1 Fundamentals
1.1 Definition and concept
Conjugate heat transfer is the combined analysis of heat flow in a solid and a fluid that are in direct thermal contact. The two domains cannot be treated independently because the temperature distribution in the solid affects the fluid, while the fluid flow and temperature field alter the wall heat transfer. The term is commonly used when conduction through a wall, fin, or other structure is coupled with convection in the adjacent moving fluid.
1.2 Heat transfer modes involved
1.2.1 Conduction in solids
Within the solid region, heat is transferred primarily by conduction. The rate depends on the material’s thermal conductivity, geometry, and temperature gradient. In many engineering devices, the solid acts as a barrier, path, or spreading medium for heat before it reaches the fluid.
1.2.2 Convection in fluids
In the fluid region, heat is transported by a combination of bulk motion and microscopic diffusion. Convection may be forced by pumps or fans, or it may arise naturally from density differences caused by temperature variation. The local heat transfer rate is strongly influenced by velocity, turbulence, and boundary-layer development.
1.2.3 Radiation effects
Radiation may also contribute, especially at high temperatures or across gaps where view factors are significant. In many conjugate problems, radiation is secondary and may be neglected for simplicity, but in furnaces, hot surfaces, and thermal protection systems it can be an important part of the overall energy balance.
1.3 Thermal coupling at interfaces
The interface between solid and fluid is the key coupling surface. Heat leaving one medium must enter the other, so the interface conditions link temperature, heat flux, and flow behavior. Because the wall temperature influences the fluid boundary layer, and the fluid heat transfer coefficient depends on the wall condition, the interface must be solved as part of the complete system.
1.4 Governing physical principles
The analysis rests on conservation of energy, together with the laws governing momentum and mass transport in the fluid. In the solid, heat conduction is described by a diffusion equation. In the fluid, the transport of heat depends on the velocity field, which itself is determined by mass and momentum conservation. The result is a coupled multiphysics problem.
2 Mathematical formulation
2.1 Energy conservation equations
The general formulation is based on the first law of thermodynamics applied to each domain. Heat storage, conduction, convection, and generation terms are balanced over a differential control volume. In conjugate problems, the energy equations in the solid and fluid are linked by common boundary conditions at their interface.
2.2 Solid-domain heat conduction equation
In the solid, the temperature field is typically described by the transient or steady conduction equation. For homogeneous materials with no internal flow, the governing relation expresses the balance between thermal diffusion and any volumetric heat generation. The exact form depends on whether material properties are treated as constant or temperature dependent.
2.3 Fluid-domain transport equations
2.3.1 Continuity equation
The continuity equation expresses conservation of mass in the fluid. It ensures that the velocity field is physically consistent and is especially important when density varies with temperature or pressure.
2.3.2 Momentum equation
The momentum equation determines the fluid velocity distribution. It accounts for pressure forces, viscous effects, body forces, and inertial terms. Since convection depends on flow structure, the momentum balance is a central part of conjugate heat transfer modeling.
2.3.3 Energy equation
The fluid energy equation describes transport of thermal energy by advection and diffusion, with possible internal heat generation. The wall temperature enters this equation through boundary conditions, while the resulting fluid temperature field feeds back on the wall heat flux.
2.4 Boundary and interface conditions
2.4.1 Temperature continuity
At a perfect contact interface, the temperature is continuous across the solid-fluid boundary. This means the wall temperature on the solid side equals the adjacent fluid-side surface temperature.
2.4.2 Heat flux continuity
The normal heat flux must also be continuous across the interface. The heat conducted to the boundary in the solid must match the heat removed or supplied by the fluid, accounting for the same physical transfer rate on both sides.
2.4.3 Contact resistance
If the interface is imperfect, an additional thermal resistance may appear. This can occur because of surface roughness, gaps, or contact pressure effects. Contact resistance reduces heat transfer and causes a temperature drop across the interface.
2.5 Dimensionless groups
2.5.1 Reynolds number
The Reynolds number characterizes the ratio of inertial to viscous forces in the fluid. It is widely used to distinguish laminar and turbulent flow regimes and to estimate the structure of convection near a wall.
2.5.2 Prandtl number
The Prandtl number compares momentum diffusivity to thermal diffusivity. It helps indicate the relative thickness of velocity and thermal boundary layers and is important in correlating convective heat transfer.
