure the system lay-out without having to change any computer code

[6] NETWORK APPROACH:

• With the flow network approach elements such as pipes, compressors, valves are denoted by circles while nodes, which are the end points of elements, are denotes as squares. Elements can be connected in any arbitrary way at common nodes to form a network as shown. An important feature of the model is that elements such as pipes, heat exchangers and the reactor, although depicted on the systems level as single elements or pairs of elements, can be discretized into sub-networks. Networks can therefore be embedded within networks thereby enabling the model to treat complex elements as distributed systems rather than lumped systems.

[7] MAIN POWER SYSTEM NETWORK MODEL:

• This slide shows a representation of the network model that was set-up to simulate the behaviour and performance of the PBMR under various conditions. The model is also used to test various control strategies.

[8] SHELL-AND-TUBE SUBNETWORK:

• As had been said, an important feature of the model is that elements such as pipes, heat exchangers and the reactor, although depicted on the systems level as single elements or pairs of elements, can be discretized into sub-networks.

• The network approach therefore allows for the discretization of more complex components such as shell and tube heat exchangers. The figure shows the discretization of part of a shell and tube heat exchanger. Larger circles and squares denote flow elements and nodes respectively while smaller circles and squares denote convective heat transfer links and metal temperatures respectively. The metal nodes can also be linked to form conductive paths.

[9] + [10] NETWORK SOLUTION METHODS:

3

• Flow network problems can be characterized in terms of their time dependence as steady-state or transient problems while transients in turn can be characterized as fast or slow. Flow problems can also be characterized in terms of fluid type i.e. gas or liquid flow. Another classification is adiabatic, non-adiabatic or isothermal flow. Methods for solving flow network problems can be broadly classified as explicit or implicit.

• Explicit methods such as the Method of Characteristics (MOC) are especially suited for solving fast transients in isothermal liquid flows while the Lax-Wendroff method are suitable for solving fast transients in gas flow problems. The stability of explicit methods is, however, governed by the ∆t-∆x relationship, which makes it very slow when solving steady-state or slow transient problems. The ∆t-∆x relationship also pose serious limitation when solving networks where both liquids and gasses can be present in different parts of the network.

• Implicit methods on the other hand are not governed by the ∆t-∆x relationship which means that longer time steps can be used. This makes implicit methods especially suited for solving steady-state or slow transient problems. A draw-back of implicit methods, however, is that they are generally much slower than explicit methods per time step which make them slower than explicit methods when solving fast transient.

• Method used in Flownex is known as the Implicit Pressure Correction Method (IPCM) and is suitable for both steady-state and transient analysis; for both slow and fast transients; for both liquid and gas flows; and for both adiabatic and non-adiabatic flows.

[11] GOVERNING EQUATIONS: • The equations governing transient flow through a variable area duct are the

continuity, the momentum and the energy equations. These equations are given in conservative form where h0 is the total enthalpy.

[12] GOVERNING EQUATIONS (for compressible flows): • In order to eliminate the convective acceleration term in the momentum

equation we first use the mass conservation equation to write the momentum equation in the non-conservative form. We then use principles in gas dynamics to replace the static pressure with the total pressure and total temperature.

[13] SOLUTION ALGORITHM:

4

• The discretized equations are obtained by integrating the equations for the conservation of mass, momentum and energy (which for a pipe are partial differential equations) over control volumes.

• The slide shows the control volume for the conservation of mass at nodal point.

[14] SOLUTION ALGORITHM: • This slide shows the control volume for the conservation of momentum over

an element. [15] SOLUTION ALGORITHM: • This slide shows the control volume for the conservation of energy. • The iterative solution procedure is then as follows:

i) Derive a set of algebraic equation using second order time integration of the mass, momentum and energy equations.

ii) Guess an initial pressure field. iii) Calculate the mass flows in all elements using the initial pressures in the

momentum equation. iv) Test for continuity at all nodes. v) Adjust the nodal pressures to ensure that continuity is satisfied at all

nodes. vi) Update the mass flows using the new updated pressure field. vii) Treat the updated pressure field as new initial pressures and repeat steps

(iii) to (vi) until convergence. viii)Solve the energy equation. ix) Repeat (iii) to (viii) until convergence. x) Move to next time step and repeat (ii) to (ix).

[16] REACTOR MODEL: • Flownex currently contained a simplified model for the pebble bed nuclear

reactor. The purpose of the model was not to do detail reactor design, but rather to allow for the integrated simulation of the reactor together with the PCU within acceptable computer simulation times.

• Recent developments in the design and layout of the PBMR reactor have given rise to the need for the simulation of a wider range of reactor phenomena, even within a simplified model, since these will influence the boundary values for the integrated plant simulations.

5

• Therefore, a need exists for the development of a new, more comprehensive pebble-bed reactor model that can still provide relatively quick integrated plant simulations, but including the phenomena listed in the following slide.

