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DEVELOPMENT OF A CORE THERMO-FLUID ANALYSIS CODE FOR PRISMATIC GAS COOLED REACTORS
http://dx.doi.org/10.5516/NET.02.2014.020
DEVELOPMENT OF A CORE THERMO-FLUID ANALYSIS CODE FOR PRISMATIC GAS COOLED REACTORS NAM-IL TAK*, SUNG NAM LEE, MIN-HWAN KIM, HONG SIK LIM, and JAE MAN NOH Korea Atomic Energy Research Institute Daedeok-Daero 989-111, Yuseong-gu, Daejeon 305-353, Korea * Corresponding author. E-mail : [email protected] Received February 25, 2014 Accepted for Publication June 16, 2014
A new computer code, named CORONA (Core Reliable Optimization and thermo-fluid Network Analysis), was developed for the core thermo-fluid analysis of a prismatic gas cooled reactor. The CORONA code is targeted for whole-core thermo-fluid analysis of a prismatic gas cooled reactor, with fast computation and reasonable accuracy. In order to achieve this target, the development of CORONA focused on (1) an efficient numerical method, (2) efficient grid generation, and (3) parallel computation. The key idea for the efficient numerical method of CORONA is to solve a three-dimensional solid heat conduction equation combined with one-dimensional fluid flow network equations. The typical difficulties in generating computational grids for a whole core analysis were overcome by using a basic unit cell concept. A fast calculation was finally achieved by a block-wise parallel computation method. The objective of the present paper is to summarize the motivation and strategy, numerical approaches, verification and validation, parallel computation, and perspective of the CORONA code. KEYWORDS : Prismatic Core, Fuel Temperature, Fluid Flow Network, Gas Cooled Reactor, CORONA, VHTR
1. INTRODUCTION A prismatic gas cooled reactor is a candidate design for national research programs such as the next generation nuclear plant (NGNP) project of the U. S. [1], and the nuclear hydrogen production and demonstration (NHDD) project of Korea [2]. For the smooth success of the NHDD project, the Korea Atomic Energy Research Institute (KAERI) has been developing key technologies for the design of a prismatic very high temperature reactor (VHTR) [3]. As one of the crucial technologies for the NHDD project, the development of a core thermo-fluid analysis code, named CORONA (Core Reliable Optimization and thermo-fluid Network Analysis), started in 2009. The CORONA code is targeted for a whole core thermo-fluid analysis of a prismatic gas cooled reactor with fast computation and reasonable accuracy. Figs. 1 and 2 show the conceptual view of a typical prismatic reactor core and its fuel block respectively [4]. The active part of the prismatic core consists of a large number of vertical stacks of a fuel block. Accurate prediction of the fuel temperature is crucial to ensure the barrier against the release of fission products into the primary coolant. However, the complex geometry of a prismatic fuel block hinders accurate evaluations of the temperature distribution without elaborate numerical calculations. NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
Therefore, during the development of existing prismatic gas-cooled reactor designs (e.g., MHTGR of the General Atomics [5] and HTTR of Japan [6]), efficient numerical methods were applied to analyse the thermo-fluid behaviour of prismatic cores. The key idea of these efficient methods is to solve a three-dimensional solid heat conduction equation combined with one-dimensional fluid flow equations. A combined method was adopted in the DEMISE code of General Atomics [7] and the FLOWNET/TRUMP codes of Japan [8]. Recently the existing idea was improved by the present authors [9] and implemented into the CORONA code. Moreover, the fluid flow model described in [9] has been further improved through a one-dimensional network model. The present paper summarizes the motivation and strategy, major outcomes, and perspective of the development of the CORONA code.
2. DEVELOPMENT MOTIVATION AND STRATEGY The recent rapid development of computational fluid dynamics (CFD) technology provides a powerful tool to carry out elaborate numerical calculations with complex geometries for nuclear applications. Some CFD codes such as CFX [10] and STAR-CD [11] are commercially available. Thanks to the commercial CFD codes, the ap641
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Fig. 1. Conceptual View of Typical Prismatic Reactor Core
Fig. 2. Standard Fuel Block of Typical Prismatic Reactor
plication areas of CFD technology have been growing. Commercial CFD codes are user-friendly and well validated against most industrial problems which have a single phase fluid. In spite of the remarkable development of CFD technology, however, a whole prismatic core CFD analysis with geometric details is still a challenging task. Since fine meshes are required for accurate CFD calculations, most CFD applications are limited to local behaviours (e.g., single fuel column [12,13], and seven fuel columns [14]) in the design of a prismatic reactor. In 2009, Pointer and Thomas [15] published the STAR-CD results for a whole core of a prismatic reactor. It is clear that such a CFD analysis requires tremendous computational efforts. Additional significant efforts have to be devoted to generate computational grids of high quality for a good convergence. Such efforts are valuable for a final calculation but are too taxing for a designer who wants a large number of calculations with various design options. The other available option at KAERI could be the use 642
of system codes such as MARS-GCR [16] and GAMMA+ [17]. It should be noted, however, that the system codes are targeted for system transients. Their meshes are too coarse for a detailed thermo-fluid analysis of a prismatic fuel block. The accuracy of the system code calculations is not satisfactory for the core thermal design of a prismatic reactor (e.g., hot spot analysis). Therefore, a new computer code named CORONA has been developed to overcome the difficulties in the available tools at KAERI, i.e., CFD codes and system codes. The CORONA code is intended for whole-core thermofluid analysis of a prismatic gas cooled reactor with fast computation and reasonable accuracy. Fig. 3 summarizes the thermo-fluid analysis codes and their application areas for the design of prismatic gas-cooled reactors at KAERI. The position of CORONA lies in between CFD and system codes. Its position is similar to that of a subchannel analysis code such as COBRA [18] used in a light water cooled reactor design. NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Fig. 3. Thermo-fluid Analysis Codes and Their Areas for the Design of Prismatic Gas-cooled Reactors at KAERI
In order to achieve the target of the CORONA code (i.e., fast computation of whole core thermo-fluid analysis with reasonable accuracy), three fundamental strategies were established and applied to the development of CORONA: (1) an efficient numerical method, (2) efficient grid generation, and (3) parallel computation. ○ Efficient Numerical Method: The use of an efficient numerical method is the most important requirement for a new code to achieve its goal. The CORONA code adopts the existing idea of the efficient numerical method used for the design of MHTGR and HTTR (e.g., DEMISE and FLOWNET/TRAMP). This efficient method solves a three-dimensional heat conduction equation for a solid like a CFD code and one-dimensional conservation equations for a fluid flow like a system code. In 2012, Travis and El-Genk [19] verified the efficiency of this combined method using STAR-CD calculations. ○ Efficient Grid Generation: For a whole prismatic core thermo-fluid simulation, grid generation can also be a time-consuming process. Therefore, an efficient grid generation method using the concept of basic unit cells was developed by the present authors [9,20]. This concept of the basic unit cells adopted in the CORONA code enables the effective generation of unstructured grids for prismatic fuel blocks without a special mesh generator. The revision of the generated grids is also simple and convenient. ○ Parallel Computation: Reliance on a parallel computing technique is inevitable for a whole prismatic core simulation since significant numbers of meshes (e.g., in the order of ten million) are required. Parallel computation leads not only to a speed up of the calculations but also allows the use of a normal personal computer (PC). NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
Major applications of the CORONA code are (1) a steady-state hot spot analysis, (2) steady-state analysis for off-design conditions such as a coolant channel blockage accident, (3) some transient scenarios (e.g., rod ejection), (4) a coupled analysis (e.g., with neutronics) for high fidelity, (5) design optimization calculations. The CORONA code is written in C++ and can be run on either Windows or Linux operating system.
