The relentless pursuit of sustainable transportation has cemented the position of Electric Vehicles (EVs) at the forefront of the automotive industry’s future. Central to the performance, safety, and longevity of any EV is its battery pack. Effective thermal management of this EV battery pack is non-negotiable; excessive heat from charge-discharge cycles accelerates degradation and poses severe safety risks like thermal runaway. Among cooling strategies, liquid cooling plates offer superior heat transfer coefficients and temperature uniformity compared to air or phase-change materials. The design of the internal cooling channels within these plates is therefore a critical engineering challenge. Traditional design relies on parametric studies of predefined shapes (e.g., serpentine, grid), which inherently limits the search for optimal flow paths. This work presents a robust topology optimization framework for designing high-performance cooling channels, explicitly targeting the demanding thermal environment of an EV battery pack.
Topology optimization, a free-form material distribution method, transcends shape and size optimization. It defines the optimal layout within a given design domain. For fluid flow problems, methods like the density (SIMP) and level-set methods are common. However, I employ a more geometrically intuitive approach: the Moving Morphable Component (MMC) method, later refined using a density-based filter. The MMC method describes potential fluid channels using explicit geometric components, offering direct control and efficient parameterization. In this study, I define the cooling channel layout for a plate serving a segment of a larger EV battery pack module. The optimization goal is to minimize the average temperature of the plate under constraints of fluid volume and hydraulic dissipation, ensuring both cooling efficacy and manageable pump power.
1. Methodology and Mathematical Framework
The physical model is simplified from a 3D battery-cooling plate assembly to a 2D plane, assuming minimal temperature variation through the plate’s thickness. The design domain, representing the cooling plate, has specified inlet and outlet ports. A constant heat flux simulates the heat generated by the lithium-ion cells in the EV battery pack. The optimization framework is built upon the coupled solution of fluid flow, heat transfer, and a material interpolation scheme.
1.1 Governing Equations
The flow is assumed to be steady, laminar, and incompressible. The governing equations are the Navier-Stokes and energy equations:
Continuity:
$$\nabla \cdot \mathbf{u} = 0$$
Momentum:
$$\rho (\mathbf{u} \cdot \nabla) \mathbf{u} = \nu \nabla \cdot (\nabla \mathbf{u} + \nabla \mathbf{u}^T) – \nabla p + \alpha_f \mathbf{u}$$
Energy:
$$\rho c_p (\mathbf{u} \cdot \nabla T) = \nabla \cdot (k \nabla T) + Q$$
where \(\mathbf{u}\) is the velocity vector, \(p\) is pressure, \(T\) is temperature, \(\rho\) is density, \(\nu\) is kinematic viscosity, \(c_p\) is specific heat, \(k\) is thermal conductivity, and \(Q\) is the volumetric heat source from the battery. The term \(\alpha_f \mathbf{u}\) represents a volumetric force for the Brinkman penalization method, used to model porous media and distinguish fluid from solid regions. The boundary conditions are: fixed velocity and temperature at inlets (\(\Gamma_{in}\)), no-slip and adiabatic walls (\(\Gamma_{wall}\)), and zero pressure at outlets (\(\Gamma_{out}\)).
1.2 Material Interpolation
A design variable field \(\gamma(\mathbf{x})\) is defined, where \(\gamma = 1\) denotes fluid and \(\gamma = 0\) denotes solid. The material properties at any point are interpolated as functions of \(\gamma\):
$$ \alpha_f(\gamma) = \alpha_{max} \frac{q_\alpha (1-\gamma)}{q_\alpha + \gamma} $$
$$ k(\gamma) = k_f + (k_s – k_f) \frac{q_k (1-\gamma)}{q_k + \gamma} $$
Here, \(k_f\) and \(k_s\) are the fluid and solid thermal conductivities, \(q_\alpha\) and \(q_k\) are penalty factors (set to 0.01), and \(\alpha_{max}\) is the maximum inverse permeability, representing a solid. To ensure numerical stability and gradual convergence, \(\alpha_{max}\) is increased progressively from a small initial value \(\alpha_0\) according to an adaptive scheme, preventing the optimizer from getting trapped in local minima early on. The relevant material properties for the coolant (water) and the plate (aluminum) are summarized below.
