VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX

Transcription

1 2nd International Topical Meeting on HIGH TEMPERATURE REACTOR TECHNOLOGY Beijing, CHINA, September 22-24, 2004 #Paper C30 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX Jan P. van Ravenswaay **, Gideon P. Greyvenstein *, Willem M.K. van Niekerk * and Johan T. Labuschagne ** * School of Mechanical and Materials Engineering North West University, Private Bag X6001, Potchefstroom 2520, South Africa ** M-Tech Industrial (Pty) Ltd, South Africa Corresponding author J.P. van Ravenswaay: Fax: jvr@mtechindustrial.com 1. ABSTRACT Regulatory requirements prescribe extensive verification and validation (V&V) of computer codes that are used in the design and analysis of accident conditions in nuclear plants. Flownex is a dynamic systems CFD code used as the primary thermal-fluid simulation code by the Pebble Bed Modular Reactor Company (PBMR). Stringent quality assurance processes have been implemented to ensure that the code conforms to the set standards. These processes include the comparison of Flownex with analytical results as well as with experimental data. The results of this process will be summarized in this paper. Analytical solutions are used to verify Flownex s element models so as to ensure that the basic theory is correctly implemented in the computer code. As part of the analytical V&V effort various well-defined problems are solved using numerical methods implemented in independent computer codes. Comparison with experimental and plant data is a very important feature of the V&V program to validate that the chosen theory is fit for purpose. For this, validation data from the Pebble Bed Micro Model (PBMM) is used. In addition to the PBMM experimental data Flownex is compared to a number of small thermal-fluid experiments in which certain specific component phenomena is validated. These experiments were developed in collaboration with North West University (Previously Potchefstroom University) 2. INTRODUCTION Flownex [1, 2] is a dynamic systems CDF code that enables users to perform detail analysis and design of complex thermal-fluid systems such as power plants and thermal-fluid networks. It is based on an implicit pressure correction method (IPCM) [3] that solves the momentum equation in each element and the continuity and energy equation at each node in large arbitrary structured networks for both steady-state and dynamic analysis. This gives the code a pseudo CFD capability, which allows it to predict complex phenomena such as pressure and temperature waves in pipes and buoyancy effects in packed beds. The code has the ability to perform both steady-state and dynamic analyses [4]. The solver, that is optimized for steady-state and transient flows, can deal with both fast and slow transients. Fast simulation speeds, on standard desktop computers, allows for real time simulations to be performed. It features the ability to simultaneously solve multiple 1

2 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C30 gas and liquid networks that are connected through heat exchangers. The fundamental principle approach that has been used allows the prediction of phenomena such as choking, natural convection and Joule heating. It features a model builder that enables users to build advanced discretized re-usable models of complex components or sub-systems such as gascooled nuclear reactors and heat exchangers. Standard component models in Flownex include: A comprehensive pipe model, a reservoir model, orifice models, a turbine model, a compressor model, various heat exchanger models (i.e. recuperator, shell and tube and finned tube), two reactor models (pebble and block fuel), a PID controller, valve models and pump models. Flownex Nuclear is being developed to perform thermal-fluid analyses on a high temperature gas cooled reactor coupled to a direct, recuperated Brayton cycle in an implicit way. Since it is the first software product of its kind, a diverse number of verification and validation methods are used to qualify the software. The fact that Flownex Nuclear is the first software product of its kind should be stressed as it impacts on the availability of independent software products and experiments that can be used for V&V activities. At PBMR, Flownex Nuclear is used to predict mass flows, heat transfer and pressures in the reactor core and the Brayton cycle during expected operational modes and states, as well as under accident conditions. It is used for both steady-state and transient simulations. Simulation results are fed back into the design process, where it dictates plant layout, material selection and operating philosophies. The accuracy of these simulations is crucial in determining the safety of operation, economic viability and protection of the plant. In order to ensure the accuracy of these simulations a rigorous verification and validation (V&V) process has been implemented to guarantee the integrity of engineering analyses and to satisfy statutory requirements regarding the licensing and operating of nuclear plants in South Africa and abroad. The process developed for the V&V of the code includes various control mechanisms and procedures that ensure that the whole development process is of the highest quality. This includes regression testing, automated testing, code reviews and validation of the code against benchmark data. The V&V process will described next. 3. OVERVIEW OF THE FLOWNEX V&V PROCESS To ensure that all phenomena for each component in Flownex Nuclear are validated for the various extremities is a comprehensive exercise. Furthermore, V&V of the individual components as well as integrated systems of components for both steady-state and dynamic analysis are required. The main focus of the Flownex V&V process is to perform effective V&V of complicated phenomena using the least possible validation cases. This is all done in a transparent and traceable manner according to regulatory guidelines and relevant international standards. V&V forms part of the overall Flownex Nuclear development process and includes the verification activities that form part of the software engineering process, as well as all related verification that is done as part of the derivation and implementation of the theory for component models or model enhancements. 2

