In an integrated coal gasification combined cycle plant, cooling pipes are installed in the gasifier reactor and water cooling is executed to avoid reaching an excessively high temperature. To accelerate the design, it is necessary to develop an analysis system that can simulate the cooling operation within the practical computational time. In the present study, we assumed the temperature fields of the cooled object and the cooling water to be governed by the three-dimensional (3D) heat equation and the one-dimensional (1D) convection-diffusion equation, respectively. Although some existing studies have employed similar modeling, the applications have been limited to simple-shaped structures. However, our target application has a complex shape. The novelty of the present study is to develop an efficient numerical analysis system that can handle cooling analysis of complicated-shaped structures, of which modeling needs a huge number of degrees of freedom (DOFs). To solve the thermally coupled problem between the cooled object and cooling water, we employed a partitioned approach with non-matching meshes. For the heat transfer analysis of the cooled object, we employed an open-source large-scale parallel solver based on the 3D finite element method, named ADVENTURE_Thermal. For the convective heat transfer analysis of the cooling water in pipes, a 1D discontinuous Galerkin method-based solver of a convection-diffusion equation was developed and used. The proposed analysis system was first verified by solving a problem on water cooling of concrete, for which an analytical solution is already available. Then, using the supercomputer “Fugaku”, we performed a cooling analysis of a laboratory-scale coal gasifier reactor, which has complicated geometry and is modeled by over 20 million DOFs, and demonstrated the practical performance of the proposed system.

Recently, an integrated coal gasification combined cycle (IGCC) plant [

For solving coupled problems between conduction phenomena in solids and convection phenomena in fluids, a conjugated heat transfer approach is often used. For example, it was applied to the cooling of rocket engines [

In the cooling problem of our interest, the size of the pipes is not enough small to be neglected. Unlike the above studies [

One of the methods for solving coupled problems is partitioning [

The rest of this paper is organized as follows. In

where

where

Note that

The boundary of

In this section, heat transfer models for the cooled object and the cooling water are explained. Then, the modeling of heat exchange between the two models is explained.

The temperature field of the cooled object, denoted by

where

where

The temperature field of the cooling water in each pipe is governed by a 1D convection-diffusion equation. The derivation of the equation is explained here. We have some assumptions for the cooling water. First, on the cross section

We consider the energy balance in a small volume

where

where

where

where

where we use

The heat flux on

where

In the heat transfer analysis of the cooled object,

where

where

For the time integration scheme, the backward Euler method is employed. As explained in

In the heat transfer analysis of the cooling water, the local DG method [

where

In the DG method, the boundary terms associated with

For an arbitrary function with an input

Each element has

For the time integration scheme, the forward Euler method is employed. The procedures for solving

In

where

where

In the present study, cooling problems are considered as coupled problems between cooled objects and cooling water in pipes. For coupled analysis methods, we employ a partitioned method, which executes multiple sub-solvers sequentially. The advantage of the partitioned methods is that they allow the use of existing solvers, which makes it easy to develop parallel analysis systems by choosing parallel solvers as sub-solvers. Because a gasifier reactor has a complex geometry, a huge number of DOFs is needed for modeling, and parallel computing is imperative. It is thus reasonable to adopt a partitioned method.

For the 3D heat transfer FE analysis of the cooled object, we employed ADVENTURE_Thermal [

In contrast, for 1D analysis of cooling water, in-house code for a DG method was developed and used. Implicit time integration is employed for the 3D FE analysis. Conversely, explicit time integration is employed for the 1D DG analysis. Therefore, the time step size for 1D DG analysis has to be much smaller than that for the 3D FE analysis. Hence, we employed a subcycling technique.

The flowchart of the proposed analysis system is shown in

Regarding the implementation, the functions corresponding to II) to V) in

For verification, the water cooling of concrete was considered [

Material | Concrete | Water |
---|---|---|

Density ( |
1997 | 1000 |

Specific heat ( |
1050 | 4187 |

Thermal conductivity ( |
2.33 | 0.599 |

For the discretization of the concrete, the number of nodes was 1,527,987, and the number of elements was 7,487,914. For the discretization of the water, the number of nodes was 1500 and the number of elements was 500. The time step size of the 3D heat transfer analysis by ADVENTURE_Thermal was 3600 s. The number of subcycles was 360,000.