3. NUMERICAL APPROACHES 3.1 Grid Generation Using Basic Unit Cells
The first step in the numerical calculation of thermofluid phenomena is to set up a grid. The unique feature of the CORONA code is the use of the basic unit cells for the generation of computational grids. The concept of basic unit cells provides an efficient generation of unstructured computational grids to model an entire fuel block. The present authors noticed that the prismatic fuel blocks have regular arrangement of hexagonal or pentagonal unit cells as shown in Fig. 4. Prismatic fuel blocks can be represented by several types of basic unit cells and are shown in Fig. 5. The unstructured computational grids are defined within the basic unit cells. Using the combination of the basic unit cells, unstructured grids can be efficiently generated to simulate the thermo-fluid behavior of the fuel blocks. Fig. 6 is provided in order to improve the understanding of the relation between the basic unit cells and the computational grids. It shows the assignment of the basic unit cells to generate the computational grids of the standard fuel block. The change in generated grids is also simple and convenient. Fig. 7 illustrates an example of computational grids generated using the basic unit cells. Fig. 8 shows the modeling procedure from the basic unit cells to a whole prismatic core. Initially, computational grids are defined within each type of basic unit cell. 643
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Fig. 4. Typical Unit Cell Arrangements in Prismatic Fuel Blocks
Fig. 5. Basic Unit Cell Types and Examples of Computational Grids
Fig. 6. Basic Unit Cell Assignment to Generate Computational Grids of Standard Fuel Block
Fig. 7. Example of Computational Grids Generated Using the Basic Unit Cells for the Standard Fuel Block
Then, using an axial combination of the basic unit cells, a hexagonal (or pentagonal) body called a ‘pin’ is made. Three-dimensional computational grids for a hexagonal fuel (or reflector) block are generated using a two-dimensional combination of pins. Modeling of a whole core is finally finished using a combination of hexagonal fuel and reflector blocks. The number of computational grids for a solid used in
the CORONA code has a similar level as the CFD analysis. However, it should be noted that the computational expense of a solid heat conduction equation (i.e., Eq. (1)) is much less than that of fluid conservation equations in a whole core CFD analysis of a prismatic core.
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3.2 Numerical Model for Solid
In order to simulate the heat conduction through the
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Fig. 8. Modeling Procedure from Basic Unit Cell to Whole Prismatic Core
complex geometry of the fuel block, the CORONA code solves the following three-dimensional heat conduction equation: (1) The CORONA code adopts the finite volume method to solve Eq. (1). The key step of the finite volume method is the integration of Eq. (1) over a control volume to yield a discretised equation: (2) The obtained discretised equation can be expressed for a general nodal point P as follows: (3) (4)
(5) where the superscript 0 represents the known value at the previous time step and the subscript i indicates the neighboring node. In Eq. (5), kˆs,i and δVˆi represent the thermal conductivity and the node volume at the node face between the node P and the neighboring node i. Equation (3) is a well-known form for unstructured grids and sinon is the cross-diffusion term resulting from the non-orthogonality of grids. This term disappears if the grids are fully NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
orthogonal. At a fluid boundary, the following equation is governed: (6) Equations (3) and (6) can be rearranged into matrix form as (7) Equation (7) can be solved simultaneously for all nodes if the material properties and qs,iconv are known. The detailed derivation from Eq. (1) to Eq. (7) is not given in this paper since it is lengthy and widely available in a CFD textbook dealing with unstructured meshes (e.g., see Versteeg and Malalasekera [21]).
3.3 Numerical Model for Fluid Flow
In contrast to the solid geometry of the fuel blocks, the fluid in the prismatic fuel blocks flows through simple circular channels. A fraction of the core flow bypasses the fuel block coolant channels and passes through the gaps between the hexagonal columns. Such gaps can also be considered as rectangular channels connected to each other. For the fluid flow within a prismatic core, a one-dimensional approach can therefore be a good approximation. Such an approximation avoids fine meshes near the walls as well as turbulence conservation equations which are one of the major challenges in a CFD analysis. A onedimensional approximation for the duct flows enables a fast calculation with reasonable accuracy. In addition, there is potential for an exchange of coolant between the bypass gaps and the coolant channels due to cross flow gaps which are small spaces between the horizontal faces of the blocks. Therefore, a one-dimensional network model has been widely used for a fluid flow in a prismatic core [7]. 645
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Fig. 9. Concept of Node and Junction for Fluid Network
A fluid flow network model implemented into the CORONA code is based on the pressure correction method [22~24]. One-dimensional conservation equation for a fluid flow can be written as:
The first step to solve Eqs. (11) and (12) is to guess the pressures at all nodes. These values are treated as prelimi¯ The correction δP is denary values and donated by P. fined as the difference between the correct pressure field ¯ and the guessed pressure field P.
(8)
(13) Similarly, the volumetric flow rate correction δQ and density correction δρ are defined as: (14)
(9)
(15) The density of the gas can be obtained using the ideal gas law. (16)
(10) Under normal operating conditions, the thermo-fluid behavior in a prismatic gas-cooled reactor core can be described using the simplified forms of Eqs. (8)~(10) based on the steady-state, constant flow area, and negligible buoyance assumptions. Fig. 9 shows the concept of a node and junction to derive the mathematical formulation for an arbitrary network of one-dimensional fluid flows. In this paper, the simplified equations which are applicable to normal operating conditions are described in the following derivations since a steady-state hot spot analysis is the most crucial concern of the CORONA calculations. Initially, the steady-state continuity equation is applied at a one-dimensional fluid node (i) having an arbitrary number of initiating junctions (Ii) and terminating junctions (Ti ), as shown in Fig. 9(a). (11)
Where R is the gas constant and zc is the compressibility factor. Substitution of Eqs. (13)~(16) into the momentum equation (i.e., Eq. (12)) and neglecting the terms involving the products of the corrections, yield the following equation for the volumetric flow rate correction.
(17) where (18)
(19)
For a junction (j) having upstream fluid node (iuj) and downstream fluid node (idj) shown in Fig. 9(b), the simplified momentum equation can be applied as
Finally, the substitution of Eqs. (14), (15) and (17) into the continuity equation (i.e., Eq. (11)) yields the following equation for the pressure correction.
(12)
(20)
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Where aj, dj, cj, and dj are the rearranged coefficients in terms of each δP node respectively and ei is the source term. With the known field of δPi, δQj can be obtained using Eq. (17). Then, new values for Pi, Qj, and ρf,i are determined using Eqs. (13), (14), and (16). The newly updated values are now considered as the preliminary values for the next iteration and the whole process is repeated a number of times after which the fluid energy conservation equation is solved. The steady-state one-dimensional form for the energy conservation equation governing the fluid flow network can be derived as
(21)
where hi is the enthalpy of the fluid node i and αj is the parameter for the donor property defined as (22) The central differencing scheme can be used as an alternative option for Eq. (21). In the case of a single pipeline, Eqs. (20) and (21) can be solved using the tridiagonal matrix algorithm (TDMA), while sparse matrix techniques can be applied in the case of complex flow networks.
3.4 Interfacing of Solid and Fluid Models
The coupling between the three-dimensional solid model and the one-dimensional fluid model uses a typical Nusselt number correlation such as: (23) (24)
Eq. (23) is the correlation of McEligot et al. [7, 25] which is one of the most popular correlations for a prismatic gas-cooled reactor core under normal operating conditions. The entrance effect is neglected due to little interest in the entrance region. Eqs. (6) and (21) are combined as follows: (25)
(26)
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Fig. 10. Overall Computational Procedure for a Steady-State Analysis
where i represents each surface of the solid contacted with the fluid node j and As,i,j is the heat transfer area between the solid surface i and fluid node j. The overall computational procedure for a steadystate analysis is shown in Fig. 10. When the maximum value of the fluid temperature change during the iteration is less than the user supplied parameter, the convergence is regarded as having been achieved.
