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Topology Optimization of Two Fluid Heat Exchangers
Lukas Christan Høghøj, Daniel Ruberg Nørhave, Joe Alexandersen, Ole Sigmund, Casper Schousboe Andreasen
TL;DR
The paper addresses heat-exchanger design beyond classical layouts, where increasing surface area can also increase flow resistance and reduce convection. It uses pressure-constrained topology optimization with a single design field and an erosion-dilation interface method. The three-dimensional shell-and-tube-like designs improve heat transfer by up to 113% compared with the best-performing baseline.
Problem
Classical heat-exchanger designs remain restricted to relatively simple layouts, while increasing surface area through geometric complexity can raise flow resistance and reduce the convection coefficient.
Source: §1 · pp. 1–2Method
A single-field topology-optimization approach identifies separated fluid domains and a solid interface, maximizing transferred heat subject to pressure-drop constraints.
Source: §7 · p. 13Results
Up to 113% improvement compared with the best-performing baseline was reported for the three-dimensional shell-and-tube-like example.
Source: §7 · p. 15Takeaways & Limitations
The three-dimensional optimized designs exhibit microvilli-like features that enhance surface area and perturb flow near them.
Source: §7 · p. 15In the two-dimensional results, the pressure constraint has little influence while the flow remains mostly porous, before continuation reaches β = 4.
Source: §6.1.2 · p. 10
Abstract
from arXiv · showhide
A method for density-based topology optimization of heat exchangers with two fluids is proposed. The goal of the optimization process is to maximize the heat transfer from one fluid to the other, under maximum pressure drop constraints for each of the fluid flows. A single design variable is used to describe the physical fields. The solid interface and the fluid domains are generated using an erosion-dilation based identification technique, which guarantees well-separated fluids, as well as a minimum wall thickness between them. Under the assumption of laminar steady flow, the two fluids are modelled separately, but in the entire computational domain using the Brinkman penalization technique for ensuring negligible velocities outside of the respective fluid subdomains. The heat transfer is modelled using the convection-diffusion equation, where the convection is driven by both fluid flows. A stabilized finite element discretization is used to solve the governing equations. Results are presented for two different problems: a two-dimensional example illustrating and verifying the methodology; and a three-dimensional example inspired by shell-and-tube heat exchangers. The optimized designs for both cases show an improved heat transfer compared to the baseline designs. For the shell-and-tube case, the full freedom topology optimization approach is shown to yield performance improvements of up to 113% under the same pressure drop.
1. Introduction
Heat exchanger design has largely relied on classical layouts, while increasing surface area can also raise flow resistance and reduce convection. The paper proposes simulation-driven, pressure-drop-constrained topology optimization that directly measures exchanger efficiency.
1. Introduction
- Heat exchangers transfer thermal energy between fluids, commonly separated by solid walls, and are used in applications including engine cooling, air conditioning, power production, and refrigeration.Established designs commonly use parallel-flow, counter-flow, and cross-flow layouts.Source: §1 · p. 1, Figure 1 · p. 2
- Additive manufacturing has enabled more geometrically complex exchangers, including designs that use Triply Periodic Minimal Surfaces to increase internal surface area.The paper notes that increasing area through complexity can reduce the convection coefficient as flow resistance rises.Source: §1 · p. 1
- The proposed optimization maximizes heat-exchanger efficiency measured in simulations, subject to pressure-drop constraints, rather than optimizing surface area as a proxy.The motivating trade-off is that added geometric complexity can increase resistance, slow flow, and decrease the convection coefficient.Source: §1 · pp. 1–2
- Topology-optimization studies of heat exchangers are sparse, and several prior works address flow-guiding features rather than the exchanger solid geometry itself.The review identifies studies of actual solid geometry by Papazoglou, Haertel, Tawk et al., Saviers et al., and Kobayashi et al. [32].Source: §1 · p. 2
2. Parametrization
The design problem arranges a solid interface between two fluid domains to maximize transferred heat at a prescribed operating power, proportional to pressure drop. A relaxed domain representation permits topology optimization while an identification technique seeks exclusive assignment of each point to one fluid or solid.
2. Parametrization
- The objective is to maximize heat transferred at a given operational power, which is proportional to the pressure drop across the exchanger.Source: §2 · p. 3
- The computational domain is partitioned into fluid domains Ω1 and Ω2 and a solid domain Ωs separating the fluids.Figure 2 depicts the two fluid regions on opposite sides of the solid interface.Source: §2 · p. 3, Figure 2 · p. 3
- The relaxed representation permits both fluids throughout the domain, while an identification technique seeks to assign each point exclusively to one fluid or the solid.Accordingly, each fluid's velocity should be zero outside its own domain.Source: §2 · p. 3
3. Governing equations
The governing model couples two separately solved fluid-flow problems to one heat-transfer problem over the full design domain. Its equations assume steady state, constant fluid properties, incompressibility, and negligible viscous-dissipation heating.
