Surrogate Model for Heat Transfer Prediction in Impinging Jet Arrays using Dynamic Inlet/Outlet and Flow Rate Control

This study develops and validates high-accuracy, real-time CNN-based surrogate models trained on CFD data to predict Nusselt number distributions in flexible impinging jet arrays with dynamic inlet/outlet configurations, enabling efficient thermal management and model-based control strategies.

The Gist

Imagine you are trying to cool down a giant, hot pizza. You have a row of five fans (jets) that can blow air onto the pizza. But here's the twist: these fans are magical. They can do three things:

1. Blow cold air (Inlet) to cool the pizza.
2. Suck hot air away (Outlet) to pull heat off the surface.
3. Turn off (Shut) completely.

You can change which fan does what, and how hard it blows, every single second. The goal is to keep the pizza at the perfect temperature, everywhere, at the same time.

The Problem: The "Super-Computer" Bottleneck

To figure out the perfect setting for your fans, you usually need to run a massive, high-tech physics simulation (called CFD). Think of this like a video game with ultra-realistic graphics that simulates every single air molecule.

The problem? These simulations are slow. Running one takes hours or even days on a supercomputer. If you want to control the pizza temperature in real-time (changing the fans every second), waiting hours for a calculation is useless. You need an answer instantly.

The Solution: The "Smart Apprentice" (Surrogate Model)

The authors of this paper built a Surrogate Model. Think of this as a "Smart Apprentice" or a "Weather Forecaster" for your cooling system.

1. The Training Phase: First, the researchers ran about 100 of those slow, expensive, high-fidelity simulations. They taught the Apprentice: "When you have Fan 1 blowing hard and Fan 2 sucking, the heat looks like THIS. When you switch them, it looks like THAT."
2. The AI Brain: They used a special type of Artificial Intelligence called a Convolutional Neural Network (CNN). If a normal computer program is like a calculator, a CNN is like a human eye. It's really good at looking at a picture (the heat map) and recognizing patterns, just like you recognize a face in a crowd.
3. The Result: Once trained, this Apprentice can predict the cooling pattern instantly. It doesn't need to simulate every air molecule; it just looks at the fan settings and says, "I've seen this before, here's what the temperature will look like."

The Magic Trick: Predicting the Future (Extrapolation)

There was one catch. The Apprentice was only trained on "gentle breezes" (low air speeds, or low Reynolds numbers). But in the real world, you might need "hurricane-force winds" (high Reynolds numbers) to cool something really fast.

Usually, an Apprentice trained on gentle breezes would fail at predicting hurricanes. But the authors taught the Apprentice a mathematical rule (based on a famous scientist named Martin). They showed the AI: "If you know how the heat behaves at low speeds, you can mathematically scale it up to predict high speeds."

It's like teaching a child how to ride a bike at 5 mph, and then showing them the physics of how to ride at 20 mph. They can extrapolate the knowledge to handle the faster speed without needing to practice at 20 mph first.

Did It Work?

The researchers tested their "Smart Apprentice" in two ways:

1. The Simulation Test: They compared the AI's predictions against new, expensive simulations it had never seen before.
   • Result: The AI was incredibly accurate. For the 3x3 fan setup, it was 99.4% accurate. Even for the more complex 5-fan setup, it was over 98% accurate.
2. The Real-World Test: They built a physical device with real fans and a hot metal plate. They used the AI to control the fans and cool the plate.
   • Result: The AI's predictions matched the real-world temperature measurements within 6%. That is a huge success for such a complex system!

Why Does This Matter?

This isn't just about cooling pizza. This technology is crucial for:

• Electric Vehicles: Keeping batteries from overheating while driving fast.
• Factories: Cooling down injection molds so parts can be made faster.
• Solar Panels: Managing heat to keep them efficient.

In a nutshell: The authors created a "crystal ball" for cooling systems. Instead of waiting days for a supercomputer to tell you how to cool a hot surface, this AI gives you the answer in a split second, allowing for real-time, smart temperature control that was previously impossible.

Technical Summary

1. Problem Statement

Impinging jet arrays are critical for advanced thermal management in applications like gas turbines, electric vehicle batteries, and injection molding. A specific challenge arises in active cooling systems where individual jets can dynamically switch between acting as inlets, outlets, or being shut off.

• Complexity: The number of possible jet arrangements (states) grows exponentially with the number of jets. For a $5 \times 5$ array, there are roughly $3^{25}$ ($\approx 1$ trillion) possible configurations.
• Limitation of Current Methods:
  • Empirical Correlations: Typically assume fixed configurations (all jets as inlets) and fail to capture complex jet-jet interactions or dynamic state changes.
  • Computational Fluid Dynamics (CFD): While high-fidelity CFD (specifically Implicit Large-Eddy Simulation, ILES) can accurately model these dynamic interactions, the computational cost is prohibitive for real-time applications such as model-based temperature control. Simulating high Reynolds numbers ($Re > 10,000$) requires massive computational resources (e.g., 25 core-years for 100 simulations at $Re=10,000$ vs. 3 core-years at $Re=2,000$).

