Topology optimization of two-fluid turbulent heat exchangers: A Darcy flow-based multifidelity approach

This paper presents a multifidelity topology optimization framework that calibrates a computationally efficient Darcy flow-based low-fidelity model against a high-fidelity RANS model to design two-fluid turbulent heat exchangers, achieving up to a 22% improvement in performance over conventional designs by balancing enhanced heat transfer with manageable pressure drops.

The Gist

Imagine you are trying to design the ultimate heat exchanger—a device that acts like a thermal handshake between two fluids (like hot water and cold water) flowing through pipes. The goal is to make them swap heat as quickly as possible without making the fluids struggle too much to get through (which would waste energy).

For decades, engineers have tried to improve these devices by twisting metal ribbons inside the pipes or adding fins. But these methods are like trying to sculpt a masterpiece with a hammer; they are limited by what traditional manufacturing can bend and twist.

This paper introduces a new way to design these devices using a computer "brain" called Topology Optimization. Think of this as a digital sculptor that can carve out any shape imaginable, provided it fits inside the pipe. However, simulating how fluids swirl and mix at high speeds (turbulence) is like trying to predict the weather in a hurricane: it's incredibly accurate but takes a supercomputer years to run.

The Problem: The "Perfect" vs. The "Fast"

The researchers faced a dilemma:

1. The High-Fidelity (HF) Model: This is the "weather forecaster." It uses complex physics (RANS equations) to predict exactly how turbulent fluids behave. It's accurate but so slow that running it thousands of times to find the best design is impossible.
2. The Low-Fidelity (LF) Model: This is the "quick sketch." It uses a simplified math model (Darcy flow) that treats the fluid like it's moving through a sponge. It's incredibly fast but often gets the details wrong, especially regarding how much pressure the fluid loses.

If you only use the sketch, you might design a beautiful pipe that collapses under real pressure. If you only use the weather forecaster, you'll never finish the design.

The Solution: The "Multifidelity" Approach

The authors created a clever two-step strategy, which they call a Multifidelity Approach. Think of it like training for a marathon:

1. The Training Run (Optimization): You use the "quick sketch" (the LF model) to run thousands of practice races. You tweak the design, change the speed, and try different shapes to find promising candidates. Because the sketch is fast, you can explore hundreds of different "what-if" scenarios quickly.
2. The Calibration: Before the training runs, they "calibrated" the sketch. They adjusted the sponge's density in the math so that the sketch's results matched the weather forecaster's results for a standard pipe. This made the sketch much smarter.
3. The Race Day (Evaluation): Once the computer found a bunch of interesting designs using the fast sketch, they took the top contenders and ran them through the "weather forecaster" (the HF model) just once each. This is the final, accurate test to see which design actually wins.

What They Found

They applied this method to a "double-pipe" heat exchanger (one pipe inside another) where the fluids were moving very fast (turbulent flow).

• The Results: The computer-designed shapes were wild and complex, looking nothing like standard pipes. They created intricate internal walls that forced the fluids to swirl and mix intensely, much like a chef vigorously stirring a sauce to cool it down faster.
• The Comparison: They compared their new designs against a standard pipe with a "twisted tape" (a common industry trick to improve heat transfer).
  • The twisted tape improved heat transfer but caused a massive "traffic jam" (high pressure drop), making it inefficient overall.
  • The new computer-designed shapes improved heat transfer by up to 66% compared to a plain pipe.
  • Crucially, they managed the "traffic jam" much better. When you look at the overall score (balancing heat gain vs. energy cost), their designs were up to 22% better than the twisted tape.

The Takeaway

The paper proves that you don't need to simulate every single swirl of a hurricane to find a great design. By using a fast, calibrated "sketch" to explore the possibilities and a slow, accurate "forecaster" to verify the winners, engineers can design high-performance heat exchangers that are far superior to what we can currently build with traditional methods.

The study specifically notes that these designs work well across a wide range of speeds, suggesting they are robust and ready for real-world use, provided they can be manufactured (likely using 3D printing, which the authors mention as a key enabler for such complex shapes).

Technical Summary

Technical Summary: Topology Optimization of Two-Fluid Turbulent Heat Exchangers

Problem Statement
The design of two-fluid heat exchangers (HXs), particularly double pipe heat exchangers (DPHXs), traditionally relies on passive enhancement techniques such as twisted tape insertion, fins, or corrugated tubes. While effective, these methods are constrained by conventional manufacturing limitations and often result in suboptimal trade-offs between heat transfer enhancement and pressure drop penalties. Topology optimization (TO) offers a pathway to generate innovative geometries, yet applying TO to turbulent two-fluid systems presents significant computational challenges. Directly incorporating high-fidelity (HF) turbulence models (e.g., Reynolds-Averaged Navier-Stokes, RANS) into the optimization loop is computationally prohibitive due to the nonlinearity of turbulence, the need for extremely fine meshes, and numerical instability. Furthermore, existing TO studies for two-fluid HXs have predominantly focused on laminar or low-Reynolds-number regimes, leaving the optimization of turbulent flows largely unaddressed.