4. VERIFICATION AND VALIDATION 4.1 Heat Conduction in Unit Cells
In order to verify the multi-dimensional heat conduction within unit cells of the prismatic fuel block, two conceptual problems using unit cells were investigated [20]. The verification study started from a single unit cell problem which is one of the simplest test cases. The single unit cell consists of a fuel compact, helium gap, and graphite region. A uniform heat generation in the fuel compact was assumed 647
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Fig. 11. Result of Verification of CORONA Using Single Unit Cell Problem
Fig. 13. Result of Verification of CORONA Using Seven Unit Cell Problem
Fig. 12. Layout of Seven Unit Cell Problem and Its Boundary Conditions
and the temperature of the outmost boundary was fixed. Since it is believed that a commercial CFD code can solve a multi-dimensional heat conduction problem accurately, a commercial CFD code, CFX, is used for the verification of the CORONA code for the multi-dimensional heat conduction problem. Fig. 11 compares the predicted temperature profile by CORONA with that by CFX. It shows that CORONA accurately solves the multi-dimensional heat conduction equation. It is obvious that a finer mesh provides a more accurate result. The second case considers seven unit cells composed of one coolant cell (designated as ‘C’) surrounded by six fuel cells (designated as ‘F’) as shown in Fig. 12. A uniform heat generation in the fuel compacts was assumed and the temperature at the coolant boundary was fixed. In addition, the adiabatic boundary condition was imposed on the outmost boundary of the unit cells. Fig. 13 shows the predicted temperature profiles along Line A in Fig. 12. It was observed that when a fine mesh is adopted, the accuracy of CORONA is close to that of CFX. 648
Fig. 14. Schematic of Kaburaki and Takizuka Experimental Apparatus for Bypass Flow and Crossflow [25]
4.2 Bypass Flow and Crossflow in Fuel Column
An air flow experiment performed by Kaburaki and Takizuka [26] was selected to validate the one-dimensional fluid flow network model implemented in CORONA [27]. Fig. 14 shows a schematic of their experimental apparatus. Four graphite blocks were stacked up in a column and surrounded by a steel shroud to simulate the bypass gap (~1.2mm) between fuel columns. The hexagonal fuel block has 12 coolant holes 20 mm in diameter. One artificial cross flow gap (~1 mm) was created in the middle of the column to simulate a crossflow gap between fuel blocks. The air at atmospheric pressure was drawn from the top of the test section and flowed down the channels and the bypass gap. The bottom of the bypass gap was completely sealed and the coolant leakage flow, which entered the bypass gap, was merged into the main coolant channels through the crossflow gap. The mass flow rate NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
and the static pressure distribution in each coolant channel were measured. The measurement uncertainties of the experiment were not published and the sensitivity calculations of CORONA confirmed later that the effect of the entrance and exit form losses was small. Fig. 15 shows a comparison of the CORONA prediction results with the measured pressures. A good agreement can be seen along all of the stacks. This was achieved by applying the correlation of Kaburaki and Takizuka developed based on their experiment. This means that the described one-dimensional fluid flow model is correctly implemented into the CORONA code and is able to simulate the bypass flow and crossflow phenomena reasonably.
4.3 Conjugate Heat Transfer in Fuel Column
In order to verify a conjugate heat transfer in a prismatic fuel block, two conceptual problems with single fuel columns were investigated [28, 29]. Two types of fuel blocks (i.e., standard and reserved shutdown control (RSC) fuel blocks) were considered. Since it is believed that a commercial CFD code is well verified against a single phase conjugate heat transfer problem, the verification of the CORONA code is made using a comparison with the results of CFX. However, it should be noted that a commercial CFD code also needs a sufficient validation for prismatic core conditions (i.e., high temperature helium flow at a high heat flux condition in a prismatic fuel block geometry). Table 1 shows the major thermo-fluid parameters for the considered single fuel column problems under consideration. Fig. 16 shows the CFX meshes used to simulate the single fuel column problems. Using symmetry, 1/12 and 1/2 sections of the geometries were modeled for CFX calculations of the standard and RSC fuel blocks, respectively.
Fig. 15. Result of Validation of CORONA Using the Kaburaki and Takizuka Experiment Table 1. T hermo-fluid Parameters for Single Fuel Column Problems Standard Fuel Column
RSC Fuel Column
Column thermal power (MW)
3.118
2.832
Column flow rate (kg/s)
1.207
1.268
490
490
System pressure (MPa)
7
7
Number of stacked fuel blocks
6
6
Active core height (m)
4.758
4.758
Bypass gap size (mm)
2
2
Crossflow gap size (mm)
1
1
Uniform
Uniform
Parameter
Coolant inlet temperature ( C) o
Axial power profile
Fig. 16. CFX Meshes to Simulate Single Fuel Column Problems NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
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Fig. 17. Comparison of Predicted Axial Temperature Distributions at the Center of the Hottest Fuel Compact for the Standard Fuel Column Problem
Fig. 19. Comparison of Predicted Coolant Velocities at the Control Hole for the RSC Fuel Column Problem
fair comparison of the computation time is not feasible due to differing convergence criteria and a differing number of meshes. However, it is obvious that significant reduction in the computation time can be achieved by CORONA and the difference in the computation time would dramatically soar in the case of a whole core analysis.
5. PARALLEL COMPUTATION 5.1 Block-Based Parallel Computation
Fig. 18. Comparison of Predicted Axial Temperature Distributions at the Center of the Hottest Fuel Compact for the RSC Fuel Column Problem
Figs. 17 and 18 compare the predicted axial temperature distribution at the center of the hottest fuel compact. They clearly show good agreements. Non-continuing regions between fuel blocks are generated due to the non-fuel zones (i.e., graphite plug and seat). A comparison of the coolant velocities at the control hole of the RSC fuel column is shown in Fig. 19. It also shows a good agreement. Figs. 17~19 obviously indicate that the CORONA code can reliably simulate a conjugate heat transfer in the prismatic fuel blocks. Although the predicted temperature profiles by CORONA are comparable to those of CFX, the used computation time is significantly different. In order to obtain the result in Fig. 18, for example, about 46 hours were spent by CFX using four CPUs whereas about 4 minutes were spent by CORONA using a single CPU of the same computer. A 650
Parallel computing capability is important for the CORONA code since a whole prismatic core simulation requires a huge computing cost and a significant amount of computer memory. For a simple implementation of parallel computing capability, a block-based parallel computation method was developed for the CORONA code [30]. The idea of the block-based parallel computation method is based on the fact that each fuel (or reflector) block is completely enclosed by the fluid boundaries due to the existence of bypass and cross flows as shown in Fig. 20. Therefore, the heat conduction matrix shown in Eq. (7) can be naturally decomposed into much smaller matrixes that represent each fuel (or reflector) block. The decomposed matrix can be solved independently using a parallel computation library.
