# Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Serpentine Model

#### 2.1. Model Description

_{h}) of 80 mm and an aspect ratio (AR) of 2.0. The length of all the straight passages is 245 mm, resulting in a length-to-hydraulic diameter of 3.06. Before entering the heating section, entrance and exit sections extended with a length of 300 mm are created at the channel inlet and outlet, respectively. The first turning region is designed as a rectangular shape with an inner radius of 20 mm since this region near the tip of the turbine blade suffers from a high heat load, which needs a large heat transfer area to cool. Meanwhile, the second bend near the blade platform receives a lower heat load; hence it is designed as an arc shape with outer and inner diameters of 140 and 20 mm, to reduce the total pressure loss in the channel [5].

_{in}) is set as uniform velocity (U

_{in}) with a prescribed temperature (T

_{in}). The inlet velocity is defined based on the experimental Reynolds number (Re), as follows:

_{out}) is set as a zero static pressure (p

_{0}). The boundary condition of the heating surfaces on the top and bottom is imposed with uniform heat flux (q

_{0}). The remaining walls are thus set as no-slip conditions (Γ

_{wall}) to surround the serpentine channel. Figure 1b also shows the design domains, in which all the heating regions inside design domains (Ω

_{id}) are evolved. Meanwhile, the extended entrance and exit of the channel set as outside design domains (Ω

_{od}) are not considered for optimization.

#### 2.2. Thermofluid Modeling

#### 2.2.1. Fluid-Flow Modeling

**u**indicates the fluid velocity vector; $\nabla $ signifies the gradient operator; p denotes the pressure field;

**I**is the identical tensor; μ

_{T}is the turbulent dynamic viscosity; * is the transpose; and

**F**represents the Brinkman friction term introduced by Borrvall and Petersson [34], calculated from Equation (4). Here, the body force is not exerted on the fluid outside the design domain (

**F**= 0 in Ω

_{od}).

_{u}

_{,max}is the maximum inverse permeability of the porous medium, while α

_{u}

_{,min}is the minimum one. I

_{u}(γ) denotes the inverse permeability interpolated function stated in Section 3.1.

_{u}

_{,max}can be defined depending on Darcy number (Da):

#### 2.2.2. Turbulence Modeling

_{k}signifies the turbulent kinetic energy source term.

_{0}and ε

_{0}for the solid regions in the design domain. They are defined as follows [26]:

_{k,}

_{max}and α

_{ε,}

_{max}are the maximum limits of the penalization for k and ε, respectively, while α

_{k,}

_{min}and α

_{ε,}

_{min}are the minimum ones. I

_{k}(γ) and I

_{ε}(γ) are material interpolated functions in both Equations (8) and (9). Here, the values of α

_{k,}

_{min}, α

_{ε}

_{,min}, k

_{0}and ε

_{0}are assigned to zero, while the value of α

_{k,}

_{max}and α

_{ε,}

_{max}are set equal to the value of α

_{u}

_{,max}. The material interpolations are detailed in Section 3.1.

#### 2.2.3. Heat Transfer Modeling

_{f}denotes the fluid heat capacity, k

_{f}indicates the fluid thermal conductivity and T is the temperature field in the serpentine channel.

_{c}(γ) is the function used to activate/deactivate the convective term for fluid (I

_{c}(γ) = 1) and solid (I

_{c}(γ) = 0). I

_{K}(γ) is used to interpolate between the thermal conductivity of the fluid and the thermal conductivity of the solid (k

_{s}). Lastly, the term q

_{0}indicates the constant bottom wall heat flux.

_{0}is the uniform heat flux applied to the heating boundaries in the 2D model. Additionally, the energy equation is weakly coupled to the fluid flow problem to obtain

**u**; then, it substitutes the energy equation to solve T. Furthermore, the Reynolds number and heat flux applied in this work are fixed at Re = 10,000 and q

_{0}= 1500 W/m

^{2}, respectively, because varying the velocity or the heat input in the topology optimization could change the topology of the structure in the final result [11,14].

## 3. Optimization Procedures

#### 3.1. Topology Optimization

#### 3.1.1. Material Distribution

_{u}, q

_{k}and q

_{ε}are tuning parameters to control the function curvature.