3. Governing equations
- The governing equations assume steady state, constant fluid properties, and incompressibility, and neglect heat generated by viscous dissipation.The resulting coupling is one-way: mass transfer affects heat transfer, but not vice versa.Source: §3 · p. 3
3.1. Mass transfer
- Each fluid's Navier–Stokes mass-transfer problem is solved over the entire domain, with Brinkman friction penalizing flow outside that fluid's subdomain.The impermeability is taken as infinite outside the corresponding fluid in the model, but a large finite value is used in practice for numerical reasons.Source: §3.1 · p. 3
3.2. Heat transfer
- Heat transfer is modeled by a convection–diffusion equation over the entire domain, using both fluid flows and a spatially varying conductivity coefficient.The solid Peclet number relates fluid convection to diffusion in the solid; inlet temperatures are prescribed, and the remaining boundary is externally insulated.Source: §3.2 · pp. 3–4
4. Finite element formulation
The finite-element formulation solves the two flow problems and the global heat problem sequentially. Stabilization is used for equal-order elements and to address steep gradients associated with convection.
4. Finite element formulation
- The governing problems are discretized with structured meshes of regular trilinear hexahedral elements and solved using the finite element method.Source: §4 · p. 4
- PSPG stabilization facilitates equal-order elements, while SUPG stabilization is applied to both flow problems and heat transfer to alleviate steep-gradient problems from convection.Source: §4 · p. 4
- The two mass-transfer residual systems are solved separately, and the heat-transfer system uses a global velocity field combining both fluid flows.The heat system matrix includes thermal diffusion, convection, and SUPG stabilization contributions.Source: §4 · p. 4
5. Topology optimization
The optimization represents the design with a relaxed, elementwise scalar field and seeks heat transfer subject to pressure-drop constraints. Filtering, erosion-dilation identification, interpolation, adjoint sensitivities, and parameter continuation shape the resulting design and its optimization.
5.1. Optimization problem
- The relaxed design field minimizes the difference between outlet enthalpies to maximize heat transfer, subject to a separate pressure-drop constraint for each fluid.The optimization uses a nested formulation and is solved with MMA.Source: §5.1 · pp. 4–5
5.2. Filtering and projecting
- A PDE filter and smooth Heaviside projection regularize the design field and can enforce a minimum length scale when multiple design realizations are considered.The projection uses threshold η and sharpness β.Source: §5.2 · p. 5
5.3. Modelling solid interfaces
- One design variable generates two fluid fields and a solid interface; erosion-dilation identification guarantees a separating solid wall whose thickness can be controlled.Filtering and projection precede re-filtering and projections at low and high thresholds.Source: §5.3 · p. 5, Figure 3 · p. 5
5.4. Interpolation functions
- RAMP interpolates fluid impermeabilities, while a SIMP-inspired conductivity interpolation penalizes the solid phase to avoid high relative conductivities at intermediate fluid-field values.The impermeability upper limit applies where its fluid is absent; qα controls curvature and gives a linear function when qα = 0.Source: §5.4 · p. 6
5.5. Sensitivity analysis
- Adjoint sensitivities come from a Lagrangian built with the finite-element residuals and multipliers, with the multipliers obtained by solving three weakly coupled adjoint problems.Adjoint Dirichlet conditions are homogeneous.Source: §5.5 · p. 6
5.6. Continuation of parameters
- Continuation sharpens interfaces and improves physical interpolation as optimization progresses, adjusting β and conductivity penalization together while relaxing qα in a single step.Updates occur every 40 iterations in 2D and 20 in 3D when constraints were met in the preceding three iterations.Source: §5.6 · p. 6, Table 1 · p. 6
6. Results
The two-dimensional counter-flow example demonstrates the method under varying pressure-drop constraints, while the three-dimensional shell-and-tube-inspired example improves on both baselines. In the latter, performance and channel penetration depend on the initial design’s projection sharpness.