Methodology
The authors propose a multifidelity topology optimization framework that decouples the design search from the high-fidelity evaluation to balance computational efficiency and accuracy.

1. Low-Fidelity (LF) Optimization Model:

   • The optimization process utilizes a Darcy flow-based model rather than a full turbulence model. This approach, referred to as the "poor man's approach," replaces the Navier-Stokes equations with the Darcy equations, which are linear, computationally inexpensive, and numerically stable.
   • A single design variable field ($\gamma$) represents the distribution of three phases: Fluid 1, Fluid 2, and the solid wall.
   • Interpolation Schemes: The permeability for each fluid is interpolated using RAMP functions to ensure that in the solid wall region (intermediate $\gamma$ values), permeability is sufficiently low to prevent fluid mixing. Thermal conductivity is interpolated using a Gaussian function to smoothly transition between fluid and solid properties.
   • Calibration: To mitigate discrepancies between the simplified Darcy model and reality, the LF model parameters (maximum permeability and thermal conductivity sharpness) are calibrated against a reference HF model for a straight pipe configuration. This calibration aligns the LF model's pressure drop and temperature difference predictions with the HF model.

2. Multifidelity Strategy:

   • Instead of running a single optimization, the framework employs a seeding parameter strategy. Multiple independent optimizations are performed using the calibrated LF model, varying inlet velocities and trade-off parameters (balancing heat transfer vs. pressure drop) to generate a diverse set of candidate designs.
   • High-Fidelity (HF) Evaluation: The optimized designs extracted from the LF process are evaluated using a RANS-based HF model (specifically the $k-\omega$ turbulence model with wall functions). This step occurs outside the optimization loop.
   • Selection: The HF model assesses the performance of all candidates based on the Performance Evaluation Criterion (PEC), which normalizes the overall heat transfer coefficient against the pressure drop penalty. The best-performing design is selected based on these accurate evaluations.

3. Numerical Implementation:

   • The LF model is solved using COMSOL Multiphysics with a density filter and Heaviside projection to ensure clear fluid-solid boundaries.
   • The HF evaluation utilizes body-fitted meshes generated from the optimized design variables, employing the $k-\omega$ model for turbulence and validating results against empirical correlations (Dittus-Boelter, Blasius, and Manglik & Bergles for twisted tapes).

Key Contributions

• Framework Development: The study establishes a multifidelity TO framework specifically for two-fluid turbulent heat exchangers, addressing the gap in existing literature which largely focuses on laminar flows or single-fluid systems.
• Calibration Strategy: A novel calibration strategy is introduced to align the Darcy-based LF model with RANS-based HF responses in two-fluid turbulent scenarios, improving the consistency of pressure drop and temperature predictions.
• Quantitative Validation: The paper provides a rigorous quantitative comparison between the optimized designs and a conventional reference design (smooth pipe with twisted tape insertion) across a wide range of Reynolds numbers (up to 13,000).

Results

• Design Generation: The LF model successfully generated diverse, complex wall structures that promote fluid mixing and increase heat exchange surface area while maintaining streamlined flow paths.
• Performance Improvement: Compared to the smooth pipe reference, the optimized designs achieved overall heat transfer coefficient improvements of up to 66.7%.
• PEC Enhancement: When compared to the twisted tape reference design (a standard enhancement technique), the optimized designs achieved up to 22% higher Performance Evaluation Criteria (PEC) values. This indicates that the optimized geometries provide superior heat transfer enhancement relative to the pressure drop penalty.
• Model Discrepancies: The LF model was found to reasonably predict temperature distributions but tended to underestimate pressure drops (mean error ~72% for Fluid 1) due to the lack of viscous shear and turbulence modeling. However, the multifidelity approach effectively compensated for this by filtering candidates through the HF model.
• Robustness: Designs optimized at higher Reynolds numbers (e.g., Re = 13,000) demonstrated robustness, maintaining high PEC values across a wide range of operating conditions.

Significance and Claims
The paper claims that the proposed multifidelity approach effectively overcomes the computational barriers associated with optimizing two-fluid heat exchangers under turbulent conditions. By leveraging a computationally cheap Darcy model for the search phase and a high-fidelity RANS model for the evaluation phase, the method enables the discovery of high-performance configurations that would be infeasible to find using direct optimization with turbulence models.

The authors emphasize that the optimized configurations outperform conventional twisted tape insertions, particularly in terms of the trade-off between heat transfer and pressure loss. The study concludes that while the Darcy-based LF model has limitations in predicting absolute pressure drops, the multifidelity framework is a viable and effective strategy for designing high-performance double pipe heat exchangers, showcasing the potential of this approach for complex turbulent thermal-fluid problems. The work acknowledges limitations, such as the manual selection of seeding parameters and the lack of manufacturing constraints in the current study, suggesting future directions toward data-driven automation and the inclusion of manufacturability constraints.