[17] REACTOR MODEL: • The presence of a central reflector column that implies that the core itself has

an annular rather than a cylindrical shape. • The addition and extraction of gas via purpose provided channels and/or leak

flow paths along the inner or outer perimeters of the core. • The simulation of heat transfer and fluid flow through porous and solid core

structures surrounding the core. • The simulation of fluid flow and heat transfer, including radiation and natural

convection, in purpose provided cavities between core structures with a two-dimensional rather than one-dimensional nature.

• The ability to take into account variations in porosity throughout the core. • The ability to specify normalised radial power distribution profiles within the

different axial layers in the core. • The ability to take heat generation that may occur in any of the core

structures into account. [18] GOVERNING EQUATIONS (integral): • The conservation equations in integral form for mass, momentum, and energy

for the fluid in the pebble-bed reactor can written as shown in the slide. [19] GOVERNING EQUATIONS (cylindrical mass): • If it is assumed that the properties of the fluid are continuous and sufficiently

differentiable, then the conservation equations in integral form can be transformed into an equivalent set of partial differential equations through the divergence theorem. The equation for the conservation of mass can be expressed in axi-symmetric cylindrical coordinates as shown in the slide.

[20] GOVERNING EQUATIONS (cylindrical momentum): • The equation for the conservation of momentum in the radial direction can be

obtained as axi-symmetric cylindrical coordinates. The continuity equation can be extracted from it, the dynamic head can be added and subtracted and the static pressure can be converted to the total pressure. This then leads to the equation shown in the slide.

• The same can be done to obtain the corresponding equation for the conservation of momentum in the axial direction.

[21] GOVERNING EQUATIONS (cylindrical energy):

6

• The equation for the conservation of energy in terms of the total specific enthalpy can be obtained in axi-symmetric cylindrical coordinates as shown in this slide.

[22] GOVERNING EQUATIONS (porous momentum): • For flow through a porous medium, such as the pebble-bed, it has been found

that the convective terms and diffusive or shear stresses terms may be neglected if dimensionless particle (pebble) resistance is much larger then one. For the pebble-bed the value is approximately 1470.

• The momentum equations may be reduced to the equations shown in the slide.

• Taking a close look at the convective terms in the momentum equations shown earlier reveals that these terms have been written in terms of the vorticity. The above conclusion therefore implies that the average (or macroscopic) flow can be assumed to be irrotational.

[23] GOVERNING EQUATION (pb vs pipe): • This slide shows the correspondence between the momentum equations for

the pebble-bed and a pipe. [24] GOVERNING EQUATIONS (reduced energy): • It can be shown that the contributions of the viscous dissipation terms are

negligible compared to the other terms in the energy equation. The energy equation can therefore be simplified to become as shown in the slide.

[25] GOVERNING EQUATIONS (pebble and convective): • It is assumed that the temperature distribution in a pebble is the same in all

radial directions. The equation for the conservation of energy in the pebble can therefore be written in one-dimensional spherical coordinates as shown in the slide.

• The heat transfer between the surfaces of the pebbles due to contact, convection and radiation can be written in axi-symmetric cylindrical coordinates as shown in the slide.

[26] GOVERNING EQUATIONS (reflector blocks): • Finally the equation for the conservation of energy in the (porous) reflector

blocks is given in axi-symmetric cylindrical coordinates as shown in the slide. • The Zehner-Schlünder correlation is employed to determine effective

conductivity effk between the pebbles, whilst the correlation for the surface heat transfer coefficient proposed by Kugeler et al. is used to determine the heat transfer between the pebbles and the coolant. The heat transfer between

7

the porous reflector blocks and the coolant is determined using the well-known Dittus-Boelter relationship.

• A study of all the equations reveal that they are one-dimensional, or can be written as the sum of two one-dimensional equations. This formulation of the equations allows the reactor to be discretized into a collection of one-dimensional elements (models) that can be incorporated in a general network approach.

[27] DISCRETIZATION (core, structures and solids): • The slide shows a simplified network representation of the solids in the

pebble bed and core structures of the reactor. In the network both the pebble bed and the core structures are each represented by two control volumes in the radial direction and four control volumes in the axial direction.

• In the case of the core structures, the large light grey squares (nodes) represent the mass of the solid material and therefore also the temperatures of the core structure material that is assumed to be homogeneous throughout a control volume. The large grey circles (elements) represent the radial and axial conduction respectively within the core structures.

• In the case of the pebble bed, the large grey squares represent only the outer layer of the pebbles in that control volume and therefore also the surface temperature of all the pebbles in that specific control volume. The inner layers of pebble mass is represented by the smaller grey squares. It will therefore be possible to calculate a temperature distribution profile within the pebbles of each control volume. The small grey circles represent the one-dimensional spherical heat conduction within the pebble while the large grey circles represent the effective conduction and radiation between the pebble surfaces within the bed.

