Conjugated Heat Transfer: Best Practices & Applications

Although we may prefer a constant temperature for our comfort, it's important to acknowledge the reality of temperature variations across different spaces and times. The temperature gradient is a natural aspect of the temperature field surrounding us, even within our bodies. While we think of the temperature field as being restricted to a specific "domain," such as the air in a room or the surface of a metal object, it extends beyond these boundaries, even when there's a sharp contrast between solid and fluid domains. The study of conjugate heat transfer (CHT) provides a comprehensive approach to understanding temperature fields across fluid and solid domains.

The CHT concept involves the simultaneous consideration of heat transfer processes within both solids and fluids, treating them as a unified system.

The result is a comprehensive analysis, where continuous heat flux interactions between the two domains are accounted for, yielding insights into complex thermal behaviors.

This article will expose the core physics and mathematics of the CHT (Conjugate Heat Transfer) problem. We'll explain the basic equations, introduce essential quantities, and provide insights into real-world applications of conjugate heat transfer with AI technologies.

Last but not least, we'll see how AI contributes to the diffusion of this type of analysis to all engineers in product design.

What Are the Modes of Heat Transfer? A Juicy Introduction...

Before delving into the specialist discussion of CHT, let's take the everyday succulent example of cooking using a metal pot or pan on a stovetop. Let's break down how conjugate heat transfer is involved in this scenario:

Cooking on a Stovetop: You initiate the process by placing a metal pot or pan on a stovetop burner and turning on the heat. The burner's flames or the electromagnetic heating element are a heat source that releases energy, transferred to the pot or pan.

Heat Flux Within the Metal: The heat energy from the stovetop is conducted through the metal of the pot or pan. Metal is a good heat conductor, so the heat rapidly travels through the walls of the cooking vessel, raising its temperature.

Natural Convection and Radiation Heat Transfer: As the pot or pan heats up, it also starts transferring heat to the surrounding air through natural convection and radiation. The hot metal surface warms up the air around it, causing it to rise and creating a convective energy transfer process. Additionally, the hot metal emits thermal radiation - a form of electromagnetic heat flux.

CHT: In this scenario, conjugate heat transfer occurs at the interface between the metal pot/pan and the food being cooked. The heat from the metal is transferred to the food through conduction, while the food's heat is transferred back to the metal through convection and radiation. The combined heat flux mechanisms at the metal-food interface illustrate conjugate heat transfer in the specific case of conjugate convective heat transfer. Conjugate convective heat transfer highlights that no forced flow convection was present, such as with a fan or an extractor hood.

Food Cooking Process: As heat is transferred from the pot/pan to the food, its temperature increases, causing it to cook. The heat from the metal to the food cooks the food's bottom or sides. Simultaneously, the food's heat is transferred back to the metal, which can change the pot's or pan's temperature.

What Is Conjugate Heat Transfer?

The practical example demonstrated that solid structures and adjacent fluids interact through various mechanisms. This type of analysis considers the interplay between heat conduction in solids and fluids, which are distinct energy and mass transfer mechanisms.

The continuity between solids and fluids is a crucial aspect of CHT analysis, wherein the heat flux is treated as continuous across the interface. This means the mathematical representation of heat flux remains smooth and consistent as it moves from the solid to the fluid domain. This continuity assumption is vital for accurately modelling and simulating heat transfer between solids and fluids. It enables the seamless integration of heat transfer mechanisms in both domains, resulting in a more realistic representation of thermal behavior at the interface. Engineers can use this approach to predict temperature distributions, heat flux, and other critical thermal parameters relevant to product design optimization, such as durability and energy efficiency.

The following section will provide an introduction to heat transfer in solids.

Heat Transfer in Solids - Intuitive Introduction

Heat conduction in solids occurs when thermal energy moves from hotter to cooler regions within the material. Different materials exhibit varying insulation levels for heat transfer, for example, wood, brick, wool, and steel. Let's explore basic mathematics using a fundamental "governing equation" called Fourier's Law.

Fourier's Law describes the heat flux rate in solids, specifically related to heat conduction. It's a fundamental principle in heat transfer that explains how heat flows through a solid material due to temperature difference.

This venerable law states that the heat flux (Q) through a solid is directly proportional to the temperature gradient (grad_T), which represents how much the temperature changes over a certain distance, and the material's thermal conductivity (k), indicating its ability to conduct heat under uniform conditions.

Mathematically, this relationship can be expressed as:

Q = -k × grad_T

The negative sign is to ensure that heat flows from higher temperature to lower temperature regions. The law can also be written more shortly as:

Q= -k ∇T

In case we don't trust that thermal conductivity (k) is constant, we'd write the equation as follows:

Q= -∇ (kT)

Fourier's Law quantitatively explains how heat "moves" through solids, i.e. the heat flux. It allows us to predict and analyze heat transfer in various materials, helping engineers and scientists design efficient thermal systems and understand the behavior of materials under different temperature conditions.