3 HTR 2004 Beijing, CHINA, FIG. 1 shows a diagram of the V&V process for Flownex Nuclear. Tools used in the verification of Flownex include an implemented ISO 9001:2000 quality management system at M-Tech Industrial, the code developers, test plans and procedures, code reviews, user testing, automated testing and regression testing. Validation of Flownex Nuclear is performed by comparing the results of the implemented theoretical models in Flownex Nuclear with benchmark data obtained from appropriate methods or sources such as analytical data, experimental data, plant data and data obtained form other codes such as Spectra, XNet and Star CD. Flownex Analysis Flownex V&V Flownex Software V&V Validation Verification BENCHMARKS Hand calculations Analytical solutions Numerical methods Other codes Experimental data Plant data REVIEWS Test Plans and Procedures Code reviews Various checklists Manual Testing Automated testing Independent review REPORTS Validation Manual Benchmark Programs Benchmark Data REPORTS Acceptance test Report Anomaly reports Completed checklists 4. ACCEPTANCE CRITERIA FIG. 1: Flownex Nuclear software V&V diagram. In order to ensure that the results obtained are valid, acceptance criteria need to be stipulated. The comparisons of differences between the benchmark results and the Flownex modeling results are based on two different definitions: A normalized point difference and a Euclidian difference. 4.1 Normalized Point difference The normalized point difference gives the difference between Flownex and the benchmark for each point divided by the total range of the variable for the system under investigation. The benchmark data could be measured (experiment or plant data), calculated (analytical) or modeled (other software codes). The following equation is used to calculate the normalized point difference for variable y: 3

4 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C30 e NPD where y = the Flownex variable FNX BM y SYS = y FNX y y SYS y = the benchmark variable = the total range of the variable for the system under investigation. BM (1) 4.2 Euclidian difference The Euclidian difference gives the fractional difference between the Flownex results and the benchmark data over the whole range of data points. The following equation is used to calculate the Euclidian difference: e Euclidian = ( yfnx ybm ) 2 ( y ) BM 2 (2) The Euclidian difference is calculated cumulatively and at any point will give the fractional difference. The following acceptance criteria are defined by M-Tech Industrial (Pty) Ltd for the different comparisons of Flownex with benchmark data: 4 For analytical and numerical comparisons, the normalized point difference must be smaller than 1 percent. For comparisons with other codes the Euclidian difference must be smaller than 5 percent and if possible the normalized point difference as well. For comparisons with experimental or plant data the Euclidian difference must be smaller than 10 percent and if possible the normalized point difference as well. 5. COMPARISON WITH EXPERIMENTAL DATA Comparison with experimental data plays a very important part of the V&V of Flownex. The process of gathering high quality experimental data is delicate and one needs to ensure that all instrumentation used in the data gathering have been calibrated within required standards. For all the thermal-fluid experiments presented in this paper the repeatability of the data has been proven by repeating the experiment three times and ensuring that the normalized point difference between the three sets of data is less that 1 percent. Pre-test simulations on Flownex have also been done before performing the experiments. These Flownex models were then adapted using the as-built experimental facility data and measured ambient conditions. 5.1 Steady-state network balancing experiment In this validation case a complex flow network is presented. The purpose of the network balancing experiment, shown schematically in FIG. 2, is to validate the Flownex node component for the mass flow balancing behavior of a compressible gas in a complex pipe network. The network consists of 6 plenum chambers, numbered A to F, each having a volume of 0.033m 3. The plenums are interconnected through 15 pipes all with a diameter of m