5.2 Application to Whole Core Analysis
In order to examine the improved performance by the parallel computation, an example problem for a whole prismatic core was investigated. PMR600 [2] was selected as a reference core. The major thermo-fluid boundary conditions are shown in Table 2. The reactor power profile obtained by the neutronic analysis [31] was applied. The applied power distribution is shown in Fig. 21. The heat loss to the reactor cavity cooling system (RCCS) was neglected. Fig. 22 shows the calculation model for the CORONA NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
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Fig. 20. Fuel Blocks Enclosed by Fluid Boundaries Table 2. M ajor Thermo-fluid Parameters for PMR600 Benchmark Calculation Parameter
Value
Core thermal power (MW)
600
No. of fuel columns
108
No. of fuel blocks per column
9
No. of reflector columns with control rod hole
36
Core flow rate (kg/s) Coolant inlet temperature (oC)
Fig. 21. Axial Power Distribution Used for PMR600 Benchmark Calculation
251.3 490
System pressure (MPa)
7
Active core height (m)
7.929
Bypass gap size (mm)
2
Crossflow gap size (mm)
0
calculation. The total numbers of solid and fluid nodes used were ~39,000,000 and ~200,000, respectively. The calculations were carried out on two PCs with Microsoft Windows 7 Enterprise K. Table 3 shows the specification of the two PCs used. The communications between the two PCs were made using Gigabit Ethernet connections. Intel ® MPI [32] was used for parallel computation library on the Windows operating system. Fig. 23 shows the predicted maximum temperature within each fuel and reflector column based on a serial calculation. It shows that the predicted temperatures are reasonable. The same calculations were repeated based on parallel computations. Nine cases were tested to investigate the parallel performance and Fig. 24 summarizes the result for the test problem. It was thoroughly checked whether all the cases produced the same results. Fig. 24 shows that the computing speed is significantly improved by the parallel computation. The computing time was NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
Fig. 22. CORONA Model for 1/6 Core of PMR 600
Table 3. S pecification of Two PCs for Parallel Computation Benchmark CPU
No. of CPUs
RAM
PC 1
Intel Xeon X5690, Six Core, 3.47 GHz
2
144 GB
PC 2
Intel Xeon X5690, Six Core, 3.47 GHz
2
96 GB
reduced with the number of processors. The maximum number of processors per PC was limited to 4 due to the random access memory (RAM) capacity of the PCs. 651
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blocks. The usefulness of the CORONA code was proved by the example calculations using a whole prismatic core model. Further research is on-going to improve the physical models and increase the capability of the CORONA code. The major items are (1) coupling with the DeCART code [33] for a high fidelity calculation, (2) variable bypass (and crossflow) gap size model for consideration of the change in the size with the location as well as the burnup, and (3) transient simulation capability. It is believed that the CORONA code can be practically used in a whole core thermo-fluid analysis for the design of a prismatic gas cooled reactor. In addition, it is envisaged that the excellence of the CORONA code will also be highlighted in a high fidelity analysis combined with a whole core neutron transport calculation. Fig. 23. Predicted Maximum Temperature within Each Fuel and Reflector Column of PMR 600
Fig. 24. Result of Parallel Computation Performance Test Using PMR600 Problem
6. CONCLUSIONS AND OUTLOOK The development of the CORONA code was started in KAERI for the establishment of the key technologies for the design of a prismatic VHTR. The CORONA code is targeted for a whole core thermo-fluid analysis of a prismatic gas cooled reactor with fast computation and reasonable accuracy. In order to achieve the target, the development of CORONA focused on (1) an efficient numerical method using the combination of a three-dimensional heat conduction and one-dimensional fluid flow network, (2) efficient grid generation using basic unit cells, and (3) parallel computing capability. The results of the verification and validation studies show that the CORONA code can provide a reasonably accurate result for the thermo-fluid analysis of prismatic fuel 652
NOMENCLATURE
ai ai* aj a0p ap* As bj bp
= Influence coefficient at node i, defined in Eq. (5) = Rearranged influence coefficient of ai, used in Eq. (7) = Coefficient for junction j, used in Eq. (20) = Influence coefficient at node P, defined in Eq. (5) = Rearranged influence coefficient of ap, used in Eq. (7) = Solid surface area [m2] = Coefficient for junction j, used in Eq. (20) = Source term for node P at temperature matrix, used in Eq. (7) cj = Coefficient for junction j, used in Eq. (20) Cj = Resistance coefficient at fluid node j, defined in Eq. (18) Cs = Specific heat of solid [J/kg K] dj = Coefficient for junction j, used in Eq. (20) D = Hydraulic diameter [m] ei = Source term for node i at pressure matrix, used in Eq. (20) f = Friction factor [-] g = Gravitational acceleration constant [m/s2] G = Parameter defined in Eq. (19) h = Fluid enthalpy [J/kg] Ii = Initiating junction of node i kf = Thermal conductivity of fluid [W/m K] ks = Thermal conductivity of solid [W/m K] K = Form loss factor [-] L = Fluid channel length [m] N = Number of meshes Nu = Nusselt number [-] P = Pressure [Pa] Pr = Prandtl number [-] qs‴ = Power density generated in solid [W/m3] qfconv = Convection heat transfer at fluid [W] qsconv = Convection heat transfer at solid [W] Q = Volumetric flow rate [m3/s] R = Gas constant of helium [J/kg K] Re = Reynolds number [-] sinon = Cross-diffusion term t = Time [s] Tf = Fluid temperature [K] NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
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Ti Ts V w z zc
= Terminating junction of node i = Solid temperature [K] = Volume [m3] = Fluid velocity [m/s] = One-dimensional coordinate = Compressibility factor [-]
Greek
α ρ
= Parameter defined in Eq. (22) = Density [kg/m3]
ACKNOWLEDGEMENTS
This work was supported by Nuclear R&D Program of the National Research Foundation of Korea which is grant funded by the Korean government (Grant code: NRF2012M2A8A2025679).
REFERENCES_______________________________ [ 1 ] Idaho National Laboratory, “Summary for the Next Generation Nuclear Plant Project in Review,” INL/EXT-10-19142, Revision 1 (2010). [ 2 ] J. Chang, Y. W. Kim, K. Y. Lee, Y. W. Lee, W. J. Lee, J. M. Noh, M. H. Kim, H. S. Lim, Y. J. Shin, K. K. Bae, and K. D. Jung, “A Study of a Nuclear Hydrogen Production Demonstration Plant,” Nucl. Eng. Technol., vol. 39, no. 2, pp. 111-122 (2007). [ 3 ] J. M. Noh et al., “Development of Very High Temperature Reactor Design Technology,” KAERI/RR-3462/2011 (Written in Korean), Korea Atomic Energy Research Institute (2012). [ 4 ] C. K. Jo, H. S. Lim, and J. M. Noh, “Preconceptual Designs of the 200MWth Prism and Pebble-bed Type VHTR Cores,” PHYSOR 2008, Interlaken, Switzerland, Sep. 14-19, 2008. [ 5 ] A. J. Neylan, D. V. Graf, and A. C. Millunzi, “The Modular High Temperature Gas-Cooled Reactor (MHTGR) in the U.S.,” Nucl. Eng. Des., vol. 109, pp. 99-105 (1988). [ 6 ] S. Shiozawa, S. Fujikawa, T. Iyoku, K. Kunitomi, and Y. Tachibana, “Overview of HTTR design features,” Nucl. Eng. Des., vol. 233, pp. 11-21 (2004). [ 7 ] G. Melese and R. Katz, Thermal and Flow Design of HeliumCooled Reactors, American Nuclear Society, Illinois USA (1984). [ 8 ] S. Maruyama, N. Fujimoto, Y. Sudo, Y. Kiso, H. Hayakawa, “Fuel Temperature Analysis Method for Channel-Blockage Accident in HTTR,” Nucl. Eng. Des., vol. 150, pp. 69-80 (1994). [ 9 ] N. I. Tak, M. H. Kim, H. S. Lim, and J. M. Noh, “A Practical Method for Whole Core Thermal Analysis of Prismatic Gas-Cooled Reactor”, Nucl. Tech., vol. 177, pp. 352-365 (2012). [ 10 ] ANSYS Inc., http://www.ansys.com (2014). [ 11 ] CD-adapco, http://www.cd-adapco.com (2014). [ 12 ] N. I. Tak, M. H. Kim, and W. J. Lee,” Numerical Investigation of a Heat Transfer within the Prismatic Fuel Assembly of a Very High Temperature Reactor,” Ann. Nucl. Energy, vol. 35, pp. 1892-1899 (2008). [ 13 ] M. H. Kim, N. I. Tak, and J. M. Noh, “CFD Analysis of Hot Spot Fuel Temperature in the Control Fuel Block Assembly of a VHTR core,” Transactions of the Korean Nuclear Society Autumn Meeting, Jeju, Korea, Oct. 21-22, 2010. NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