Conduction in Fluids

Fluids are also thermally conducting, although the convective part of energy transfer (natural or forced convection) is generally prevalent. In many practical situations, convective heat transfer dominates over conductive heat transfer in fluids due to the fluid's ability to carry and distribute heat more efficiently through bulk motion (mass transfer).

For example, in forced convection, water convective heat transfer coefficients can range from about 100 to 10,000 W/(m²·K), depending on flow rate, temperature, and surface conditions.

In contrast, the conductive heat transfer in liquids like water tends to be lower, often around 0.5 to 5 W/(m²·K) for typical materials.

The Fundamental Mechanisms of Heat Transfer in Solids

Heat conduction is the process by which heat energy flows through solids. It's a fundamental concept in thermodynamics and physics that explains how thermal energy is transmitted in materials such as metals and insulators. Over the centuries, scientists have made significant progress in understanding heat transfer in solids, thanks to groundbreaking research by pioneers like Joseph Fourier. Fourier's Law has become a cornerstone of heat transfer theory.

Jean-Baptiste Biot's experiments in the early 19th century laid the foundation for understanding thermal conductivity. His apparatus measured heat flow through materials, providing insights into factors influencing heat conduction. Thomas Young and John Leslie further advanced the concept. Fourier's Law established a quantitative relationship between heat flux, temperature gradient, and thermal conductivity, paving the way for practical applications in engineering and geophysics. Rudolf Peierls and David Bohm's phonon theory explained how quantized lattice vibration waves contribute to heat propagation in solids.

Engineering Applications and Beyond

Materials selection, thermal insulation, and the design of heat exchangers are just a few examples of how heat conduction enhances technological innovation. Moreover, the study of heat transfer in Earth's interior has yielded insights into geological processes, such as the heat movement through the Earth's crust.

Challenges and Modern Research

Despite significant progress, challenges remain. For example, understanding anisotropic materials, where thermal conductivity varies with direction, and exploring heat transfer at nanoscales are active research areas, also thanks to novel experimental techniques, such as laser-induced thermal measurements.

Heat Transfer in Fluids

In fluids, heat transfer primarily occurs through convection, where the fluid particles transport heat by moving from hotter regions to cooler ones. Convection is described by the convective transfer equation for heat:

Q = h A ΔT

Where:

• Q is the heat transfer rate
• h is the convective heat transfer coefficient
• A is the heat transfer surface area
• ΔT is the temperature difference between the fluid and the surface

Convection in Fluids: In-Depth Exploration

In heat transfer, convection serves as a fundamental mode through which thermal energy is exchanged within fluids. Convection is characterized by the bulk movement of fluid particles, which enables heat redistribution from regions of higher temperature to those of lower temperature. This process plays a pivotal role in natural and engineered systems, and its quantitative description relies on the convective heat transfer equation we have seen before Q = h A ΔT.

What Exactly Is the Convective Heat Transfer Coefficient?

The convective heat transfer coefficient, often called "h", is central to modelling natural convection and forced convection. This coefficient embodies a conglomeration of physical phenomena that collectively govern the heat transfer process in fluids. The "h" coefficient includes the combined effects of fluid motion, boundary layer behavior, turbulence, and other interactions that occur within the fluid. Thus, in a simplified representation, "h" encompasses many factors contributing to the heat transfer rate.

Fluid Motion and Boundary Layer Phenomena

Fluid motion is a necessary aspect of natural convection and forced convection, and its characterization is rooted in fluid dynamics principles. As fluid flows over a surface, it establishes a boundary layer—a thin layer of fluid near the surface with distinctive flow characteristics. This boundary layer behavior, whether a laminar flow or turbulent, significantly influences the heat transfer rate.

Thus, "h" considers the interplay between the boundary layer thickness, flow velocity, and other factors, encapsulating the impact of fluid motion on heat transfer.

Turbulent Flow and Enhanced Heat Transfer

Turbulence, an intricate and chaotic form of fluid motion, can significantly enhance heat transfer rates. Turbulent air flow exhibits vigorous mixing and increased fluid interaction, promoting effective heat exchange between air flow and the solid surface. The convective heat transfer coefficient "h" encapsulates the turbulence effects by incorporating the dynamic nature of flow field turbulence intensity, length scales, and other turbulence-related parameters.

Temperature Gradients and Heat Transfer Area

The temperature difference (ΔT) between the fluid and the surface is a driving force behind convection— a larger ΔT results in a higher heat transfer rate. The heat transfer surface area (A) is crucial in determining the total amount of heat exchanged. The convective heat transfer coefficient "h" works in conjunction with ΔT and A to quantitatively describe the heat transfer rate and its dependence on these factors.