For parallel computing of ADVENTURE_Thermal, the domain was decomposed into 12 parts, each of which had 100 subdomains. For the domain decomposition, ADVENTURE_Metis [

The distribution of the temperature of the cooling water is shown in

Zhu provided the analytical solution of the cooling problem as

where

The computational time of the 3D FE analysis per time step was 61.9 s. That of the 1D DG analysis for 360,000 subcycles was 232 s.

The results described in this section qualitatively and quantitatively verified the developed analysis system.

This section demonstrates that the proposed analysis system can handle very large-scale problems with complicated geometries and can be operated in a supercomputer environment.

The Central Research Institute of the Electric Power Industry (CRIEPI) has been developing a laboratory-scale coal gasifier reactor [

The cross section of the reactor is shown in

Material | Steel | W40 | LWI-26 | Water |
---|---|---|---|---|

Density ( |
7800 | 2650 | 1375 | 1000 |

Specific heat ( |
440 | 836.8 | 1673.6 | 4187 |

Thermal conductivity ( |
84 | 1.5 | 0.442 | 0.599 |

The water velocity

The heat transfer coefficient can be written as

where

In the present study, the Reynolds number

For the discretization of the reactor, the number of nodes was 25,510,852, and the number of elements was 155,999,061. For pipe A, the water was discretized by 516 nodes and 172 elements. For pipes B, C, and D, the water was discretized by 522 nodes and 174 elements. The time step size for ADVENTURE_Thermal was 0.1 s. The number of subcycles was 1000.

For parallel computing of ADVENTURE_Thermal, the domain was decomposed into 480 parts, each of which had 400 subdomains. For this computation, we used 40 nodes with 12 cores of Fugaku [

The computational time of the 3D FE analysis per time step was 8.16 s. That of the 1D DG analysis for 1000 subcycles was 7.61 s.

This section described how our proposed analysis was applied to the cooling problem for a gasification reactor. The results we obtained were reasonable and confirmed that our analysis system can handle large-scale problems with complex geometry.

The present study developed a new coupled analysis system for problems on water cooling in pipes. The system employs a partitioned approach with non-matching meshes and a subcycling technique. For the heat transfer analysis of a cooled object, we employed an opensource 3D FEM large-scale parallel solver, ADVENTURE Thermal. For the convective heat transfer analysis of the cooling water in pipes, a 1D DG method-based solver of a convection-diffusion equation was developed and used. To make it easy to compute heat exchange, we proposed a Dirac delta function-based interpolation. The system was verified by solving a problem with water cooling of concrete. Then, we confirmed the system can handle a large-scale problem with complicated geometry and can be operated in a supercomputer environment.

Our final goal is to develop a coupled analysis considering combustion, water cooling, heat conduction in the vessel and structural deformation of the vessel for evaluation of the structural integrity of the reactor vessel in actual operation conditions. In the subsequent study, we plan to integrate a combustion analysis solver with the proposed analysis system.

The present study was supported in part by the Program for Promoting Research on the Supercomputer Fugaku (Digital Twins of Real World’s Clean Energy Systems with Integrated Utilization of Super-Simulation and AI) and JSPS KAKENHI, and used computational resources of the supercomputer Fugaku provided by the RIKEN Center for Computational Science. The authors thank Drs. Akio Makino and Shiro Kajitani for providing technical advice and information and data on CRIEPI’s laboratory-scale coal gasification plant.

The present study was supported by the Program for Promoting Research on the Supercomputer Fugaku (Grant Number JPMXP1020200303) and JSPS KAKENHI (Grant Number JP22K17902).

The authors confirm contribution to the paper as follows: study conception and design: S. Kaneko, N. Mitsume; data collection: S. Kaneko; analysis and interpretation of results: S. Kaneko, S. Yoshimura; draft manuscript preparation: S. Kaneko. All authors reviewed the results and approved the final version of the manuscript.

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.

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