_{c}(γ) and I

_{K}(γ) are calculated using solid isotropic material penalization (SIMP) [8] in Equations (15) and (16), respectively:

_{c}and n

_{K}are penalization power coefficients; δ is fixed at δ = 10

^{−15}to avoid numerical problems when solving Equation (11) [22].

#### 3.1.2. Problem Formulation

_{out}is the outlet boundary, and

**n**is the normal vector. With this expression, the heat transfer in the channel can be constrained (g) as follows:

_{ref}is the reference value calculated from the conventional serpentine channel.

**S**is the vectors with discrete state fields,

**R**indicates the residual vector function evaluated from the discretization of the governing equations, g represents inequality constraints and

**x**denotes the design variable vector.

#### 3.1.3. Density Filter and Projection

_{f}signifies a smoothed design variable. R

_{min}is a filter radius calculated by = 1.5 × maximum mesh size in the considered domain.

_{p}is the projected design variable, β is a parameter used to control the steepness of the projection, and η denotes the projection threshold, fixed at η = 0.5 in this study.

#### 3.1.4. Summary of Topology Optimization Process

^{−4}, or after every 100 iterations, the optimization proceeds to the next continuation steps. The continuation approach is used to avoid the low-quality local minima result and ease the convergence in this optimization process [18]. Eventually, the optimization is considered to be completed when all the continuation steps are computed.

#### 3.2. Infilling Lattice Structures

## 4. Results and Discussion

#### 4.1. 2D Optimization

#### 4.1.1. Validation of the 2D Model

_{0}) in the 2D model is equivalent to the uniform heat flux boundary in the 3D system.

#### 4.1.2. Optimized Structure

_{0}= 0.6. The model is evaluated through the continuation parameters, as given in Table 2, to find the optimal structure. After the simulation, the design converges to the optimized structure, as presented in Figure 7. It can be noticed that the model leaves the intermediate fields to satisfy the temperature constraint by distributing the material in the channel. The intermediate results are due to the high conductivity difference between the copper and air used in this work [19]. For the 2D optimized design, the total pressure loss between inlet and outlet boundaries decreases by about 33.9% compared to the original 2D model, while the constraint temperature in the channel is maintained.

^{3}.

#### 4.2. 3D Model Comparisons

_{0}= 1500 W/m

^{2}according to the design settings in the conventional topology optimization, as mentioned in Section 2. For all cases, the channel dimensions are consistent with the baseline serpentine channel, as described in Figure 1a. The inlet and outlet boundaries are the same as the baseline channel. Here, the red zones are treated by the uniform heat flux, including the top, bottom and extruded areas from the optimization and TPMS structures. The remaining surfaces are set as no-slip walls.

#### 4.2.1. 3D Model Validation

^{−4}for the continuity, velocity and turbulence quantities and less than 10

^{−6}for the energy equation [5].

_{tot}denotes the total heat transfer, A

_{w}is the wetted area, and ΔT signifies the log mean temperature difference of the heating wall, calculated as follows:

_{w}is the average wall temperature, while T

_{in}and T

_{out}are the inlet and outlet fluid temperatures.

#### 4.2.2. Pressure Loss

_{p}) is employed to characterize the pressure loss from different models. The relative pressure coefficient along the channel flow path is calculated as follows:

_{loc}is the local static pressure at measuring points and p

_{ref}indicates the pressure value at the channel inlet.

#### 4.2.3. Heat Transfer

_{h}is the flat heated area.

#### 4.2.4. Thermal Performance

_{0}is the Blasius correlation, defined in Equation (30):

#### 4.2.5. Temperature Uniformity

_{i}denotes the local temperature, and A

_{i}is the local area. T

_{avg}is the average temperature of the considered region, calculated as follows:

#### 4.3. Detailed Flow and Heat Transfer Characteristics of the Optimal Serpentine Channel

#### 4.3.1. Flow Characteristics

_{in}

^{2}) and streamlines on different cross-sections of the serpentine channel, as presented in Figure 22. For both models, the turbulent kinetic energy and streamlines exhibit the same characteristics in the first passage. The higher turbulent kinetic energy values can be observed in the second and third passages.