Applications and Practical Significance

The coefficient "h" holds immense importance in various fields.

In mechanical engineering, "h" is pivotal for designing efficient heat exchanger systems, cooling systems, and thermal management solutions.

Meteorology relies on convection to explain atmospheric processes, while geophysics utilizes it to model heat transfer within Earth's interior.

Advancements in computational fluid dynamics have enabled precise predictions of convective heat transfer in complex geometries and turbulent flow regimes.

Conjugate Heat Transfer Equations

When analyzing conjugate heat transfer, the objective is to simultaneously solve the heat conduction equation in solids and the conjugate convective heat transfer equation in fluids. The governing equation for heat conduction is the celebrated Fourier unsteady-state equation:

∇(k * ∇T) = ρ c * ∂T/∂t

Where material properties are three:

• k = the thermal conductivity of the solid material
• ρ = the material density
• c = the specific heat capacity

While the field to be solved, i.e. T=T(x,y,z;t), is the temperature distribution (over space with its spatial gradient ∇T and over time with time derivative ∂T/∂t). If k is constant, the equation can be simplified as k∇ ²T=ρc∂T/∂t.

The boundary conditions for solid and fluid domains must be appropriately defined to ensure a consistent and accurate analysis.

Transient Heat Transfer

Transient phenomena exhibit varying time scales in solids and fluids. Solids typically exhibit slower thermal responses due to higher thermal diffusivities, whereas fluids respond more rapidly. In the summertime, a house with massive stone walls remains cool due to the stone's high thermal inertia. During the day, the solid walls absorb heat from the sun, but due to the slow thermal response of the solid, the interior remains cooler than the outside air. The walls gradually release stored heat at night, maintaining a comfortable interior temperature.

Why?

In fluid heat transfer, transient conduction describes the temporal evolution of temperature through a fluid medium due to temperature differences.

Transient Conduction Equations

The transient conduction equation is a partial differential equation representing this phenomenon is the known ∇(kᵩ ∇T)= ρᵩ cᵩ (∂T/∂t) (we have just seen it in the previous chapter), where the "phi" letter stands for fluid and where ρᵩ, cᵩ and kᵩ are the usual material properties (density, specific capacity and conductivity) for the fluid. By adding heat sink or source term "q", it becomes:

∇(kᵩ ∇T) + q = ρᵩ cᵩ (∂T/∂t)

The unsteady-state heat conduction equation for the solid is, in a similar fashion, with the possibility of a heat sink or heat source term "q":

∇(kₛ ∇T) + q = ρₛ cₛ (∂T/∂t) (where the "s" index now stands for solid)

Now, transient phenomena in heat transfer manifest distinct time scales in solids and fluids, with their thermal responses governed by thermal diffusivity and material properties. This phenomenon finds practical illustration in the cooling behavior of a house with massive stone walls during summertime, shedding light on the interplay between heat transfer mechanisms.

Due to their comparatively higher thermal diffusivities, solids exhibit slower thermal responses.

Thermal Diffusivity

Thermal diffusivity (α) is a material property that relates the heat conduction rate to the temperature change rate. The diffusivity definition is (for a solid) αₛ = ρₛ / cₛ kₛ, where ρₛ, cₛ and kₛ are the usual material properties (density, specific capacity, and conductivity).

On the other hand, fluids possess their thermal diffusivities, which are generally higher than in solids:

αᵩ = ρᵩ / cᵩ kᵩ

For instance, air thermal diffusivity is approximately 19 m2/s, while for standard brick, it's 0.52 m2/s, and for wood, it's about 0.1 m2/s.

Thermal diffusivity is not a strict rule governing whether it's higher in solids or fluids. Instead, it's a property that depends on the specific material's composition, structure, and other factors. Generally, while solids often have higher thermal conductivities k than fluids, thermal diffusivity also considers heat capacity c and density ρ, which can lead to varying results.

This results in more rapid heat transfer responses in fluids than in solids. The variation in thermal diffusivity between solids and fluids plays a pivotal role in shaping their respective transient heat transfer behaviors.

Explanation of Introductory Case on Transient Conjugated Heat Transfer

The thermal response disparity comes into play in our example of a house with massive stone walls during summertime.

During the day, when the sun's radiant energy impinges upon the stone walls, they absorb heat. However, owing to the higher thermal diffusivity αᵩ of air (fluid) than αₛ of stone (solid), the house's interior exhibits slower temperature changes. Consequently, the stone walls act as a thermal buffer, preventing rapid heat transmission to the interior. This lag in heat transfer allows ambient temperature on the inside to remain cooler than the outside air, maintaining a comfortable environment.

As external temperatures drop, the stone walls gradually release the stored heat at night. This gradual heat release further stabilizes the indoor temperature, preventing sudden fluctuations.