5 HTR 2004 Beijing, CHINA, with variable lengths as shown in Table 1. All the pipes are insulated to atmosphere. A fan supplying kg/s of air at a total pressure of kPa is positioned at the inlet of plenum A. Orifices manufactured according to ISO :1991 were positioned in each pipe to measure the air mass flow rate and also to provide a higher local flow resistance than the entrance- and exit loss effects of the pipes. FIG. 2: Solid model of the Network balancing experiment Table 1: Network balancing experiment input data. Pipe Length Pipe Length Pipe Length PIPE AB 1.955m PIPE BC 1.874m PIPE CD 1.890m PIPE AF 1.955m PIPE BE 2.930m PIPE CE 1.985m PIPE AC 3.950m PIPE BD 3.950m PIPE DE 1.895m PIPE AE 4.020m PIPE BF 2.020m PIPE DF 3.965m PIPE AD 5.695m PIPE CF 2.865m PIPE EF 1.955m The flow network configuration can be altered by opening and closing valves in each of the fifteen pipes resulting in different network flow configurations. Two network balancing cases are presented in this paper. FIG. 3 and FIG. 5 show the two configurations presented in this paper as modeled in Flownex. 5

6 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C30 B C A D F 248 FIG. 3: Flownex model of the network balancing experiment Case 1. FIG. 4 shows the comparison of the mass flow rates between Flownex and the experimental results for Case 1. For this case the maximum normalized point difference for the system, calculated using Eq. (1), is 1.35 percent in Pipe CD E Mass Flow [kg/s] INLET BC AD OUTLET FE AB ED CF FD AC Pipe number CD BD BE AE CE BF AF EXP Flownex FIG. 4: Mass flow rates for network balancing experiment Case 1. 6

7 HTR 2004 Beijing, CHINA, B C A D F E FIG. 5: Flownex model of the network balancing experiment Case 2. FIG. 6 shows the comparison of the mass flow rates between Flownex and the experimental results for Case 2. For Case 2 the maximum normalized point difference for the mass flow rates is 2.64 percent at Pipe ED Mass Flow [kg/s] INLET BC AD OUTLET FE AB ED CF FD AC Pipe number CD BD BE AE CE BF AF EXP Flownex FIG. 6: Mass flow rates for network balancing experiment Case 2. 7

8 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C Transient blow down experiment The purpose of the volume blow down experiment, shown schematically in FIG. 7, is to validate the Flownex node for the transient pressure behavior using three inter-connected pressure vessels at different initial pressures. The Flownex model of the experiment is shown in FIG. 8. FIG. 7: Schematic of the blow down experiment. T P 100 C T P B T P FIG. 8: Flownex model of blow down experiment. A The three vessels are inter-connected with piping and have square-edge orifices installed between them. The purpose of the square edge orifice is to throttle the gas flow in order to obtain the required mass flow rates and thus transient time span for the blow down process. The square-edge orifices between the tanks were characterized through independent tests to determine the pressure ratio over the orifice versus the mass flow rate through the orifice. A corrected mass flow rate was then calculated using Eq. (3). The square edge orifice characteristics are shown in Table 2. 8

9 HTR 2004 Beijing, CHINA, Corrected mass flow rate. m Toi = (3) P oi Where:. m = mass flow rate [kg/s] T oi = upstream temperature [ºC] P oi = upstream total pressure [kpa] Table 2: Square edge orifice characteristics. Corrected Pressure Mass Flow Ratio rate Two blow-down cases are considered in this paper. For both cases air is used as fluid. The initial pressures for the three vessels are shown in Table 3. Table 3: Transient blow down experiment input data. Vessel Volume Initial Pressures Case 1 Initial Pressures Case 2 Vessel A m kpa 2.72 kpa Vessel B m kpa kpa Vessel C m kpa kpa FIG. 9 shows the comparison of the pressure transient in all three vessels between Flownex and the experimental results for Case 1. The maximum normalized point differences for the experiment are 3.04, 1.08 and 1.63 percent for Vessels A, B and C respectively. The Euclidian difference calculated with Eq. (2) for this sub-case is 2 percent. FIG. 10 shows the comparison of the pressure transient in all three vessels between Flownex and the experimental results for Case 2. 9