[ 14 ] O. Cioni, M. Marchand, G. Geffraye, and F. Ducros, “3D Thermal-Hydraulic Calculations of a Modular Block-type HTR Core,” Nucl. Eng. Des., vol. 236, pp. 565-573 (2006). [ 15 ] W. D. Pointer and J. W. Thomas, “Steady-State, WholeCore Prismatic VHTR Simulation Including Core Bypass,” Proceedings of ICAPP ’10, San Diego, CA, USA, June 1317, 2010, Paper 10310. [ 16 ] W. J. Lee, J. J. Jeong, S.W. Lee and J. Chang, “Development of MARS-GCR/V1 for Thermal-Hydraulic Safety Analysis of Gas-Cooled Reactor Systems,” Nucl. Eng. Technol., vol. 37, no. 6, pp. 587-594 (2005). [ 17 ] H. S. Lim and H. C. NO, “GAMMA Multidimensional Multicomponent Mixture Analysis to Predict Air Ingress Phenomena in an HTGR,” Nucl. Sci. Eng., vol. 152, pp. 87-97 (2006). [ 18 ] C. L. Wheeler, C. W. Stewart, R. J. Cena, D. S. Rowe, and A. M. Sutey, “COBRA-IV-I: An Interim Version of COBRA for Thermal-Hydraulic Analysis of Rod Bundle Nuclear Fuel Elements and Cores,” BNWL-1962, Battelle Pacific Northwest Lab. (1976). [ 19 ] B. W. Travis and M. S. El-Genk, “Thermal–Hydraulics Analyses for 1/6 Prismatic VHTR Core and Fuel Element with and without Bypass Flow,” Energy Conversion and Management, vol. 67, pp. 325-341 (2013). [ 20 ] N. I. Tak , M. H. Kim, H. S. Lim, “Development of a UnitCell Based Multi-dimensional Heat Conduction Model for a Prismatic Fuel Block,” Transactions of the Korean Nuclear Society Spring Meeting, Pyeongchang, Korea, May 27-28, 2010. [ 21 ] H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, The Finite Volume Method, Second Edition, Pearson Education Limited (2007). [ 22 ] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York (1980). [ 23 ] G. P. Greyvenstein and D. P. Laurie, “A Segregated CFD Approach to Pipe Network Analysis,” Int. J. Numer. Meth. Engng, vol. 37, pp. 3685-3705 (1994). [ 24 ] G. P. Greyvenstein, “An Implicit Method for the Analysis of Transient Flows in Pipe Networks,” Int. J. Numer. Meth. Engng, vol. 53, pp. 1127-1143 (2002). [ 25 ] D. M. McEligot, G. E. McCreery, R. R. Schultz, J. Lee, P. Hejzlar, P. Stahle, P. Saha, “Investigation of Fundamental Thermal-Hydraulic Phenomena in Advanced Gas-Cooled Reactors,” INL/EXT-06-11801, Idaho National Laboratory (2006). [ 26 ] H. Kaburaki and T. Takizuka, “Effect of Crossflow on Flow Distribution in HTGR Core Column,” J. Nucl. Sci. Tech., vol. 24 (7), pp. 516-525 (1987). [ 27 ] N. I. Tak, M. H. Kim, and W. J. Lee, “A Benchmark CFD Calculation for a Cross Flow between Fuel Blocks of a Prismatic VHTR,” Transactions of the Korean Nuclear Society Spring Meeting, Jeju, Korea, May 22, 2009. [ 28 ] M. H. Kim, “CFD Model for Hot Spot Analysis of Standard Fuel Block,” NHDD-RD-CA-11-005, Korea Atomic Energy Research Institute (2011). [ 29 ] S. N. Lee, N. I. Tak, M. H. Kim, and J. M. Noh, “ThermoFluid Verification of Fuel Column with Crossflow Gap,” Transactions of the Korean Nuclear Society Autumn Meeting, Gyeongju, Korea, Oct. 24-25, 2013. [ 30 ] N. I. Tak, M. H. Kim, and H. S. Lim, “Parallel Computation for Whole Core Thermo-fluid Simulation of Prismatic 653
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Gas-Cooled Reactor,” Transactions of the Korean Nuclear Society Autumn Meeting, Gyeongju, Korea, Oct. 27-28, 2011. [ 31 ] C. K. Jo, Preliminary Result File of MASTER-GCR Calculation for PMR600, 12BH_B4C_080_R48_MAS_ AS_PF3_Case27_summary.dat (2013).
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[ 32 ] Intel Corporation, “Intel® MPI Library for Windows OS, Reference Manual,” Rev. 4.0 Update 1, http://www.intel. com (2010). [ 33 ] J. Y. Cho et al., “DeCART v1.2 User’s Manual,” KAERI/ TR-3438/2007, Korea Atomic Energy Research Institute (2007).
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DEVELOPMENT OF A CORE THERMO-FLUID ANALYSIS CODE FOR PRISMATIC GAS COOLED REACTORS NAM-IL TAK*, SUNG NAM LEE, MIN-HWAN KIM, HONG SIK LIM, and JAE MAN NOH Korea Atomic Energy Research Institute Daedeok-Daero 989-111, Yuseong-gu, Daejeon 305-353, Korea * Corresponding author. E-mail : [email protected] Received February 25, 2014 Accepted for Publication June 16, 2014
A new computer code, named CORONA (Core Reliable Optimization and thermo-fluid Network Analysis), was developed for the core thermo-fluid analysis of a prismatic gas cooled reactor. The CORONA code is targeted for whole-core thermo-fluid analysis of a prismatic gas cooled reactor, with fast computation and reasonable accuracy. In order to achieve this target, the development of CORONA focused on (1) an efficient numerical method, (2) efficient grid generation, and (3) parallel computation. The key idea for the efficient numerical method of CORONA is to solve a three-dimensional solid heat conduction equation combined with one-dimensional fluid flow network equations. The typical difficulties in generating computational grids for a whole core analysis were overcome by using a basic unit cell concept. A fast calculation was finally achieved by a block-wise parallel computation method. The objective of the present paper is to summarize the motivation and strategy, numerical approaches, verification and validation, parallel computation, and perspective of the CORONA code. KEYWORDS : Prismatic Core, Fuel Temperature, Fluid Flow Network, Gas Cooled Reactor, CORONA, VHTR
1. INTRODUCTION A prismatic gas cooled reactor is a candidate design for national research programs such as the next generation nuclear plant (NGNP) project of the U. S. [1], and the nuclear hydrogen production and demonstration (NHDD) project of Korea [2]. For the smooth success of the NHDD project, the Korea Atomic Energy Research Institute (KAERI) has been developing key technologies for the design of a prismatic very high temperature reactor (VHTR) [3]. As one of the crucial technologies for the NHDD project, the development of a core thermo-fluid analysis code, named CORONA (Core Reliable Optimization and thermo-fluid Network Analysis), started in 2009. The CORONA code is targeted for a whole core thermo-fluid analysis of a prismatic gas cooled reactor with fast computation and reasonable accuracy. Figs. 1 and 2 show the conceptual view of a typical prismatic reactor core and its fuel block respectively [4]. The active part of the prismatic core consists of a large number of vertical stacks of a fuel block. Accurate prediction of the fuel temperature is crucial to ensure the barrier against the release of fission products into the primary coolant. However, the complex geometry of a prismatic fuel block hinders accurate evaluations of the temperature distribution without elaborate numerical calculations. NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
Therefore, during the development of existing prismatic gas-cooled reactor designs (e.g., MHTGR of the General Atomics [5] and HTTR of Japan [6]), efficient numerical methods were applied to analyse the thermo-fluid behaviour of prismatic cores. The key idea of these efficient methods is to solve a three-dimensional solid heat conduction equation combined with one-dimensional fluid flow equations. A combined method was adopted in the DEMISE code of General Atomics [7] and the FLOWNET/TRUMP codes of Japan [8]. Recently the existing idea was improved by the present authors [9] and implemented into the CORONA code. Moreover, the fluid flow model described in [9] has been further improved through a one-dimensional network model. The present paper summarizes the motivation and strategy, major outcomes, and perspective of the development of the CORONA code.