The higher thermal inertia of the massive stone walls ensures that the interior doesn't experience immediate temperature shifts, even in response to external temperature variations.

Best Practices in Conjugate Heat Transfer Simulation

Ensuring accurate simulations in conjugate heat transfer relies on a systematic approach that follows precise guidelines. By adhering to these practices, engineers can effectively utilize this analytical method and gain insightful information about heat transfer interactions in solid and fluid domains.

Mesh Generation: Building the Computational Foundation

Mesh generation forms the basis of any simulation. In conjugate heat transfer, a well-structured mesh is crucial for capturing the geometry and resolving temperature and velocity gradients. A balanced mesh is necessary — sufficiently detailed to represent complex features without high computational costs.

For complex geometries, adaptive meshing is essential. This technique refines the mesh in regions of interest, like boundary layers, providing an accurate thermal pattern representation. A well-designed mesh enables precise simulations, offering a comprehensive view of temperature distribution and heat transfer.

More on Mesh Generation: Thin-Layer Meshing for Solids

Recently, CAE has progressed to a more sophisticated level of conjugate heat transfer simulation. The simulation model can analyze a vehicle's entire chassis to identify how heat transfers from solid cars to plastic components compromise their integrity. Generating high-quality meshes is paramount, especially when dealing with intricate geometries. One advanced technique is thin-layer meshing, which creates high-resolution mesh elements inside thin solids, enabling engineers to accurately capture thermal gradients, heat transfer, and fluid dynamics while maintaining a reasonable mesh count for computational efficiency.

Boundary Conditions: Reflecting Physical Reality

Defining boundary conditions such as wall temperature is critical for a robust simulation. Accurate conditions mirror the physical situation, ensuring the fidelity of the obtained temperature field. Solid and fluid domains need comprehensive treatment of the interface boundary condition in conjugate heat transfer.

Solid domain conditions entail setting temperatures or heat fluxes at surfaces interacting with the fluid. Modern CFD tools take care of both the flow and the solid domain as concurring domains to a larger conjugated simulation and exchange wall temperatures at interfaces between concurring domains governed by different physical laws (e.g. no Navier-Stokes in solids) rather than having two separate simulations. Similarly, convective conditions account for heat exchange with the surrounding fluid, involving the convective heat transfer coefficient and ambient fluid temperature. In fluid domains, conditions may include inflow/outflow velocities, wall temperature, and pressure.

What Are Those "Heat Sinks"?

We have used a lot of the term "sink" in contrast with a heat source, but what are heat sinks?

A heat sink is a cooling device for electronics, like CPUs or GPUs. It's made of materials like copper or aluminum and has fins for more surface area. Heat travels from the component to the fins and is released into the air, preventing overheating. In cases needing more cooling, heat sinks work with fans or liquid cooling. This is also relevant to conjugate heat transfer, where both conduction and convection are important for effective cooling.

Material Properties: Key Influencers of Heat Transfer

Material properties are pivotal for accurate conjugate heat transfer simulations. Thermal conductivity, specific heat, and other properties significantly influence conjugate heat transfer problems. These properties can exhibit temperature dependencies and anisotropy, especially in complex materials.

Precise material properties require experimental measurements and empirical correlations. Correctly identifying these parameters ensures a faithful representation of heat conduction and forced convection. Thoughtful selection of properties allows engineers to scrutinize temperature gradients and heat fluxes accurately.

Solid-Flow Interface

The temperature field and heat flux maintain continuity at the solid-flow interface. Nevertheless, within a moving fluid, the temperature field can exhibit rapid fluctuations: in proximity to the solid, the fluid's temperature closely mirrors that of the solid, whereas farther from this interface, the fluid's temperature aligns with the inflowing or ambient fluid temperature.

The thermal boundary layer is the spatial range over which fluid temperature transitions from solid-equivalent to bulk-fluid temperature. The thermal boundary layer dimensions, about the dimensions of the momentum boundary layer, are expressed with Prandtl number: achieving a Prandtl number of 1 necessitates equivalence between thermal and momentum boundary layer widths.

The definition of the Prandtl Number is simply:

(Pr) = ν / α

where ν (nu) is the kinematic viscosity of the fluid (a measure of its resistance to motion due to internal friction or viscosity) and α (alpha) is the thermal diffusivity of the fluid (a measure of how quickly heat can spread through the fluid by conduction)

A thicker momentum layer would yield a Prandtl number surpassing 1. Conversely, a Prandtl number below 1 signifies a thinner momentum boundary layer than the thermal boundary layer.

Summing Up the Physics: Coupled Heat Transfer - Flat Plate

The conjugate heat transfer occurs when heat conduction within a solid object and heat convection in the surrounding fluid interact simultaneously. A simple and classic scenario illustrating this phenomenon is the heat transfer over a flat plate. In this scenario, a flat plate is exposed to a fluid flow in a solid-flow interface, such as air or a liquid.