[28] DISCRETIZATION (flow paths): • The slide shows a simplified network representation of the riser channel, inlet

slots and packed bed flow path within the pebble bed and core structures of the reactor.

• The white squares represent the void gas volumes and the circles the flow resistance elements. The node on the bottom right also represents the inlet manifold where it is assumed that the gas is well mixed so that a homogeneous temperature will be obtained.

[29] DISCRETIZATION (control rod flow paths): • The gas flowing in the riser channels, inlet slots and through the pebble bed

does not mix with the gas flowing in the control rod channels. This means

8

that a second, separated gas flow network is required to simulate the flow in the control rod channels. This network is shown in the slide as dark grey nodes and elements.

[30] DISCRETIZATION (integrated network): • The interaction between the solid structure network and the respective flow

networks is via fluid/solid surface convection. This interaction is shown schematically in the slide. In the figure the surface convection elements are shown in black, the control rod channel flow network in dark gray, the solid network in light gray and the riser channels, inlet slots and pebble bed flow path in white.

• The slide highlights one of the distinct advantages of employing the network approach rather than the traditional CFD approach. That is that even though the simplifying assumption was made earlier of two-dimensional axi-symmetric geometry, both of the riser channel and control rod channel flow paths that is unmixed, can be simulated together with its interaction with the solid structures. This is accomplished by simply superimposing another network layer on top and adding the correct connectivity via surface convection elements.

[31] VALIDATION (reservoir wall):

• In this example a cylindrical reservoir of with water flowing through at a constant rate of 5kg/s at 323.15 K is considered. The reservoir wall consists of three layers. The outside of the reservoir is exposed to air at 373.15 K.

• Starting at steady state conditions the outside convection temperature is varied with a step function from 273.15 K to 373.15 K with 100-second intervals.

[32] VALIDATION (reservoir wall results): • The slide shows the temperature variation with time for the outside, middle

and inside of the reservoir wall respectively. The IPCM results are compared with an analytical solution. As can be seen the results correlate well.

[33] VALIDATION (pressure wave in pipe): • In this example a pipe with a length of 5000 m and a diameter of 0.5 m is

connected to a reservoir at one end and has valve at the other end. Initially the valve is fully open. The pipe filled with Helium at a pressure of 5000 kPa and a temperature of 107 oC from the reservoir. The valve is then closed in 0.1 seconds. The pressure is measured at point 2000 m downstream from the reservoir.

9

[34] VALIDATION (pressure wave in pipe results): • The slide shows the variation in pressure at the measuring point with time.

The solution obtained with Flownex is compared with code based on the Lax-Wendroff approach. The agreement between the results is good.

[35] VALIDATION (PBMM picture): • To test the feasibility of the PBMR power conversion unit and the proposed

control strategies, as well as the validity of the network simulations, a model (pebble-bed micro model) was built of the of proposed main power system. The model was designed, constructed and started-up (commissioned) in 9 months at a cost of USD 1.5 million. The slide shows a picture of the plant. This is the first three shaft closed Brayton cycle in the world that was successfully started up.

• This would not have possible without the network code Flownex. [36] VALIDATION (PBMM layout): • This slide shows a schematic layout of the PBMM cycle. [37] VALIDATION (PBMM network model): • This slide shows a representation of the PBMM network model. [38] VALIDATION (PBMM results): • This slide shows one set of the initial result that was obtained. The graph

shows the comparison between the measured and the predicted pressure difference over the system inline valve. It can be seen that the agreement is good.

• From the validation the tests that had been performed sofar it can be concluded that the network code is fit for purpose.

[39] MPS NETWORK MODEL: • We will now look at how Flownex predicts the behaviour of the PBMR main

power system when a load rejection occurs. • The slide reminds us what the network model for the PBMR main power

system looks like.

10

[40] LOAD REJECTION: • Initially the plant operates at maximum power and at time t=0.9 s the

generator power is reduces by 50 percent. • The generator speed is controlled by a PID controller that adjusts the bypass

flow by opening and closing a control valve. [41] LOAD REJECTION (turbine temperatures): • This figure shows the variation in turbine temperatures during the transient

and how the PID controller succeeds in stabilising the turbines. [42] LOAD REJECTION (compressor temperatures): • This figure shows the variation in compressor temperatures during the

transient and how the PID controller succeeds in stabilising the compressors. • It should be remembered that the compressors are each link to a turbine

through a shaft. Simultaneously with the stabilization the Flownex performs the power matching of the coupled compressor-turbine pairs.

[43] CONCLUSIONS:

• IPCM method as embodied in network code Flownex is capable of simulating complex thermal-fluid systems consisting of many interacting components.

• Flownex can predict system performance for both steady state and transient conditions.

• Flownex can predict the performance of the integrated plant.

the analysis of the pebble-bed modular reactor thermal ... · the analysis of the pebble-bed...

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