2. DEVELOPMENT MOTIVATION AND STRATEGY The recent rapid development of computational fluid dynamics (CFD) technology provides a powerful tool to carry out elaborate numerical calculations with complex geometries for nuclear applications. Some CFD codes such as CFX [10] and STAR-CD [11] are commercially available. Thanks to the commercial CFD codes, the ap641
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Fig. 1. Conceptual View of Typical Prismatic Reactor Core
Fig. 2. Standard Fuel Block of Typical Prismatic Reactor
plication areas of CFD technology have been growing. Commercial CFD codes are user-friendly and well validated against most industrial problems which have a single phase fluid. In spite of the remarkable development of CFD technology, however, a whole prismatic core CFD analysis with geometric details is still a challenging task. Since fine meshes are required for accurate CFD calculations, most CFD applications are limited to local behaviours (e.g., single fuel column [12,13], and seven fuel columns [14]) in the design of a prismatic reactor. In 2009, Pointer and Thomas [15] published the STAR-CD results for a whole core of a prismatic reactor. It is clear that such a CFD analysis requires tremendous computational efforts. Additional significant efforts have to be devoted to generate computational grids of high quality for a good convergence. Such efforts are valuable for a final calculation but are too taxing for a designer who wants a large number of calculations with various design options. The other available option at KAERI could be the use 642
of system codes such as MARS-GCR [16] and GAMMA+ [17]. It should be noted, however, that the system codes are targeted for system transients. Their meshes are too coarse for a detailed thermo-fluid analysis of a prismatic fuel block. The accuracy of the system code calculations is not satisfactory for the core thermal design of a prismatic reactor (e.g., hot spot analysis). Therefore, a new computer code named CORONA has been developed to overcome the difficulties in the available tools at KAERI, i.e., CFD codes and system codes. The CORONA code is intended for whole-core thermofluid analysis of a prismatic gas cooled reactor with fast computation and reasonable accuracy. Fig. 3 summarizes the thermo-fluid analysis codes and their application areas for the design of prismatic gas-cooled reactors at KAERI. The position of CORONA lies in between CFD and system codes. Its position is similar to that of a subchannel analysis code such as COBRA [18] used in a light water cooled reactor design. NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Fig. 3. Thermo-fluid Analysis Codes and Their Areas for the Design of Prismatic Gas-cooled Reactors at KAERI
In order to achieve the target of the CORONA code (i.e., fast computation of whole core thermo-fluid analysis with reasonable accuracy), three fundamental strategies were established and applied to the development of CORONA: (1) an efficient numerical method, (2) efficient grid generation, and (3) parallel computation. ○ Efficient Numerical Method: The use of an efficient numerical method is the most important requirement for a new code to achieve its goal. The CORONA code adopts the existing idea of the efficient numerical method used for the design of MHTGR and HTTR (e.g., DEMISE and FLOWNET/TRAMP). This efficient method solves a three-dimensional heat conduction equation for a solid like a CFD code and one-dimensional conservation equations for a fluid flow like a system code. In 2012, Travis and El-Genk [19] verified the efficiency of this combined method using STAR-CD calculations. ○ Efficient Grid Generation: For a whole prismatic core thermo-fluid simulation, grid generation can also be a time-consuming process. Therefore, an efficient grid generation method using the concept of basic unit cells was developed by the present authors [9,20]. This concept of the basic unit cells adopted in the CORONA code enables the effective generation of unstructured grids for prismatic fuel blocks without a special mesh generator. The revision of the generated grids is also simple and convenient. ○ Parallel Computation: Reliance on a parallel computing technique is inevitable for a whole prismatic core simulation since significant numbers of meshes (e.g., in the order of ten million) are required. Parallel computation leads not only to a speed up of the calculations but also allows the use of a normal personal computer (PC). NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
Major applications of the CORONA code are (1) a steady-state hot spot analysis, (2) steady-state analysis for off-design conditions such as a coolant channel blockage accident, (3) some transient scenarios (e.g., rod ejection), (4) a coupled analysis (e.g., with neutronics) for high fidelity, (5) design optimization calculations. The CORONA code is written in C++ and can be run on either Windows or Linux operating system.
3. NUMERICAL APPROACHES 3.1 Grid Generation Using Basic Unit Cells
The first step in the numerical calculation of thermofluid phenomena is to set up a grid. The unique feature of the CORONA code is the use of the basic unit cells for the generation of computational grids. The concept of basic unit cells provides an efficient generation of unstructured computational grids to model an entire fuel block. The present authors noticed that the prismatic fuel blocks have regular arrangement of hexagonal or pentagonal unit cells as shown in Fig. 4. Prismatic fuel blocks can be represented by several types of basic unit cells and are shown in Fig. 5. The unstructured computational grids are defined within the basic unit cells. Using the combination of the basic unit cells, unstructured grids can be efficiently generated to simulate the thermo-fluid behavior of the fuel blocks. Fig. 6 is provided in order to improve the understanding of the relation between the basic unit cells and the computational grids. It shows the assignment of the basic unit cells to generate the computational grids of the standard fuel block. The change in generated grids is also simple and convenient. Fig. 7 illustrates an example of computational grids generated using the basic unit cells. Fig. 8 shows the modeling procedure from the basic unit cells to a whole prismatic core. Initially, computational grids are defined within each type of basic unit cell. 643
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Fig. 4. Typical Unit Cell Arrangements in Prismatic Fuel Blocks
Fig. 5. Basic Unit Cell Types and Examples of Computational Grids
Fig. 6. Basic Unit Cell Assignment to Generate Computational Grids of Standard Fuel Block
Fig. 7. Example of Computational Grids Generated Using the Basic Unit Cells for the Standard Fuel Block
Then, using an axial combination of the basic unit cells, a hexagonal (or pentagonal) body called a ‘pin’ is made. Three-dimensional computational grids for a hexagonal fuel (or reflector) block are generated using a two-dimensional combination of pins. Modeling of a whole core is finally finished using a combination of hexagonal fuel and reflector blocks. The number of computational grids for a solid used in
the CORONA code has a similar level as the CFD analysis. However, it should be noted that the computational expense of a solid heat conduction equation (i.e., Eq. (1)) is much less than that of fluid conservation equations in a whole core CFD analysis of a prismatic core.
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3.2 Numerical Model for Solid
In order to simulate the heat conduction through the
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Fig. 8. Modeling Procedure from Basic Unit Cell to Whole Prismatic Core
complex geometry of the fuel block, the CORONA code solves the following three-dimensional heat conduction equation: (1) The CORONA code adopts the finite volume method to solve Eq. (1). The key step of the finite volume method is the integration of Eq. (1) over a control volume to yield a discretised equation: (2) The obtained discretised equation can be expressed for a general nodal point P as follows: (3) (4)
(5) where the superscript 0 represents the known value at the previous time step and the subscript i indicates the neighboring node. In Eq. (5), kˆs,i and δVˆi represent the thermal conductivity and the node volume at the node face between the node P and the neighboring node i. Equation (3) is a well-known form for unstructured grids and sinon is the cross-diffusion term resulting from the non-orthogonality of grids. This term disappears if the grids are fully NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
orthogonal. At a fluid boundary, the following equation is governed: (6) Equations (3) and (6) can be rearranged into matrix form as (7) Equation (7) can be solved simultaneously for all nodes if the material properties and qs,iconv are known. The detailed derivation from Eq. (1) to Eq. (7) is not given in this paper since it is lengthy and widely available in a CFD textbook dealing with unstructured meshes (e.g., see Versteeg and Malalasekera [21]).
3.3 Numerical Model for Fluid Flow
In contrast to the solid geometry of the fuel blocks, the fluid in the prismatic fuel blocks flows through simple circular channels. A fraction of the core flow bypasses the fuel block coolant channels and passes through the gaps between the hexagonal columns. Such gaps can also be considered as rectangular channels connected to each other. For the fluid flow within a prismatic core, a one-dimensional approach can therefore be a good approximation. Such an approximation avoids fine meshes near the walls as well as turbulence conservation equations which are one of the major challenges in a CFD analysis. A onedimensional approximation for the duct flows enables a fast calculation with reasonable accuracy. In addition, there is potential for an exchange of coolant between the bypass gaps and the coolant channels due to cross flow gaps which are small spaces between the horizontal faces of the blocks. Therefore, a one-dimensional network model has been widely used for a fluid flow in a prismatic core [7]. 645
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Fig. 9. Concept of Node and Junction for Fluid Network
A fluid flow network model implemented into the CORONA code is based on the pressure correction method [22~24]. One-dimensional conservation equation for a fluid flow can be written as:
The first step to solve Eqs. (11) and (12) is to guess the pressures at all nodes. These values are treated as prelimi¯ The correction δP is denary values and donated by P. fined as the difference between the correct pressure field ¯ and the guessed pressure field P.