10 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C Pressure [kpa] Time [s] Vessel C FNX Vessel B FNX Vessel A FNX Vessel C EXP Vessel B EXP Vessel A EXP FIG. 9: Vessel pressures for Transient blow down experiment Case 1. For Case 2 the normalized point differences for the experiment is 1.15, 1.07 and 2.59 percent for Vessels A, B and C respectively. The Euclidian difference for this case is 1.53 percent Pressure [kpa] Time [s] Vessel C FNX Vessel B FNX Vessel A FNX Vessel C EXP Vessel B EXP Vessel A EXP FIG. 10: Vessel pressures for Transient blow down experiment Case 2. 10

11 HTR 2004 Beijing, CHINA, Transient pipe network experiment In this validation case a simple branching pipe network is presented. FIG. 11 shows the Flownex model for the branching pipe network. The network consists of 5 pipes with lengths as shown in Table 4. All pipes have a diameter of m and a pipe internal roughness of 10µm was assumed. PIPE 300, 303 and 304 have fast actuating valves at there ends. Air is used as fluid in the validation cases. Table 4: Branching pipe network input data. Pipe Length PIPE m PIPE m PIPE m PIPE m PIPE m Two validation cases are considered in this paper. For both validation cases the system is initially pressurized to kpa with all valves closed. For the first case considered the valve (VAL-301) at the end of PIPE-304 is opened and for the second case the valve (VAL-302) at the end of PIPE-303 is opened. PT P P VAL-300 PIPE-300 PIPE-301 PIPE PIPE-303 PIPE VAL-302 VAL-301 P P 9 10 PT-305 PT-304 FIG. 11: Flownex simulation model for branching pipe network. The results for Case 1 are shown in FIG. 12. The pressure at the valve being opened (VAL- 301) as well as the pressure at the downstream node of PIPE-300 (VAL-300) is shown. For Case 1 the maximum normalized point differences are 5.44 and 15.7 percent for PT-303 and PT-304 respectively. This high point differences occur at the onset of the transient and is high because a small time difference or lag exists between the measured and Flownex data. Although the time difference is very small the gradient of the graph is very high resulting in large point differences. The Euclidian difference for this case is 2.0 percent for PT-303 and 5.0 percent for PT-304 which is within the acceptance criteria. For Case 2 the pressure at the valve being opened (VAL-302) as well as the pressure at the downstream node of PIPE-300 (VAL-300) is shown in FIG

12 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C Pressure [kpa] Time [s] EXP_PT-303 FNX_PT-303 EXP_PT-304 FNX_PT-304 FIG. 12: Branching pipe network results Case 1. For Case 2 the maximum normalized point differences for PT-303 and PT-305 are 9.2 and 17.6 percent respectively. The Euclidian difference for this case is 2.0 percent for PT-303 and 6.0 percent for PT-305 which is within the acceptance criteria Pressure [kpa] Time [s] EXP_PT-303 FNX_PT-303 EXP_PT-305 FNX_PT-305 FIG. 13: Branching pipe network result Case 2. 12

13 HTR 2004 Beijing, CHINA, Heat exchanger transient experiment In this validation case a pipe-in-pipe counter flow heat exchanger is considered. A schematic of the experiment is shown in FIG. 14 and the Flownex model in FIG. 15. The heat exchanger consists of two lengths of copper piping with outside diameters of 22mm (wall thickness 0.6mm) and 15mm (wall thickness 0.5mm) respectively, drawn into each other using spacers. The outer flow path is considered as the primary flow channel. The heat exchanger length is 32m and it is insulated to atmosphere. Water is used as the fluid. The input data for this validation case is provided in Table 5. FIG. 14: Schematic of the heat exchanger experiment. Table 5: Heat exchanger input data Parameter Value Primary side inlet area m 2 Secondary side inlet area m 2 Primary side inlet pressure 467 kpa Primary side outlet pressure 452 kpa Secondary side inlet pressure 232 kpa 13