#### 4.3.2. Heat Transfer Characteristics

## 5. Conclusions

- The conventional topology optimization model achieves a lower pressure loss and provides higher heat transfer than the baseline channel. Infilling the diamond- and gyroid-sheet structures in the optimal solution, including the intermediate results of the conventional topology optimization model, further enhance high heat transfer and maintains lower pressure loss than the baseline channel.
- The 3D optimized models with the TPMS structures provide lower pressure loss by about 19.0%–30.8% and higher total heat transfer by 36.6%–45.8%, compared to the 3D baseline channel. The optimized mode infilled with the diamond-TPMS structure is selected as the optimal serpentine channel in this study since it provides the best thermal performance, up to 64.8%, superior to the baseline channel. The temperature uniformity on the surface is also improved, particularly in the second passage.
- The 3D optimized model with the diamond-TPMS structure significantly eliminates the recirculation flow and the low heat transfer regions at the second and third passages. This model also reduces the influence of Dean’s vortices but maintains high turbulence kinetic energy, leading to better uniform flow and heat transfer distributions.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A_{h} | Flat heated area (m^{2}) |

A_{i} | Local area (m) |

A_{w} | Wetted area (m^{2}) |

AR | Aspect ratio of the channel |

C | Constraint function |

c_{f} | Fluid heat capacity (J/(kg K)) |

C_{p} | Relative pressure coefficient |

C_{ref} | Reference value in constraint function |

Da | Darcy number |

D_{h} | Hydraulic diameter of the channel (m) |

F | Fictitious body-force (N/m^{3}) |

f | Friction factor |

f_{0} | Blasius correlation |

g | Inequality constraints |

I_{u}, I_{k}, I_{ε}, I_{c}, I_{K} | Material interpolated functions |

J | Objective function |

k, k_{0} | Turbulent kinetic energy of the fluid and solid domains (m^{2}/s^{2}) |

k_{f}, k_{s} | Thermal conductivity of the fluid and solid (W/(m K)) |

L | Total length of the serpentine channel (m) |

n | Normal vector |

n_{c}, n_{K} | Penalization power coefficients in material interpolated functions |

$\overline{\overline{Nu}}$, ${\overline{\overline{Nu}}}_{tot}$ | Globally and total Nusselt numbers |

${\overline{\overline{Nu}}}_{0}$ | Dittus–Boelter equation |

P_{k} | Turbulent kinetic energy source term (W/m^{3}) |

p_{0} | Zero static pressure (Pa) |

p_{loc} | Local static pressure at measuring points (Pa) |

p_{ref} | Pressure value at the channel inlet (Pa) |

Pr | Prandtl number |

Q_{tot} | Total heat transfer (W) |

q_{0} | Uniform heat flux (W/m^{2}) |

q_{u}, q_{k}, q_{ε} | Tuning parameters in material interpolated functions |

R | Residual vector function |

Re | Reynolds number |

R_{min} | Filter radius (m) |

S | Vectors with discrete state fields |

TP | Thermal performance of cooling channels |

T_{avg} | Average temperature of the considered surface (m^{2}) |

T_{i} | Local temperature at the consider area (K) |

T_{in}, T_{out} | Inlet and outlet temperatures (K) |

T_{w} | Average wall temperature (K) |

u | Fluid velocity vector |

U_{in} | Inlet velocity (m/s) |

x | Design variable vector |

y+ | Nondimensional distance from the endwall to the first element node |

Greek Letters | |

α_{u,max}, α_{u,min} | Maximum and minimum inverse permeabilities of the porous medium (Pa s/m) |

β | Projection slope |

γ | Design variable |

γ_{f}, γ_{p} | Smoothed and projected design variables |

ε, ε_{0} | Turbulent energy dissipation for fluid and solid regions (m^{2}/s^{3}) |