(8)
(13) Similarly, the volumetric flow rate correction δQ and density correction δρ are defined as: (14)
(9)
(15) The density of the gas can be obtained using the ideal gas law. (16)
(10) Under normal operating conditions, the thermo-fluid behavior in a prismatic gas-cooled reactor core can be described using the simplified forms of Eqs. (8)~(10) based on the steady-state, constant flow area, and negligible buoyance assumptions. Fig. 9 shows the concept of a node and junction to derive the mathematical formulation for an arbitrary network of one-dimensional fluid flows. In this paper, the simplified equations which are applicable to normal operating conditions are described in the following derivations since a steady-state hot spot analysis is the most crucial concern of the CORONA calculations. Initially, the steady-state continuity equation is applied at a one-dimensional fluid node (i) having an arbitrary number of initiating junctions (Ii) and terminating junctions (Ti ), as shown in Fig. 9(a). (11)
Where R is the gas constant and zc is the compressibility factor. Substitution of Eqs. (13)~(16) into the momentum equation (i.e., Eq. (12)) and neglecting the terms involving the products of the corrections, yield the following equation for the volumetric flow rate correction.
(17) where (18)
(19)
For a junction (j) having upstream fluid node (iuj) and downstream fluid node (idj) shown in Fig. 9(b), the simplified momentum equation can be applied as
Finally, the substitution of Eqs. (14), (15) and (17) into the continuity equation (i.e., Eq. (11)) yields the following equation for the pressure correction.
(12)
(20)
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Where aj, dj, cj, and dj are the rearranged coefficients in terms of each δP node respectively and ei is the source term. With the known field of δPi, δQj can be obtained using Eq. (17). Then, new values for Pi, Qj, and ρf,i are determined using Eqs. (13), (14), and (16). The newly updated values are now considered as the preliminary values for the next iteration and the whole process is repeated a number of times after which the fluid energy conservation equation is solved. The steady-state one-dimensional form for the energy conservation equation governing the fluid flow network can be derived as
(21)
where hi is the enthalpy of the fluid node i and αj is the parameter for the donor property defined as (22) The central differencing scheme can be used as an alternative option for Eq. (21). In the case of a single pipeline, Eqs. (20) and (21) can be solved using the tridiagonal matrix algorithm (TDMA), while sparse matrix techniques can be applied in the case of complex flow networks.
3.4 Interfacing of Solid and Fluid Models
The coupling between the three-dimensional solid model and the one-dimensional fluid model uses a typical Nusselt number correlation such as: (23) (24)
Eq. (23) is the correlation of McEligot et al. [7, 25] which is one of the most popular correlations for a prismatic gas-cooled reactor core under normal operating conditions. The entrance effect is neglected due to little interest in the entrance region. Eqs. (6) and (21) are combined as follows: (25)
(26)
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Fig. 10. Overall Computational Procedure for a Steady-State Analysis
where i represents each surface of the solid contacted with the fluid node j and As,i,j is the heat transfer area between the solid surface i and fluid node j. The overall computational procedure for a steadystate analysis is shown in Fig. 10. When the maximum value of the fluid temperature change during the iteration is less than the user supplied parameter, the convergence is regarded as having been achieved.
4. VERIFICATION AND VALIDATION 4.1 Heat Conduction in Unit Cells
In order to verify the multi-dimensional heat conduction within unit cells of the prismatic fuel block, two conceptual problems using unit cells were investigated [20]. The verification study started from a single unit cell problem which is one of the simplest test cases. The single unit cell consists of a fuel compact, helium gap, and graphite region. A uniform heat generation in the fuel compact was assumed 647
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Fig. 11. Result of Verification of CORONA Using Single Unit Cell Problem
Fig. 13. Result of Verification of CORONA Using Seven Unit Cell Problem
Fig. 12. Layout of Seven Unit Cell Problem and Its Boundary Conditions
and the temperature of the outmost boundary was fixed. Since it is believed that a commercial CFD code can solve a multi-dimensional heat conduction problem accurately, a commercial CFD code, CFX, is used for the verification of the CORONA code for the multi-dimensional heat conduction problem. Fig. 11 compares the predicted temperature profile by CORONA with that by CFX. It shows that CORONA accurately solves the multi-dimensional heat conduction equation. It is obvious that a finer mesh provides a more accurate result. The second case considers seven unit cells composed of one coolant cell (designated as ‘C’) surrounded by six fuel cells (designated as ‘F’) as shown in Fig. 12. A uniform heat generation in the fuel compacts was assumed and the temperature at the coolant boundary was fixed. In addition, the adiabatic boundary condition was imposed on the outmost boundary of the unit cells. Fig. 13 shows the predicted temperature profiles along Line A in Fig. 12. It was observed that when a fine mesh is adopted, the accuracy of CORONA is close to that of CFX. 648
Fig. 14. Schematic of Kaburaki and Takizuka Experimental Apparatus for Bypass Flow and Crossflow [25]
4.2 Bypass Flow and Crossflow in Fuel Column
An air flow experiment performed by Kaburaki and Takizuka [26] was selected to validate the one-dimensional fluid flow network model implemented in CORONA [27]. Fig. 14 shows a schematic of their experimental apparatus. Four graphite blocks were stacked up in a column and surrounded by a steel shroud to simulate the bypass gap (~1.2mm) between fuel columns. The hexagonal fuel block has 12 coolant holes 20 mm in diameter. One artificial cross flow gap (~1 mm) was created in the middle of the column to simulate a crossflow gap between fuel blocks. The air at atmospheric pressure was drawn from the top of the test section and flowed down the channels and the bypass gap. The bottom of the bypass gap was completely sealed and the coolant leakage flow, which entered the bypass gap, was merged into the main coolant channels through the crossflow gap. The mass flow rate NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
and the static pressure distribution in each coolant channel were measured. The measurement uncertainties of the experiment were not published and the sensitivity calculations of CORONA confirmed later that the effect of the entrance and exit form losses was small. Fig. 15 shows a comparison of the CORONA prediction results with the measured pressures. A good agreement can be seen along all of the stacks. This was achieved by applying the correlation of Kaburaki and Takizuka developed based on their experiment. This means that the described one-dimensional fluid flow model is correctly implemented into the CORONA code and is able to simulate the bypass flow and crossflow phenomena reasonably.
4.3 Conjugate Heat Transfer in Fuel Column
In order to verify a conjugate heat transfer in a prismatic fuel block, two conceptual problems with single fuel columns were investigated [28, 29]. Two types of fuel blocks (i.e., standard and reserved shutdown control (RSC) fuel blocks) were considered. Since it is believed that a commercial CFD code is well verified against a single phase conjugate heat transfer problem, the verification of the CORONA code is made using a comparison with the results of CFX. However, it should be noted that a commercial CFD code also needs a sufficient validation for prismatic core conditions (i.e., high temperature helium flow at a high heat flux condition in a prismatic fuel block geometry). Table 1 shows the major thermo-fluid parameters for the considered single fuel column problems under consideration. Fig. 16 shows the CFX meshes used to simulate the single fuel column problems. Using symmetry, 1/12 and 1/2 sections of the geometries were modeled for CFX calculations of the standard and RSC fuel blocks, respectively.
Fig. 15. Result of Validation of CORONA Using the Kaburaki and Takizuka Experiment Table 1. T hermo-fluid Parameters for Single Fuel Column Problems Standard Fuel Column
RSC Fuel Column
Column thermal power (MW)
3.118
2.832
Column flow rate (kg/s)
1.207
1.268
490
490
System pressure (MPa)
7
7
Number of stacked fuel blocks
6
6
Active core height (m)
4.758
4.758
Bypass gap size (mm)
2
2
Crossflow gap size (mm)
1
1
Uniform
Uniform
Parameter
Coolant inlet temperature ( C) o
Axial power profile
Fig. 16. CFX Meshes to Simulate Single Fuel Column Problems NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
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Fig. 17. Comparison of Predicted Axial Temperature Distributions at the Center of the Hottest Fuel Compact for the Standard Fuel Column Problem
Fig. 19. Comparison of Predicted Coolant Velocities at the Control Hole for the RSC Fuel Column Problem
fair comparison of the computation time is not feasible due to differing convergence criteria and a differing number of meshes. However, it is obvious that significant reduction in the computation time can be achieved by CORONA and the difference in the computation time would dramatically soar in the case of a whole core analysis.