Heat is transferred from the plate's surface into the fluid through both conduction within the plate and convection at the fluid-solid interface. The interplay between these two modes of heat transfer gives rise to interesting thermal behavior. A boundary layer develops at the fluid-solid interface as the fluid flows over the flat plate. This boundary layer consists of two primary regions: the thermal and momentum boundary layers.

The thermal boundary layer is characterized by a gradual transition of fluid temperature from the plate's temperature at the surface to the fluid's bulk temperature. Close to the solid surface, the fluid temperature closely follows the plate temperature. However, as you move away from the surface, the fluid temperature approaches the bulk fluid temperature. Simultaneously, the momentum boundary layer forms as the fluid's velocity adjusts from zero at the solid surface to its bulk velocity. This layer is responsible for the transport of momentum within the fluid due to viscosity effects.

The Prandtl number, a dimensionless quantity, plays a significant role in this conjugate heat transfer process. Defined as (Pr) = ν / α, it reflects the ratio of momentum diffusivity ν to thermal diffusivity α. The thermal and momentum boundary layers are of similar thickness for a Prandtl number near unity. However, when the Prandtl number deviates from unity, one of the layers becomes more dominant. For example, in the case of air (with Pr around 0.7), the thermal and momentum boundary layers exhibit comparable thicknesses, leading to balanced heat and momentum transfer. In contrast, substances like water (with a higher Pr around 7) emphasize the thermal boundary layer, resulting in sharper temperature gradients compared to velocity gradients.

The study of conjugate heat transfer over a flat plate is crucial in engineering applications such as heat exchangers, cooling systems, and aerodynamic designs. Engineers can optimise thermal performance and devise efficient designs for various practical systems by comprehending the intricate interplay between conduction and convection.

Mathematics of Conjugate Heat Transfer: Looking for Analytical Solutions

Solving the heat diffusion equation analytically for general cases can be pretty complex, but analytical solutions are possible for simple cases and boundary conditions. Let's consider the 1D heat conduction equation with a constant thermal conductivity (k), where one end of the rod is held at a fixed temperature (Dirichlet boundary condition) and the other end is insulated (Neumann boundary condition). We'll solve it for the temperature (T) along the rod.

The equation is:

k ∂²T/∂x² = ρc ∂T/∂t

Let's consider a rod with length L initially at a uniform temperature of T0.

To solve this equation, we employ the method of separation of variables.

By introducing the variables T(x, t) = X(x) * T(t), we initiate the analysis and through a rigorous derivation, we arrive at a final solution represented by a series expansion:

T(x, t) = T0 + ∑[An exp(-((n π / L)^2) (k / (ρc)) t) sin(n π * x / L)], where n = 1, 2, 3, ...

This equation illustrates the temperature profile along the rod as a sum of terms. Each term corresponds to a distinct Fourier mode (n) contributing to the temperature profile over time.

The coefficients An denote the amplitude of each mode, and the exponential term involving -((n π / L)^2) (k / (ρc)) t accounts for the decay of each mode due to heat conduction over time.

Meanwhile, the sine term sin(n π * x / L) captures the spatial variation of each mode along the rod's length.

However, while this equation provides an analytical solution, it's essential to recognize its limitations, mainly when dealing with complex shapes such as gas turbine blades or the under hood of a car.

The complexity of such geometries often defies the extension of these methods. In such cases, numerical solutions become imperative for accurate analyses and predictions of a temperature profile.

Numerical Methods: Solving Complex Equations

Conjugate heat transfer simulations use numerical methods to solve heat conduction and convection equations. Finite Element Analysis (FEA) models energy transport in solids, while Computational Fluid Dynamics (CFD) models air and fluid energy transport, including phase change.

Conversely, accurately representing the temperature fields within the solid domain would produce a more accurate wall temperature for the fluid thermal boundary layer.

Selecting appropriate numerical methods and solvers ensures computational efficiency and accuracy. Wise choices of discretization schemes and iterative algorithms reveal the interplay of energy transport in conjugate systems.

More Insight Into FEA and CFD and Temperature Distribution Prediction

Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD) are powerful numerical techniques with a pivotal role in simulating and understanding complex thermal phenomena. These methodologies are widely used in engineering and scientific disciplines to analyze intricate systems' temperature distribution, fluid flow, and heat exchange.

FEA

FEA involves partitioning a given domain into discrete elements, enabling precise approximations of temperature profiles. This procedure, known as discretization, allows the representation of intricate geometries and material properties. The full temperature profile of each element can be expressed using interpolation functions, with the system's behavior characterized by a set of algebraic equations.