14 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C30 Parameter Value Secondary side outlet pressure 86 kpa Primary and secondary sides roughness 15 µm Primary side mass flow rate [kg/s] 0.1 kg/s Secondary side mass flow rate [kg/s] 0.2 kg/s Fluid path length [m] 32 m Increments used 32 Wall material conductivity [W/(m.K)] 401 W/m.K Wall material heat capacity [kj/k] kj/k Initially the total inlet temperature of both streams is 26 C. Starting at steady-state conditions a temperature transient is introduced in the primary flow channel by increasing the total inlet temperature from 26 C to 60 C. T 1 P 1 M 2 M 4 2 T 3 P FIG. 15: Flownex network of counter flow heat exchanger. FIG. 16 shows the outlet temperatures of the primary and secondary flow channels. The normalized point differences are summarized in Table 6. A maximum point difference of 20.8 percent occurs at the primary flow channel inlet temperature. The Euclidian difference for this case is less than 5 percent for all the temperature measurements. The Flownex results compare reasonably well but the Flownex model is responding slower that the experiment. This is being investigated further. Normalized Point difference Table 6: Heat exchanger normalized point differences. Primary side inlet Primary side outlet Secondary side inlet Secondary side outlet 20.8 % 0.64% 0.4% 2% 14

15 HTR 2004 Beijing, CHINA, Temperature [C] Time [s] EXP (Primary inlet) EXP (Primary outlet) EXP (Secondary intlet) EXP (Secondary outlet) FNX (Primary inlet) FNX (Primary outlet) FNX (Secondary inlet) FNX (Secondary outlet) FIG. 16: Primary and secondary heat exchanger temperatures. 5.5 Pebble Bed Micro Model (PBMM) The Pebble Bed Micro Model (PBMM) [6] is a closed three-shaft, recuperative Brayton cycle. It was designed and constructed in 2002 at North-West University. Flownex was used extensively in the design of the PBMM and the prediction of its performance. The PBMM is based on the Brayton power cycle (as used in aircraft engines) but with the following distinguishing features: It uses nitrogen as the working fluid. The gas moves around in a closed circuit, which implies that no nitrogen is consumed in the power generation process it merely acts as an energy carrier. The PBMM uses single stage centrifugal compressors and turbines. It makes use of three separate shafts, one for the high-pressure (HP) compressor/turbine pair, one for the low-pressure (LP) compressor/turbine pair and one for the power turbine (PT) and generator. This allows the HP and LP compressor/turbine pairs to run at high speeds thereby reducing the size and therefore also the cost of the machines. It makes use of a recuperator to recover heat that would otherwise have been rejected to atmosphere. The recovered heat is transferred elsewhere in the system thereby reducing the heat required from the heat source and ultimately increasing the thermal efficiency of the plant. The generator is emulated by a load compressor connected to a power dissipation loop consisting of a flow control valve and a heat exchanger. Variations in load are effected by increasing or decreasing the pressure level in the load rejection loop. 15

16 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C30 Turbo machines Pre Cooler Electrical Heater Inter Cooler Recuperator FIG. 17: Solid model of the PBMM. The first phase of the PBMM project was to prove the feasibility of a three-shaft recuperative Brayton cycle. The next phase in the PBMM project was to gather experimental data for the V&V of Flownex. During this phase all instrumentation was calibrated to ensure reliable experimental data. The focus of the PBMM in the V&V effort for Flownex is to show that Flownex can accurately model an integrated thermal-fluid system. This means that the experimental data obtained from the PBMM will be extremely important for the V&V of Flownex. An integrated-effects test on the system as a whole will contribute to the validation of many advanced features of Flownex, especially the capability of the code to correctly balance the performance characteristics of all components to find the system operating point. In the PBMM validation case presented in this paper a steady-state comparison will be done with the following boundary conditions: A low pressure compressor (LPC) inlet pressure of 99kPa and an electrical heater outlet temperature of 600 C. Refer to the PBMM data pack [5] for all the component details. FIG. 18 and FIG. 19 show the pressure and temperature conditions at the inlet and outlets for the various components as well as the percentage differences between the measured data and Flownex results. As can be seen the Flownex and experimental results correlate very well for the pressures. Larger difference exists for the temperature comparisons. The origin of these differences will be explained in the next paragraph. 16