Objective: Develop a surrogate model capable of predicting the Nusselt number distribution in real-time for dynamic jet configurations, enabling efficient control strategy testing and implementation.

2. Methodology

A. Geometry and Simulation Setup

The study focuses on two active cooling systems:

1. 5-by-1 Array: 5 jets in a single row.
2. 3-by-3 Array: 9 jets in a square grid.

• CFD Solver: The open-source software Lethe was used, employing a Finite Element Method (FEM) with an Implicit Large-Eddy Simulation (ILES) strategy to capture turbulence without explicit subgrid-scale models.
• Training Data Generation:
  • Simulations were run for $Re < 2,000$ to manage computational costs.
  • Latin Hypercube Sampling (LHS) was used to generate diverse jet arrangements (inlet/outlet/shut states and flow rates).
  • Dataset Size: 83 simulations for the 5-by-1 system; 100 simulations for the 3-by-3 system.
  • Data Split: 80% for training (augmented with symmetry reflections), 20% for validation.

B. Neural Network Architecture

A Convolutional Neural Network (CNN) was developed to map jet configurations to Nusselt number maps.

• Input Encoding: The input is a multi-channel tensor representing:
  1. Normalized inlet Reynolds numbers (0.0 to 1.0).
  2. Jet states (1 for inlet, -1 for outlet, 0 for shut).
  3. Spatial coordinates (normalized distance from center).
• Architecture:
  • Upscaling: Uses transposed convolutions to expand the low-resolution input vector into a high-resolution feature map (e.g., from $4 \times 3 \times 3$ to $64 \times 256 \times 256$).
  • Downscaling: Followed by standard convolution and pooling layers to reconstruct the final Nusselt number map at the target resolution ($25 \times 101$ pixels for 5-by-1; $75 \times 75$ for 3-by-3).
  • Activation: Leaky ReLU functions are applied after convolutional layers.
• Training: Optimized using the AdamW optimizer with a grid search for hyperparameters (max channels, image dimensions, epochs).

C. Extrapolation Strategy

To address high Reynolds numbers ($Re \le 10,000$) without re-simulating:

• The model is trained on $Re < 2,000$.
• Predictions are extrapolated to higher $Re$ using a correlation-based scaling law derived from Martin [15]:
  $$Nu(Re_{pred}) = Nu(2000) \times \left( \frac{Re_{pred}}{2000} \right)^{0.574}$$

3. Key Contributions

1. First Dynamic Surrogate Model: Development of the first CNN-based surrogate model specifically for impinging jet arrays with dynamic inlet/outlet switching, a capability missing in existing literature which focuses on fixed configurations.
2. Real-Time Capability: The model predicts Nusselt distributions in real-time, overcoming the latency of CFD, making it suitable for model-based control strategies.
3. Scalability and Extrapolation: Demonstrated a method to extrapolate low-$Re$ training data to high-$Re$ operational regimes ($Re=10,000$) with acceptable error margins.
4. Experimental Validation: Validated the surrogate model against physical experiments, confirming its utility in real-world thermal management scenarios.

4. Results and Performance

Accuracy on Validation Data ($Re < 2,000$)

The surrogate models achieved high accuracy, capturing complex jet-jet interactions and cross-flow effects:

• 5-by-1 Model:
  • Normalized Mean Average Error (NMAE): 1.6%
  • RMSE: 0.64
  • Average error in global Nusselt number: 1.4%
• 3-by-3 Model:
  • NMAE: 0.57%
  • RMSE: 0.24
  • Average error in global Nusselt number: 0.2%
  • Note: The 3-by-3 model performed better due to a larger effective dataset generated via geometric symmetries.

Extrapolation Performance ($Re = 10,000$)

• When scaling predictions from $Re=2,000$ to $Re=10,000$, the NMAE increased to 11% for the 5-by-1 model.
• Despite the increase in local error, the global average Nusselt number remained well-preserved, with errors generally within acceptable limits for control applications.

Experimental Validation

• An experiment was conducted on the 5-by-1 system at $Re=10,000$ using a heat gun and thermal camera.
• The reconstructed temperature (using the surrogate model for boundary conditions) matched experimental data within 5.8% after 120 seconds.
• Discrepancies were attributed to modeling assumptions and experimental uncertainties, but the authors note that a closed-loop controller would likely compensate for these errors.

5. Significance and Future Work

This work establishes a reproducible framework for integrating high-fidelity CFD data into machine learning models for complex, multi-input/multi-output thermal systems.

• Impact: It enables the development of real-time, model-based closed-loop control strategies for active cooling systems, allowing for dynamic modulation of flow rates and jet configurations to manage spatial and temporal temperature distributions.
• Future Direction: The authors plan to integrate this surrogate model directly into a control loop to dynamically actuate individual jets in active cooling systems, optimizing performance for applications like battery thermal management and injection molding.

Availability: The data and code are publicly available via Zenodo (DOI: 10.5281/zenodo.18157225).