| Parameter | Water (Fluid) | Aluminum (Solid) |
|---|---|---|
| Thermal Conductivity, \(k\) (W/m·K) | 0.61 | 237 |
| Specific Heat, \(c_p\) (J/kg·K) | 4180 | 880 |
| Density, \(\rho\) (kg/m³) | 1000 | 2700 |
| Dynamic Viscosity, \(\mu\) (Pa·s) | 0.001 | – |
1.3 The MMC Approach for Channel Representation
The core of the initial design phase is the MMC method. Instead of a pixel-based density field, the cooling channel is described by a set of deformable components. Each component is defined by two connected quadratic Bézier curves, providing smooth and flexible geometry. A single component is parameterized by 11 variables: the center coordinates \((x_0, y_0)\), orientation \(\theta\), lengths \(L_1, L_2\), control heights \(h_1, h_2\), and width parameters \(d_1, d_2, d_3, d_4\). The component’s centerline \(\mathbf{B}_e(t)\) and variable width \(D(t)\) are given by:
$$ \mathbf{B}_e(t) = (1-t)^2 \mathbf{P}_1 + 2t(1-t)\mathbf{C}_1 + t^2 \mathbf{P}_2, \quad t \in [0,1] $$
$$ D_1(t) = (d_2 – d_1)t + d_1, \quad D_2(t) = (d_4 – d_3)t + d_3 $$
The topology description function \(\phi_j(\mathbf{x})\) for the \(j\)-th component determines if a point \(\mathbf{x}\) is inside (\(\phi>0\)), on (\(\phi=0\)), or outside (\(\phi<0\)) the component. The final pseudo-density field \(\gamma_{mmc}\) from \(m\) components is obtained by a smoothed Heaviside projection of the maximum of all component functions:
$$ \gamma_{mmc}(\mathbf{x}) = H_r \left( \max_{j=1,\ldots,m} (\phi_j(\mathbf{x})), r \right) $$
This explicit parameterization allows efficient optimization with a relatively small number of design variables (e.g., 70 for 64 components) compared to density-based methods, directly generating smooth channel geometries ideal for the cooling plates in an EV battery pack.
1.4 Optimization Problem Formulation
The objective is to minimize the average temperature of the design domain \(\Omega\), which directly correlates with the cooling performance for the EV battery pack. Constraints are placed on the maximum allowed fluid volume fraction \(V_f^*\) and the total power dissipation \(J^*\) in the flow field, which relates to the required pumping power. The topology optimization problem is stated as:
$$
\begin{aligned}
& \underset{\gamma}{\text{minimize}} & & \Psi = \frac{1}{|\Omega|} \int_{\Omega} T \, d\Omega \\
& \text{subject to} & & \frac{1}{|\Omega|} \int_{\Omega} \gamma \, d\Omega \leq V_f^* \quad \text{(Volume constraint)} \\
& & & J = -\int_{\Gamma_{in} \cup \Gamma_{out}} \left(p + \frac{1}{2} \mathbf{u} \cdot \mathbf{u} \right) \mathbf{u} \cdot \mathbf{n} \, d\Gamma \leq J^* \quad \text{(Dissipation constraint)} \\
& & & \text{Governing Equations (1)-(3)} \\
& & & 0 \leq \gamma \leq 1
\end{aligned}
$$
The optimization is performed in two steps. First, the MMC method is used to find a promising channel layout. The result from this step, \(\gamma_{mmc}\), is then used as the initial guess for a second optimization phase using a density-based method with sensitivity filtering. This hybrid MMC-Density strategy leverages the global search capability of MMC and the local refinement capability of the density method to produce a high-performance, manufacturable design.
2. Numerical Implementation and Case Studies
The optimization framework is implemented within the open-source computational fluid dynamics (CFD) platform OpenFOAM, coupled with an adjoint sensitivity solver. The design domain is a rectangular plate with two inlets and two outlets to promote temperature uniformity. A non-uniform heat flux \(Q\) is applied to simulate localized high-heat zones from the EV battery pack cells.