2.5.3 Nusselt number
The Nusselt number measures the enhancement of heat transfer by convection relative to pure conduction across a fluid layer. It is often used to express wall heat transfer in nondimensional form.
2.5.4 Biot number
The Biot number compares internal conduction resistance in a solid to surface convection resistance. It indicates whether temperature gradients inside a solid are significant and whether lumped approximations are appropriate.
3 Physical mechanisms and flow regimes
3.1 Laminar conjugate heat transfer
In laminar flow, the fluid moves in orderly layers, and heat transfer near the wall is often dominated by smooth boundary-layer development. Conjugate effects can be pronounced when the solid has low conductivity or when the wall thickness is not negligible.
3.2 Turbulent conjugate heat transfer
Turbulence increases mixing in the fluid and usually raises the convective heat transfer coefficient. In conjugate systems, the enhanced wall transfer can produce sharper coupling between fluid temperature fluctuations and solid temperature response.
3.3 Forced convection systems
Forced convection occurs when pumps, fans, or imposed pressure differences drive the fluid motion. These systems are common in heat exchangers, cooling loops, and process lines, where the fluid velocity strongly shapes the thermal field.
3.4 Natural convection systems
Natural convection arises from buoyancy forces created by temperature-driven density differences. The flow pattern depends on geometry, gravity, and thermal loading, making the coupled solid-fluid behavior sensitive to orientation and boundary shape.
3.5 Mixed convection systems
Mixed convection occurs when both forced and buoyancy-driven motion contribute significantly. In such cases, the relative importance of each mechanism may vary across the domain, and the resulting heat transfer can be strongly nonuniform.
3.6 Transient and steady-state behavior
In steady state, temperatures no longer change with time, and the heat balance is established under constant operating conditions. In transient problems, the solid and fluid temperatures evolve together, and the response may involve thermal lag, startup effects, or periodic loading.
4 Analytical and numerical methods
4.1 Analytical solutions
4.1.1 Idealized geometries
Closed-form solutions are possible for simplified systems such as flat plates, cylinders, and spheres with ideal boundary conditions. These solutions are useful for understanding basic behavior and for checking more complex calculations.
4.1.2 Approximate methods
Approximate analytical approaches include thermal resistance models, boundary-layer estimates, and integral methods. They provide rapid engineering estimates when full numerical simulation is unnecessary or impractical.
4.2 Numerical simulation
4.2.1 Finite difference methods
Finite difference methods replace derivatives with algebraic approximations on a grid. They are straightforward to implement for regular geometries and are often used in introductory or specialized heat transfer models.
4.2.2 Finite volume methods
Finite volume methods enforce conservation laws over discrete control volumes. They are especially well suited to coupled fluid and heat transfer problems because they preserve local balances of mass, momentum, and energy.
4.2.3 Finite element methods
Finite element methods divide the domain into elements and approximate the solution with basis functions. They are flexible for complex shapes and material interfaces, making them useful in solid-fluid thermal coupling problems.
4.3 Computational fluid dynamics approaches
4.3.1 Conjugate domain coupling
In computational fluid dynamics, the solid and fluid regions are solved together or exchanged iteratively through an interface. This coupling allows the wall temperature and heat flux to be computed self-consistently.
4.3.2 Meshing considerations
Accurate prediction often requires refined meshes near walls and interfaces, where temperature gradients are steep. Mesh quality affects stability, convergence, and the resolution of boundary layers and heat fluxes.
4.3.3 Solver strategies
Solution methods may use segregated or coupled solvers, steady or transient formulations, and relaxation techniques to improve convergence. The choice depends on problem stiffness, nonlinearity, and the degree of coupling.
4.4 Model validation and verification
Verification checks whether the numerical implementation solves the intended equations correctly, while validation compares predictions with experimental or benchmark data. Both steps are essential when the results will support design decisions.
5 Engineering applications
5.1 Heat exchangers
Heat exchangers are classic conjugate systems, since heat must pass through tube walls, plates, or fins between two fluids. The performance depends on wall conductivity, flow arrangement, and surface area.
5.2 Pipe flow and insulated lines
In pipes and process lines, heat transfer through the wall and insulation affects temperature losses or gains. Conjugate analysis helps estimate delivery temperatures, energy efficiency, and freeze or overheating risks.