6.1.1. Problem definition
- The two-dimensional counter-flow problem uses opposing hot- and cold-fluid inlets, parabolic inlet profiles, and excluded inlet and outlet regions to preserve undisturbed flow.The optimization compares pressure-drop limits derived from the straight-channel baseline.Source: §6.1.1 · p. 7, Figure 5 · p. 8, Figure 6 · p. 8
6.1.2. Results
- As admissible pressure drop rises in the two-dimensional case, channels narrow and flow speeds increase; the authors also observe higher heat transfer at higher admissible pressure loss.Some channels bend toward the boundary, possibly avoiding a conductive wall that could act as an unwanted regenerator.Source: §6.1.2 · p. 8
- In the two-dimensional case, optimized performance closely tracks the theoretical prediction, and at ∆Pmax = 5∆Pbase the curved-channel design outperforms the parallel-channel design by about 1%.The authors attribute the higher heat transfer to greater design freedom, including slightly curved channels.Source: §6.1.2 · pp. 8–9, Figure 8 · p. 9, Figure 7 · p. 9
- For the two-dimensional example, the eroded and dilated fields produce a separating wall, and the resulting physical fields identify that wall by higher conductivity.A few transition elements have intermediate conductivity, which the authors report has very small influence on the final design.Source: §6.1.2 · p. 9, Figure 9 · p. 10, Figure 10 · p. 10
6.2.1. Problem definition
- In the three-dimensional case, higher initial projection sharpness lets coolant channels reach farther into the design domain; loosely defined starts can leave the back region unused.The half-domain setup uses symmetry, and the two fluids enter and exit on adjacent faces.Source: §6.2.1 · pp. 10–12, §6.2.2 · p. 12, Figure 13 · p. 12
6.2.2. Results
- The three-dimensional optimized structures include shifted microvilli-like features that curve the adjacent flow and increase surface area; four-channel-baseline starts also substantially modify coolant paths and disturb cooled-fluid flow.The authors suggest that the increased cooled-flow disturbance may benefit heat transfer.Source: §6.2.2 · pp. 12–13
- 74.8%–113%: optimized three-dimensional designs improve heat transfer over the better four-channel baseline, with the best design transferring 89.9 W versus 42.2 W for that baseline.The best designs use βinitial = 4; the reported improvement range spans different initial designs and pressure-drop settings.Source: Table 6 · p. 13, §6.2.2 · p. 13
7. Discussion and conclusion
The paper presents a two-fluid topology optimization method and reports improved heat-exchanger designs. Its examples also expose practical limits from geometric complexity and CAD transfer.
7. Discussion and conclusion
- A single design-variable field and erosion-dilation interface identification define the solid interface and fluid phases, while optimization maximizes transferred heat under pressure-drop limits for both fluids.The two mass-transfer problems and one heat-transfer problem are solved sequentially across the design domain.Source: §7 · p. 13
- In the limited-freedom two-dimensional example, channels narrow as allowable pressure drop increases, matching Poiseuille-flow and ε−NTU predictions for this simple case.The improvement is related to a theoretically derived optimum for a one-dimensional optimization.Source: §7 · pp. 13, 15
- Up to 113% improvement compared to the best-performing baseline was achieved in the three-dimensional shell-and-tube example; initial flow fields mattered for early sensitivities.The optimized designs also show microvilli-like features that increase surface area and perturb nearby flow.Source: §7 · p. 15
- Microvilli-like features in the optimized designs increase surface area and perturb the flow near them.Source: §7 · p. 15
- The intricate optimized designs have been challenging to export to CAD-based engineering analysis, so their performance has not yet been evaluated there.The authors say the designs may currently remain inspirational for future heat-exchanger designs.Source: §7 · p. 15
- The study prioritizes method development and ultimate design freedom; its intricate details may be impractical, and added geometry constraints may lower achievable efficiency.Larger length-scale, overhang, and extrusion constraints are proposed for future work.Source: §7 · p. 15
Appendix B.1. Adjoint sensitivity analysis
The adjoint sensitivity analysis differentiates a Lagrangian with respect to the design variable, then uses adjoint problems to obtain sensitivities without the difficult state-derivative terms.
Appendix B.1. Adjoint sensitivity analysis
- The analysis sets up a Lagrangian with arbitrary multipliers and differentiates it with respect to the design variable using the chain rule.Source: Appendix B.1 · p. 18
- Because the two mass-transfer states are mutually independent and weakly coupled, corresponding terms in the sensitivity expression can be omitted.Source: Appendix B.1 · p. 18
- Choosing the arbitrary multipliers appropriately eliminates difficult state-derivative terms; solving the adjoint problems then yields the sensitivities.Source: Appendix B.1 · p. 18
Appendix B.2. Chain rule projection of sensitivities
This appendix maps sensitivities from the eroded and dilated design fields back to the original design variable through the projection and filtering steps.
Appendix B.2. Chain rule projection of sensitivities
- The paper reports that sensitivities of both the objective function and the constraints are found with respect to ξ1 and ξ2, as described in Section 5.5 and Appendix B.1.Source: Appendix B.2 · p. 18
- Sensitivities with respect to the eroded and dilated variables are projected back to design variable ξ using steps corresponding to the erosion-dilation process.Source: Appendix B.2 · p. 18
- The smooth Heaviside projection is differentiated, and the sensitivity transformation through the PDE filter is equivalent to filtering the sensitivities.Source: Appendix B.2 · p. 18