2.1 Grid Independence and Method Comparison
A grid independence study was conducted to ensure solution accuracy. The average plate temperature converged with a mesh of approximately 36,800 cells, which was adopted for subsequent optimizations. A key study compared the standalone density method against the MMC method. The density method, starting from a homogeneous field, tended to create a main channel early in optimization, often leading to suboptimal local minima with inefficient secondary branches. In contrast, the MMC method, initialized with a grid of components, naturally evolved interconnected networks that could adapt more globally as constraints tightened. This demonstrated the MMC method’s superior suitability for conceptual channel design in EV battery pack cooling plates.
| Method | Number of Design Variables | Average Temp. (K) | Max Temp. (K) | Key Observation |
|---|---|---|---|---|
| Density (SIMP) | ~147,200 (mesh-dependent) | 297.95 | 301.46 | Prone to local minima; generates complex, sometimes impractical, micro-channels. |
| MMC (64 components) | 704 | 297.93 | 301.85 | Generates smooth, connected network layouts; more efficient global search. |
2.2 Influence of Inlet and Outlet Locations
The placement of coolant ports is a critical practical decision. I investigated four different port configurations (Case A-D) using the MMC method under the same volume (\(V_f^* = 0.45\)) and dissipation constraints. The results, summarized below, show that port arrangement significantly impacts the optimal topology and final performance.
| Case | Port Layout Description | Average Temp. \(\Psi\) (K) | Maximum Temp. \(T_{max}\) (K) |
|---|---|---|---|
| A | Inlets top-left/bottom-right; Outlets bottom-left/top-right | 297.13 | 299.97 |
| B | Inlets top-left/top-right; Outlets bottom-left/bottom-right | 297.25 | 300.42 |
| C | Inlets left side; Outlets right side | 297.00 | 300.04 |
| D | Inlets top-left/bottom-left; Outlets top-right/bottom-right | 296.97 | 299.68 |
Case D, with inlets on the left edge and outlets on the right edge, achieved the best cooling performance. The optimized channel layout effectively distributed flow across the entire plate, particularly through the high-heat zones. The subsequent density-based refinement of Case D (resulting in Case E) further improved the design by smoothing irregularities and subtly adjusting the channel network, yielding a final average temperature of 296.74 K.
2.3 Performance Benchmark Against Conventional Designs
To validate the superiority of the topology-optimized design, Case E was compared against two traditional serpentine/grid channel layouts with the same fluid volume fraction. The results are striking and underscore the value of the optimization framework for EV battery pack thermal management.
| Design | Average Temp. \(\Psi\) (K) | Maximum Temp. \(T_{max}\) (K) | Flow Dissipation \(J\) (kg·m²/s³) | Key Flow Feature |
|---|---|---|---|---|
| Traditional Grid 1 | 303.63 | 323.30 | 2.44 \(\times 10^{-7}\) | Low-velocity stagnation zones in center, poor heat extraction. |
| Traditional Grid 2 | 303.96 | 317.00 | 2.10 \(\times 10^{-7}\) | Improved over Grid 1 but still has localized hot spots. |
| Case E (MMC-Density Optimized) | 296.74 | 299.06 | 3.40 \(\times 10^{-7}\) | Highly distributed flow network; avoids stagnation; excellent thermal uniformity. |
The performance improvement is substantial. Compared to the best traditional design, the topology-optimized channel reduced the average plate temperature by approximately 6.89 K and, more critically, lowered the peak temperature by about 24.24 K. This dramatic reduction in peak temperature is vital for preventing hotspot-induced damage within the EV battery pack. The optimized design achieves this by creating a complex, branched network that maintains adequate flow velocity throughout the domain, ensuring efficient heat advection from all heated regions.
3. Conclusion and Perspectives
This work successfully demonstrates a two-stage topology optimization strategy for designing liquid cooling channels for EV battery pack thermal management systems. The hybrid MMC-Density approach proves highly effective: the MMC method provides an excellent, smooth initial channel layout through geometric component manipulation, while the subsequent density-based refinement polishes the design for ultimate performance. The framework systematically addresses practical considerations, such as the impact of inlet/outlet port locations, guiding better system-level design decisions.
The optimized channels outperform conventional grid-based layouts by a significant margin, offering markedly lower average and, crucially, maximum temperatures. This directly translates to enhanced battery safety, longevity, and performance consistency. The methods and results presented here provide a powerful computational design tool for developing next-generation thermal management solutions for high-density EV battery packs. Future work will involve extending the optimization to full 3D models, incorporating transient thermal loads, and considering multi-objective formulations that simultaneously minimize temperature, pressure drop, and manufacturing cost.