17 HTR 2004 Beijing, CHINA, Pressure [kpa] PC PC LPC LPC IC IC HPC HPC EXP FNX % Difference RXHP RXHP HPT HPT LPT LPT FIG. 18: Component inlet and outlet pressures for the PBMM. PT PT RXLP RXLP Temperature [C] PC PC LPC LPC IC IC HPC HPC RXHP RXHP EXP FNX % Difference HPT HPT LPT LPT FIG. 19: Component inlet and outlet temperatures for the PBMM. FIG. 20 shows the comparison between Flownex and the PBMM data on a T-s diagram. The simulation was done prior to the steady-state run and no change was made to the Flownex simulation model to improve the correspondence with the measured values. PT PT RXLP RXLP 17

18 VERIFICATION AND VALIDATION OF THE HTGR SYSTEMS CFD CODE FLOWNEX #C Temperature [C] Entropy [kj/kg K] Experimental Flownex Calculated FIG. 20: Comparison opf Measured and Simulated steady state values of the PBMM. The agreement between the experimental values and the simulated values for the compressors is quite good while the agreement for the turbines is not. Part of the reason for the poor agreement is clear from the graph. From the experimental values it can be seen that the nitrogen leaving the high pressure compressor (point 1) is heated as it flows on the inside of the pressure vessel to the inlet of the recuperator (point 2). At least some of this heat comes from the exposed outside surfaces of the turbines. Such a heat loss will decrease the outlet temperatures and the entropies at the outlet of all the turbines. These heat losses were not taken into account in the Flownex simulation. We are in the process of analysing the experimental data and quantifying the heat losses. The simulation model will then be changed to incorporate these heat losses. In order to justify the above statement the following adjustment was made to the experimental line for the turbines to obtain a more meaningful comparison with Flownex. The measured temperature and pressure data for the compressors were assumed to be reliable because the measuring points could be properly placed. This can also be seen from the graph were the correspondence between the experimental and simulated values for the compressors are quite good. Therefore it was assumed that the power requirement for the compressors could be correctly calculated. Starting from the same inlet conditions for the high pressure turbine, and using the calculated power consumption of the HP compressor, the outlet temperature for the HP turbine was calculated. Using the same approach, the outlet conditions of the LP turbine and the power turbine were calculated. These are shown as the green ( Calculated ) line in FIG. 20 The correspondence between the calculated and simulated values gives an indication of Flownexs ability to simulate the turbo compressor sets. 6. CONCLUSIONS An overview of the Flownex verification and validation process as well as a number of experimental validation cases have been presented in this paper. These cases consisted of a steady-state mass balance experiment, a simple transient pipe experiment, a volume blow 18

19 HTR 2004 Beijing, CHINA, down experiment, a heat exchanger experiment and a steady-state comparison with the Pebble Bed Micro Model (PBMM). The comparison with the experimental data proved to be extremely good with average differences of less than 10 percent achieved in all cases. Future work planned for the PBMM includes steady-state experiments at higher temperatures and pressures as well as transients experiments. 7. NOMENCLATURE BM FNX HPC HPT HS IC LPC LPT PBMM PBMR PC PT RXLP RXHP V&V Benchmark Flownex High pressure compressor High pressure turbine Heat source Inter-cooler Low pressure compressor Low pressure turbine Pebble Bed Micro Model Pebble Bed Modular Reactor (Pty) Ltd Pre-cooler Power turbine Recuperator low pressure side Recuperator high pressure side Verification and validation 8. Acknowledgements This work was carried out in association with M-Tech Industrial (Pty) Ltd on contract for PBMR (Pty.) Ltd. 9. References 1. Flownex Version 6.5 User Manual, M-Tech Industrial, Potchefstroom, South Africa, Landman, W.A. and Greyvenstein, G.P., Dynamic systems simulation code for the modelling of HTGR power plants, To be published. 3. Greyvenstein, G.P., An implicit method for the analysis of transient flow in pipe networks, Int. J. Numer. Meth. Engrng., 53, , Greyvenstein, G.P. and Van Ravenswaay J.P., Dynamic modelling of heat, mass and momentum transfer in the Pebble Bed Modular Reactor, 1st International Conference on Heat Transfer, Fluid Mechanics, and Thermodynamics, Kruger Park, South Africa, April 8-10, Greyvenstein, G.P. and Labuschagne, J.T., Pebble Bed Micro Model data pack. North West University, South Africa Greyvenstein, G.P. and Rousseau, P.G., Design of a physical model of the PBMR with the aid of Flownet, Nuclear Engineering and design, 222(2003)