What kind of equations are involved in radiant heating and cooling system analysis?

The analysis of radiant heating and cooling systems.

How do you actually solve these equations using a computer?

We use spatial discretization via the Finite Element Method (FEM). This involves assembling the element stiffness matrices and constructing the global stiffness equation.

What specifically do you mean by matrix solution algorithms?

We solve the system of equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (Conjugate Gradient method, GMRES). For large-scale problems, preconditioned iterative methods are highly effective.

Radiant heating and cooling system

Q: Professor! Today's topic is radiant heating and cooling systems, right? What exactly are they?

It's heating and cooling via ceiling and floor radiant panels. It involves the relationship between mean radiant temperature and operative temperature. Condensation prevention control is critically important.

Category: 熱解析 | Integrated 2026-04-06

CAE visualization for radiant heating cooling theory - technical simulation diagram

放射冷暖房システム

Theory and Physics

Overview

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Professor! Today's topic is about radiant heating and cooling systems, right? What are they like?

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Heating and cooling via ceiling/floor radiant panels. The relationship between mean radiant temperature and operative temperature. Condensation prevention control is extremely important.

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Now I understand what my senior meant when he said, "Make sure you do it right, especially for ceiling/floor radiant panels."

Governing Equations

$$ q_r = \varepsilon\sigma(T_p^4 - T_{AUST}^4) $$

$$ T_{AUST} = \sum F_{p-i} T_i $$

Discretization Methods

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How do you actually solve these equations on a computer?

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We use spatial discretization via the Finite Element Method (FEM). We assemble the element stiffness matrices and construct the global stiffness equation.

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We perform transformation to the weak form (variational form) and use formulation via Galerkin's method using test functions and shape functions. The choice of element type (low-order elements vs. high-order elements, full integration vs. reduced integration) directly affects the trade-off between solution accuracy and computational cost.

Matrix Solution Algorithms

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What exactly are matrix solution algorithms?

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We solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). For large-scale problems, preconditioned iterative methods are effective.

Solution Method	Classification	Memory Usage	Applicable Scale
LU Decomposition	Direct Method	O(n²)	Small to Medium Scale
Cholesky Decomposition	Direct Method (Symmetric Positive Definite)	O(n²)	Small to Medium Scale
PCG Method	Iterative Method	O(n)	Large Scale
GMRES Method	Iterative Method	O(n·m)	Large Scale / Non-symmetric
AMG Preconditioner	Preprocessing	O(n)	Very Large Scale

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So, if you cut corners on the finite element method part, you'll pay for it later. I'll keep that in mind!

Implementation in Commercial Tools

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So, what software can be used for radiant heating and cooling systems?

Tool Name	Developer/Current	Main File Formats
Ansys Fluent	Ansys Inc.	.cas, .dat, .msh, .jou
COMSOL Multiphysics	COMSOL AB	.mph
Simcenter STAR-CCM+	Siemens Digital Industries Software	.sim, .java, .csv
Ansys Mechanical (formerly ANSYS Structural)	Ansys Inc.	.cdb, .rst, .db, .ans, .mac

Vendor Lineage and Product Integration History

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Is the background of each software quite dramatic?

Ansys Fluent

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Next is about Ansys Fluent. What's the content?

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Developed by Fluent Inc. Acquired by Ansys in 2006. A general-purpose CFD solver based on unstructured grids.

Current affiliation: Ansys Inc.

COMSOL Multiphysics

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Please tell me about "COMSOL Multiphysics"!

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Founded in Sweden in 1986. Started as FEMLAB with MATLAB integration, later renamed to COMSOL. Strong in multiphysics.

Current affiliation: COMSOL AB

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After hearing this, I finally understand why development is important!

Simcenter STAR-CCM+

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Next is about Simcenter STAR. What's the content?

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Developed by CD-adapco. Acquired by Siemens in 2016 and integrated into the Simcenter brand. Polyhedral meshing is a key feature.

Current affiliation: Siemens Digital Industries Software

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Wow, the story about development is super interesting! Please tell me more.

File Formats and Interoperability

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Are there any points to note when transferring data between different software?

Format	Extension	Type	Overview
STEP	.stp/.step	Neutral CAD	3D CAD data exchange format compliant with ISO 10303. Supports geometry + PMI.
IGES	.igs/.iges	Neutral CAD	Early CAD data exchange standard. Has compatibility issues with surface data. Transition to STEP is progressing.
VTK	.vtk/.vtu	Visualization	Visualization Toolkit format. Used by ParaView, etc.

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When converting models between different solvers, careful attention is needed regarding element type correspondence, material model compatibility, and differences in load/boundary condition representation. Particularly, high-order elements and special elements (cohesive elements, user-defined elements, etc.) often cannot be directly converted between solvers.

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I see... Formats seem simple at first glance, but they're actually quite profound, aren't they?

Practical Considerations

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