Mathematically, this is expressed as:

∑N k_ij dT_j/dx = q_i (the heat source/sink term at node i)

where

• ∑N stands for the summation over the total number of nodes or elements within the domain (the summation is over "j")
• k_ij represents the thermal conductivity matrix element between nodes i and j.
• dT_j/dx denotes the gradient across element j

CFD

CFD uses mesh divisions to model turbulent flows and energy transport. The Navier-Stokes and energy equations with appropriate additional models to "close" the turbulence physics by approximations govern turbulent fluid flow, accounting for heat exchange and complex interactions between motion and thermal effects.

The Navier-Stokes equations can be succinctly represented as:

ρ ∂u/∂t+ u⋅∇ u = -∇ P + μ ∇ ²u + ρ g

(newly introduced symbols: u is the velocity vector, P denotes pressure, μ is the fluid's dynamic viscosity, and g represents gravitational acceleration)

Now, turning to turbulence modelling, the k-epsilon model assumes a prominent role because of its popularity within the CFD users community. It revolves around two fundamental variables: k (turbulence kinetic energy) and ϵ (turbulence dissipation rate). The following equations govern these variables:

d(ρk)/dt + u ∇(ρk) = ∇ (μt * ∇k) + ρε - ρε

d(ρε)/dt + u ∇(ρε) = ∇ (μt * ∇ε) + Cε1 (ε/k) ρε - Cε2 (ε^2/k)

These equations describe the evolution of turbulent kinetic energy per unit volume (ρk) and turbulence dissipation rate (ρε), μt is turbulent viscosity, and Cε1 Cε2 are (suffice it to say!) "adjustable constants".

Incorporating the energy equation provides a comprehensive representation of fluid behavior and energy transport thus:

rate of change + convective transport = conductive transport + sources

i.e.

ρ c (∂T)/(∂t) + u⋅∇T = ∇⋅(k ∇T) + q

The equation describes the change in energy (ρc∂T/∂t) within the fluid, the convective transport of energy (u⋅∇T) indicated by the "u" term for velocity (clearly an inexistent component of the equation in solids where u=0), the conductive transport of energy ∇⋅(k∇T), and any heat sinks or heat sources (q). This equation is used to model the fluid behavior when considering how energy is transported through convection, conduction, and heat generation/absorption.

So you can clearly see here that the "u" term and associated transport by convection + u⋅∇T differentiates solids and fluids from solids, on top of very different thermal diffusivity. This isn't a slight difference: a fluid flow field is, by its nature, an unsteady flow. Modelling turbulent flow poses infinite challenges to scientists and engineers and is far from being solved. Here we have taken a simplified approach with two additional equations (k-ϵ) to "close" the issue and give practical answers.

FEA and CFD use numerical approaches to find reliable solutions for complex geometrical shapes encountered in daily engineering applications. These approaches offer a detailed insight into temperature distributions, fluid flow patterns, and energy transport mechanisms, enabling informed design and analysis.

The figure shows that CFD and AI can solve numerically turbulent flows over complex geometrical shapes and suggest design changes.

Validation: Confirming Simulation Reliability

Validation is integral to robust simulation practice. Comparing simulation or AI prediction outcomes with experimental data or analytical solutions substantiates accuracy and reveals system nuances. Validation establishes model reliability, aligning simulated temperature distributions and energy transport rates with real-world observations.

Validation involves benchmarking against established cases. This step identifies disparities and refines simulations. Validated models for simulation (or AI, as we will see in the next chapter) instill confidence in predictive capabilities, enabling design optimization.

In conclusion, meticulous practices can establish a foundation for successful conjugate heat transfer simulations.

A well-structured mesh, representative boundary conditions, accurate material properties, suitable numerical methods, and rigorous validation form an environment mirroring reality.

Engineers equipped with these practices can tackle complex energy transport phenomena and gain insights into coupled solid and fluid systems.

AI and CAE Simulation

In modern engineering, the convergence of artificial intelligence (AI) and computer-aided engineering (CAE) simulation redefines the boundaries of what is achievable in design, analysis, and innovation. Deep Learning, Convolutional Neural Networks (CNNs), and the remarkable possibility of utilizing computer vision techniques to decode and interpret complex Computer-Aided Design (CAD) shapes are at the heart of this shift. Neural Concept Shape (NCS), a platform that leverages AI-driven approaches to unlock new dimensions in CAE simulation and design optimization, is among the pioneers of this technological revolution. You can find more details about it below.

Deep Learning

Deep Learning, an integral subset of AI, has emerged as a cornerstone technology that emulates human-like cognitive processes. By processing datasets, deep learning algorithms can identify intricate patterns, make nuanced predictions, and even generate novel insights, surpassing traditional computational methods in accuracy and efficiency. Integrating Deep Learning within CAE simulation introduces a paradigm shift, empowering engineers to transcend conventional limitations and explore designs previously deemed unfeasible.