5. PARALLEL COMPUTATION 5.1 Block-Based Parallel Computation
Fig. 18. Comparison of Predicted Axial Temperature Distributions at the Center of the Hottest Fuel Compact for the RSC Fuel Column Problem
Figs. 17 and 18 compare the predicted axial temperature distribution at the center of the hottest fuel compact. They clearly show good agreements. Non-continuing regions between fuel blocks are generated due to the non-fuel zones (i.e., graphite plug and seat). A comparison of the coolant velocities at the control hole of the RSC fuel column is shown in Fig. 19. It also shows a good agreement. Figs. 17~19 obviously indicate that the CORONA code can reliably simulate a conjugate heat transfer in the prismatic fuel blocks. Although the predicted temperature profiles by CORONA are comparable to those of CFX, the used computation time is significantly different. In order to obtain the result in Fig. 18, for example, about 46 hours were spent by CFX using four CPUs whereas about 4 minutes were spent by CORONA using a single CPU of the same computer. A 650
Parallel computing capability is important for the CORONA code since a whole prismatic core simulation requires a huge computing cost and a significant amount of computer memory. For a simple implementation of parallel computing capability, a block-based parallel computation method was developed for the CORONA code [30]. The idea of the block-based parallel computation method is based on the fact that each fuel (or reflector) block is completely enclosed by the fluid boundaries due to the existence of bypass and cross flows as shown in Fig. 20. Therefore, the heat conduction matrix shown in Eq. (7) can be naturally decomposed into much smaller matrixes that represent each fuel (or reflector) block. The decomposed matrix can be solved independently using a parallel computation library.
5.2 Application to Whole Core Analysis
In order to examine the improved performance by the parallel computation, an example problem for a whole prismatic core was investigated. PMR600 [2] was selected as a reference core. The major thermo-fluid boundary conditions are shown in Table 2. The reactor power profile obtained by the neutronic analysis [31] was applied. The applied power distribution is shown in Fig. 21. The heat loss to the reactor cavity cooling system (RCCS) was neglected. Fig. 22 shows the calculation model for the CORONA NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Fig. 20. Fuel Blocks Enclosed by Fluid Boundaries Table 2. M ajor Thermo-fluid Parameters for PMR600 Benchmark Calculation Parameter
Value
Core thermal power (MW)
600
No. of fuel columns
108
No. of fuel blocks per column
9
No. of reflector columns with control rod hole
36
Core flow rate (kg/s) Coolant inlet temperature (oC)
Fig. 21. Axial Power Distribution Used for PMR600 Benchmark Calculation
251.3 490
System pressure (MPa)
7
Active core height (m)
7.929
Bypass gap size (mm)
2
Crossflow gap size (mm)
0
calculation. The total numbers of solid and fluid nodes used were ~39,000,000 and ~200,000, respectively. The calculations were carried out on two PCs with Microsoft Windows 7 Enterprise K. Table 3 shows the specification of the two PCs used. The communications between the two PCs were made using Gigabit Ethernet connections. Intel ® MPI [32] was used for parallel computation library on the Windows operating system. Fig. 23 shows the predicted maximum temperature within each fuel and reflector column based on a serial calculation. It shows that the predicted temperatures are reasonable. The same calculations were repeated based on parallel computations. Nine cases were tested to investigate the parallel performance and Fig. 24 summarizes the result for the test problem. It was thoroughly checked whether all the cases produced the same results. Fig. 24 shows that the computing speed is significantly improved by the parallel computation. The computing time was NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
Fig. 22. CORONA Model for 1/6 Core of PMR 600
Table 3. S pecification of Two PCs for Parallel Computation Benchmark CPU
No. of CPUs
RAM
PC 1
Intel Xeon X5690, Six Core, 3.47 GHz
2
144 GB
PC 2
Intel Xeon X5690, Six Core, 3.47 GHz
2
96 GB
reduced with the number of processors. The maximum number of processors per PC was limited to 4 due to the random access memory (RAM) capacity of the PCs. 651
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
blocks. The usefulness of the CORONA code was proved by the example calculations using a whole prismatic core model. Further research is on-going to improve the physical models and increase the capability of the CORONA code. The major items are (1) coupling with the DeCART code [33] for a high fidelity calculation, (2) variable bypass (and crossflow) gap size model for consideration of the change in the size with the location as well as the burnup, and (3) transient simulation capability. It is believed that the CORONA code can be practically used in a whole core thermo-fluid analysis for the design of a prismatic gas cooled reactor. In addition, it is envisaged that the excellence of the CORONA code will also be highlighted in a high fidelity analysis combined with a whole core neutron transport calculation. Fig. 23. Predicted Maximum Temperature within Each Fuel and Reflector Column of PMR 600
Fig. 24. Result of Parallel Computation Performance Test Using PMR600 Problem
6. CONCLUSIONS AND OUTLOOK The development of the CORONA code was started in KAERI for the establishment of the key technologies for the design of a prismatic VHTR. The CORONA code is targeted for a whole core thermo-fluid analysis of a prismatic gas cooled reactor with fast computation and reasonable accuracy. In order to achieve the target, the development of CORONA focused on (1) an efficient numerical method using the combination of a three-dimensional heat conduction and one-dimensional fluid flow network, (2) efficient grid generation using basic unit cells, and (3) parallel computing capability. The results of the verification and validation studies show that the CORONA code can provide a reasonably accurate result for the thermo-fluid analysis of prismatic fuel 652
NOMENCLATURE
ai ai* aj a0p ap* As bj bp
= Influence coefficient at node i, defined in Eq. (5) = Rearranged influence coefficient of ai, used in Eq. (7) = Coefficient for junction j, used in Eq. (20) = Influence coefficient at node P, defined in Eq. (5) = Rearranged influence coefficient of ap, used in Eq. (7) = Solid surface area [m2] = Coefficient for junction j, used in Eq. (20) = Source term for node P at temperature matrix, used in Eq. (7) cj = Coefficient for junction j, used in Eq. (20) Cj = Resistance coefficient at fluid node j, defined in Eq. (18) Cs = Specific heat of solid [J/kg K] dj = Coefficient for junction j, used in Eq. (20) D = Hydraulic diameter [m] ei = Source term for node i at pressure matrix, used in Eq. (20) f = Friction factor [-] g = Gravitational acceleration constant [m/s2] G = Parameter defined in Eq. (19) h = Fluid enthalpy [J/kg] Ii = Initiating junction of node i kf = Thermal conductivity of fluid [W/m K] ks = Thermal conductivity of solid [W/m K] K = Form loss factor [-] L = Fluid channel length [m] N = Number of meshes Nu = Nusselt number [-] P = Pressure [Pa] Pr = Prandtl number [-] qs‴ = Power density generated in solid [W/m3] qfconv = Convection heat transfer at fluid [W] qsconv = Convection heat transfer at solid [W] Q = Volumetric flow rate [m3/s] R = Gas constant of helium [J/kg K] Re = Reynolds number [-] sinon = Cross-diffusion term t = Time [s] Tf = Fluid temperature [K] NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.46 NO.5 OCTOBER 2014
TAK et al., Development of a Core Thermo-Fluid Analysis Code for Prismatic Gas Cooled Reactors
Ti Ts V w z zc
= Terminating junction of node i = Solid temperature [K] = Volume [m3] = Fluid velocity [m/s] = One-dimensional coordinate = Compressibility factor [-]
Greek
α ρ
= Parameter defined in Eq. (22) = Density [kg/m3]
ACKNOWLEDGEMENTS
This work was supported by Nuclear R&D Program of the National Research Foundation of Korea which is grant funded by the Korean government (Grant code: NRF2012M2A8A2025679).
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