5.3 Finned surfaces
Fins increase surface area and are widely used to enhance heat dissipation. Their effectiveness depends on conduction within the fin and convection from the exposed surfaces, making conjugate effects central to their design.
5.4 Chemical reactors
In reactors, heat transfer through vessel walls, jackets, coils, or internal structures can control reaction rates and selectivity. Conjugate modeling is useful when heat release or removal is strongly coupled to fluid motion and wall temperature.
5.5 Electronic and process cooling
Electronic devices and process equipment often generate concentrated heat that must be removed through solid packages, housings, or cold plates. Conjugate heat transfer helps predict hot spots and cooling performance.
5.6 Furnaces and thermal insulation
In furnaces, hot gases transfer heat to walls, loads, and insulation layers. The interaction between radiative, convective, and conductive mechanisms is important for efficiency and surface temperature control.
5.7 Drying and evaporation systems
Drying and evaporation processes involve heat transfer to moist materials or liquid films while mass transfer removes water or solvent. Conjugate modeling assists in estimating heating rates and surface evaporation behavior.
6 Design considerations
6.1 Thermal resistance networks
Thermal resistance networks provide a compact representation of conduction and convection paths. They are useful for preliminary sizing, identifying dominant resistances, and comparing design options.
6.2 Material selection
Material choice influences conductivity, thermal expansion, corrosion resistance, and allowable temperature. In conjugate systems, the selected material can strongly affect wall temperatures and thermal gradients.
6.3 Surface geometry effects
Shape affects boundary-layer development, surface area, flow separation, and local heat transfer. Curvature, roughness, ribs, and fins can all modify the conjugate response.
6.4 Fouling and deposits
Deposits on heat transfer surfaces add resistance and may alter flow patterns. Fouling reduces performance over time and can create local hot spots or thermal maldistribution.
6.5 Heat transfer enhancement
Enhancement methods include extended surfaces, turbulence promoters, inserts, and flow shaping. Such measures aim to improve thermal contact between solid and fluid while balancing pressure drop and manufacturing complexity.
6.6 Safety and thermal limits
Design must account for maximum allowable temperatures, material degradation, thermal stress, and loss of cooling capability. Conservative margins are often required where overheating could damage equipment or reduce reliability.
7 Measurement and experimental study
7.1 Temperature measurement techniques
Temperature fields are measured using thermocouples, resistance sensors, infrared methods, and optical techniques. The choice depends on temperature range, spatial resolution, and whether direct contact is acceptable.
7.2 Heat flux measurement
Heat flux can be estimated from guarded sensors, calibrated surface gauges, or inverse methods based on measured temperatures. Direct measurement is often difficult at interfaces, so multiple techniques may be combined.
7.3 Experimental test rigs
Laboratory rigs for conjugate heat transfer typically include a heated or cooled solid, a controlled fluid stream, and instrumentation for flow and temperature monitoring. These setups are used to study geometry effects, material behavior, and validation cases.
7.4 Correlation with simulation results
Experimental data are compared with numerical predictions to assess whether the model captures the correct heat transfer rates and temperature distributions. Differences may reveal missing physics, measurement uncertainty, or inadequate boundary conditions.
8 Common assumptions and limitations
8.1 Constant-property approximations
Many models treat density, viscosity, conductivity, and specific heat as constants. This simplifies analysis but may be less accurate when temperature ranges are large or when strong property variation affects flow and heat transfer.
8.2 Neglect of radiation
Radiation is often omitted in moderate-temperature applications to reduce complexity. This assumption can be reasonable when convective and conductive effects dominate, but it may fail in hot, low-flow, or highly emissive environments.
8.3 Thin-wall approximations
Some analyses assume the solid wall is thin enough to have nearly uniform temperature through its thickness. This can simplify calculations, although thicker walls may develop substantial internal gradients.
8.4 Perfect contact assumptions
A perfect thermal contact model ignores interfacial resistance and gaps. It is convenient for idealized studies, but real assemblies may exhibit measurable temperature drops at joints and interfaces.
8.5 Limitations in complex geometries
Highly detailed geometries, moving boundaries, phase change, and strong material nonlinearity can make conjugate problems difficult to solve accurately. In such cases, simplified models may be useful for preliminary design, but careful validation is needed for final engineering use.