CNNs

Convolutional Neural Networks (CNNs) are a class of deep learning models that recognize complex patterns and features within images. They're inspired by the visual processing system of the human brain and are particularly adept at handling the intricate geometries and multifaceted structures often encountered in CAD designs. Engineers can use CNNs to extract valuable information from CAD models and unravel insights that enable product designers to access simulations.

NCS Technology

The NCS technology confidently employs CNNs and computer vision to understand intricate CAD shapes accurately, enabling successful engineering simulations.

The NCS approach leverages AI technology, learning from CAD and CAE datasets to translate 3D CAD geometries into real-time predictions. Thus, engineers can gain a visual understanding and uncover potential flaws and optimization opportunities with AI's predictions that are surrogates for CFD and CHT. Optimization opportunities may be missed with traditional methods because they would be obtained at the expensive cost of specialist resources and budgets. Therefore in many cases, optimization is not carried out because of the associated cost and time, and small incremental ameliorations are considered sufficient.

Engineering Applications of Conjugate Heat Transfer

Conjugate heat transfer has widespread applications in various engineering fields, including electronics cooling, thermal management of engines, and aerospace design. It's crucial in optimizing cooling strategies, enhancing energy efficiency, and preventing thermal failures.

Conjugate heat transfer coupled with modern simulation tools is an indispensable analytical tool across engineering domains rooted in technical precision. From electronics cooling to aerospace engineering and industrial processes, this approach comprehensively elucidates fluid-solid heat interactions. We'll now review a few applications.

Cooling Systems Applications

Conjugate heat transfer (CHT) finds indispensable application in cooling systems, addressing the imperative of efficient thermal regulation in electronics and integrated circuits. The relentless progression towards miniaturization has accentuated power densities, rendering advanced cooling strategies paramount. Engineers use conjugate heat transfer simulations to analyze intricate geometries, such as heat sinks and microchannels, coupled with fluid flow dynamics.

With conjugate CHT simulation, engineers can discern temperature distributions, identify thermal hotspots, refine cooling solutions, sustain operational stability, minimize heat-induced failures, and extend component lifespans.

Heat Exchangers Applications and Conjugate Heat Transfer

The field of heat exchangers, integral to diverse industrial processes, benefits from conjugate heat transfer analysis. The effective thermal energy exchange between fluid streams is essential in applications from HVAC system design to chemical processing.

With CHT simulation, engineers evaluate heat transfer coefficients, pressure drops, and thermal gradients, enabling precise parameter tuning for optimal performance. This comprehensive analysis culminates in better energy efficiency and reduced operational expenditures.

Aerospace Applications

Under demanding thermal regimes, aerospace engineering leverages CHT to ensure vehicular safety and performance. High-speed flight conditions, like supersonic and hypersonic flows, generate formidable heat loads necessitating innovative thermal protection systems.

Conjugate heat transfer simulations model aerodynamic-solid interactions, enabling assessment of temperature distributions, thermal stresses, and material integrity.

This analytical approach guides thermal protection system design, mitigating thermal-induced failures and ensuring optimal performance under extreme flight conditions.

Industrial Process Conjugate Heat Transfer Problems

Conjugate heat transfer analysis holds significant relevance in industrial processes, pivotal in a wide array of manufacturing and energy conversion operations. This analytical approach finds essential applications in metallurgy, chemical processing, and power generation, where precise heat exchange mechanisms are vital for ensuring efficient and cost-effective operations.

The core principle of conjugate heat transfer analysis lies in its ability to model solid components, and fluid flows within process equipment comprehensively. This holistic approach allows engineers to gain insights into heat transfer rates, temperature distributions, and thermal stresses. Engineers can fine-tune operational parameters and optimize designs using these insights, resulting in heightened process efficiency, reduced energy consumption, and effective thermal management.

Examples of Conjugate Heat Transfer in Industrial Processes

In metallurgy, engineers use conjugate heat transfer analysis to optimize cooling strategies in processes such as continuous casting by simulating heat transfer between the metal and cooling water. Similarly, in chemical processing industries, engineers can control reaction kinetics and temperature gradients, improving yield, selectivity, and safety through conjugate heat transfer simulations.

In power generation, conjugate heat transfer analysis becomes indispensable for designing efficient and robust thermal systems. Take a gas turbine, for example, where high-temperature combustion gases interact with intricate cooling structures in turbine blades. Through conjugate heat transfer simulations, engineers can optimize the design of blade cooling passages, minimizing thermal stresses and enhancing the overall lifespan and performance of the turbine.