μ | Dynamic viscosity (N s/m^{2}) |

μ_{T} | Turbulent dynamic viscosity (N s/m^{2}) |

ρ | Fluid density (kg/m^{3}) |

η | Projection threshold |

θ | Uniformity index of temperature |

ω | Specific rate of dissipation in the k–ω turbulence model (1/s) |

Γ_{in}, Γ_{out} | Inlet and outlet boundary conditions |

Γ_{wall} | No-slip boundary conditions |

Ω_{id}, Ω_{od} | Inside and outside design domains |

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**Figure 1.**Numerical model of a three-short-pass serpentine channel: (

**a**) full 3D model; (

**b**) 2D simplified design from the 3D model.

**Figure 4.**A single unit cell of (

**a**) diamond-sheet structure; (

**b**) gyroid-sheet structure for fulfilling the optimal results in this study.

**Figure 6.**Comparison of the average values of the state variables between 2D and 3D at Re = 10,000: (

**a**) pressure values; (

**b**) velocity magnitudes; (

**c**) temperature values.

**Figure 8.**(

**a**) 3D optimal solution including intermediate results; (

**b**) topology optimization model infilled with lattice structures.

**Figure 9.**Models for comparison: (

**a**) optimized model extracted with density filtering of 0.5 (TO model); (

**b**) optimized model infilled with diamond-sheet structure (TO model with diamond-TPMS); (

**c**) optimized model infilled with diamond-sheet structure (TO model with gyroid-TPMS).

**Figure 11.**The spanwise-averaged temperature and grid convergence index on the surface of the conventional serpentine channel at Re = 10,000.

**Figure 12.**Comparisons of globally averaged Nusselt number in each pass for different turbulence models at Re = 10,000.

**Figure 17.**Heat transfer enhancements for all serpentine models at Re = 10,000: (

**a**) globally averaged Nusselt number ratio; (

**b**) total Nusselt number ratio.

**Figure 20.**Velocity distributions and streamlines in the middle section of the serpentine channel: (

**a**) baseline model; (

**b**) TO model with diamond-TPMS structure.

**Figure 21.**Pressure distributions in the middle section of the serpentine channel: (

**a**) baseline model; (

**b**) TO model with diamond-TPMS structure.

**Figure 22.**Local dimensionless turbulent kinetic energy and velocity streamline on the cross-sections in the multipass channel: (

**a**) baseline model; (

**b**) TO model with diamond-TPMS structure.

**Figure 23.**Nusselt number contours of the serpentine channel: (

**a**) baseline model; (

**b**) TO model with diamond-TPMS structure.

**Table 1.**An overview of density-based topology optimization in recent studies on thermofluid systems.

Author | Flow Regime | Structure | Discretization | Optimizer |
---|---|---|---|---|

Alexandersen et al. [17] | Laminar | 2D and 3D | FEM ^{1} | MMA ^{3} |

Dilgen et al. [18] | Turbulent | 2D and 3D | FVM ^{2} | MMA |

Haertel et al. [19] | Laminar | 2D | FEM | GCMMA ^{4} |

Li et al. [20,21] | Laminar | 2D | FEM | SNOPT ^{5} |

Zhang et al. [22] | Laminar | 2D | FEM | GCMMA |

^{1}finite element method.

^{2}finite volume method.

^{3}method of moving asymptotes.

^{4}globally convergent method of moving asymptotes.

^{5}sparse nonlinear optimizer.

Parameter | Da | β | η | q_{u} | q_{k} | q_{ε} | n_{c} | n_{K} |
---|---|---|---|---|---|---|---|---|

Value | 10^{−4}–10^{−6} | 1.5–16.0 | 0.5 | 0.1–10 | 0.001 | 0.1 | 3–5 | 3–5 |

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**MDPI and ACS Style**

Yeranee, K.; Rao, Y.; Yang, L.; Li, H. Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces. *Energies* **2022**, *15*, 8924.
https://doi.org/10.3390/en15238924

**AMA Style**

Yeranee K, Rao Y, Yang L, Li H. Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces. *Energies*. 2022; 15(23):8924.
https://doi.org/10.3390/en15238924

**Chicago/Turabian Style**

Yeranee, Kirttayoth, Yu Rao, Li Yang, and Hao Li. 2022. "Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces" *Energies* 15, no. 23: 8924.
https://doi.org/10.3390/en15238924