Automotive Applications

Braking System Conjugate Heat Transfer Simulation

Brake systems in vehicles are subjected to intense thermal conditions during operation. As the brakes are applied, kinetic energy is converted into heat, causing temperature spikes that must be effectively managed. Brake components, including rotors and calipers, interact with the surrounding air, leading to a complex interplay of conduction, convection, and radiation phenomena.

Braking System - CFD Flow Simulation with CHT

CFD flow simulation with CHT capability accurately captures the heat dissipation in brake systems. CFD simulation models the airflow around the brake components, while conjugate heat transfer extends the analysis to solid materials, accounting for temperature variations.

Simulation Steps

Geometry and Meshing: The brake system geometry, including rotors, calipers, and surrounding components, is meticulously created in the simulation software. A computational mesh divides the geometry into discrete cells, allowing for numerical analysis.

Flow Simulation (CFD): CFD flow simulation predicts the airflow patterns around the brake components during vehicle operation. Aerodynamic forces and temperature distributions on the brake surfaces are computed.

CHT Analysis: CHT analysis is used to simulate the temperature distribution in brake components influenced by airflow convection. CFD and CHT results are combined to simulate thermal interaction between the solid components and surrounding air, with fluid and solid domains considered separate but with interface zones. Modern CFD tools ensure uniformity of meshes at static interface boundaries.

Performance Evaluation: Temperature profiles, heat fluxes, and thermal gradients are analyzed to assess brake system performance under various conditions, such as repeated braking or emergency stops.

By conducting CHT simulations for brake systems, engineers gain valuable insights for design optimization:

1. Ventilation and Cooling: Evaluation of the effectiveness of cooling mechanisms, such as ventilation channels or ducts, to mitigate temperature rise in critical brake components.
2. Materials: Suitable materials with optimal thermal properties for enhanced heat dissipation and durability.
3. Performance Enhancement: Improved braking performance, reduced wear, and extended component lifespan.

Automotive Example 2 - Under Hood Conjugate Heat Transfer Simulation

A car's under hood area contains various heat sources, such as the engine, exhaust system, and radiators. Engineers can use conjugate heat transfer simulation to evaluate the performance of cooling systems and ensure efficient heat dissipation from critical components in the car's under hood. Simulation-guided modifications to heat shields and insulation materials can enhance thermal protection for sensitive parts. Additionally, simulation can help optimize component placement to minimize heat accumulation and ensure efficient cooling airflow paths. However, this interplay is challenging due to the limited availability of space and the simultaneous collaboration between several Tier 1 suppliers and the automotive OEM.

In the author's experience, designing for tight, confined spaces requires CAD designers and all product design engineers to think creatively while considering manufacturability, ensuring functionality, and anticipating assembly, maintenance, and performance challenges. It's a delicate balance between fitting components into the available space while providing optimal functionality and ease of operation.

A Personal Note - Thermodynamic Automata

The author and late Prof. Luigi Sertorio collaborated years ago on the "Thermodynamic Automaton" concept (see Complex Systems Volume 5 Issue 6, 1991), relevant to CHT, control theory, simulation, and AI.

The primary motivation behind introducing Thermodynamic Automata was to address the challenge of achieving a uniform temperature state at a desired set level with zero energy consumption.

A thermodynamic automaton is a finite collection of subsystems interacting on the principles of non-equilibrium thermodynamics. These subsystems are modelled using a lumped parameter approach and exchange thermal energy to drive their evolution in a defined state space. The heat flux Qij = cij hij Aij ΔTij between two subsystems "i" and "j" with temperature difference ΔTij (a replacement for gradient) and contact area Aij is precisely regulated by multiplicative controls cij, which are mathematical mappings that operate within the state space without any artificial energy sources or heat sinks (no "+qi" term in the previous equation)

Mathematically, thermodynamic automata fall within the realm of finite-dimensional dynamical systems. They provide a framework to simplify complex distributed parameter problems into more manageable lumped parameter problems using compartmental modelling. Different theoretical models governing equations can be developed based on the system's complexity.

The field of research initially explored the predictive adaptation of Automata to seasonal variations, it was abandoned when the author moved to industry research work, and Prof. Sertorio also shifted to other topics.

Thermodynamic Automata introduced a new approach to tackling thermodynamic equilibrium and energy consumption challenges in complex systems that remain relevant today.

Conclusions on CHT

In this article, we have progressed from physics to mathematics to numerical solutions and, finally, the CHT engineering applications.

CHT analysis is the ultimate solution for engineers optimizing designs with "rock-solid" reliability for thermal simulations.

By combining CFD simulations with thermal analysis in solids, this multidisciplinary approach enables engineers to understand the interaction between fluid flows and solid components and, consequently, improve the efficiency of brake systems and under hood thermal management in automotive engineering.

CHT analysis will continue to be important in sectors such as industrial processes and aerospace exploration, and we're confident that implementing AI is the way forward to involve all engineers looking to achieve the best results.

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