Temperature control is a critical concern in the development of high-energy-density lithium-ion battery systems for electric vehicles. In this work, I focus on a hybrid liquid cooling plate that integrates straight channels with pin-fin arrays, aiming to improve the single-phase flow and heat transfer performance of a traction battery pack. A battery thermal model is first established based on discharge experiments and the Bernardi heat generation approach. Then, a conjugate heat transfer model of a battery module with several cold plate configurations is solved numerically. Among square, circular, triangular, and straight channels, the circular pin-fin configuration achieves the best trade-off between cooling effectiveness and pressure loss. Next, a Kriging surrogate model combined with an NSGA-II multi-objective optimizer is adopted to optimize the mass flow rate, channel height, and pin-fin spacing. The optimization simultaneously minimizes the maximum temperature, the temperature difference, and the pressure drop across the module. Selected Pareto-optimal solutions are verified with CFD, and the best compromise design reduces the temperature difference by 23.1% and the pressure drop by 43.2% relative to a baseline design. Finally, a circular pin-fin cold plate with the optimized channel height is manufactured and tested in a single-phase flow loop. Experiments characterize the flow resistance and heat transfer behavior for different coolant flow rates and heat fluxes. This research provides a practical strategy for designing high-efficiency cold plates and improving the safety and durability of traction battery packs.
1. Introduction
The rising demand for long-range electric vehicles has pushed lithium-ion battery packs toward increasingly compact and energy-dense layouts. During high-rate discharge, a traction battery pack can generate enormous heat. If this heat is not removed promptly, the cell temperature can exceed the recommended operating window. Lithium-ion cells usually perform best between 293 K and 313 K. Once the temperature exceeds this range, capacity fades, internal resistance grows, and side reactions may accelerate, eventually leading to thermal runaway. Therefore, thermal management of a traction battery pack is not merely a performance issue, but a fundamental safety requirement.
Many cooling techniques have been studied in the literature, including air cooling, phase-change material cooling, heat pipes, and liquid cooling. Air cooling has a limited heat-transfer coefficient, while phase-change materials add weight and suffer from low thermal conductivity. Heat pipes offer excellent conductance but are expensive to integrate into a large traction battery pack. Liquid cooling using cold plates currently dominates the electric-vehicle market because of its high heat-transfer capacity, adjustable flow rate, and compact footprint. In a typical liquid cooling system, coolant flows through channels inside an aluminum plate that is mounted between cells or under the module. The coolant carries heat away from the cells and rejects it through an external radiator or chiller.
Cold-plate design determines both the thermal performance and the energy consumption of a traction battery pack. Straight rectangular channels are easy to machine, but the thermal boundary layer along the channel continuously thickens. This behavior reduces the local convective coefficient downstream, which produces non-uniform temperature distributions and a high maximum temperature. One way to disrupt the boundary layer is to install pin-fin arrays in the channel. Pin-fins increase the heat-transfer area, induce flow separation, and enhance turbulent mixing near the wall. In this work, I systematically study cold plates with pin-fin channels for traction battery pack cooling. The investigation combines numerical simulation, multi-objective optimization, and experiments. The central goal is to identify a pin-fin configuration that yields outstanding cooling performance while maintaining an acceptable pressure drop and pumping power.
2. Battery Heat Generation Model
Since the cold plate is designed for a real battery module, accurate thermal boundary conditions at the battery-cold plate interface are needed. I model the battery as a homogeneous heat source with anisotropic thermal conductivity. The internal structure of a lithium-ion cell includes positive and negative electrodes, a separator, current collectors, and electrolyte. During discharge, lithium ions migrate from the negative electrode to the positive electrode, while electrons flow through an external circuit. The chemical and electrical processes inevitably release heat. The total heat-generation rate \(Q\) can be decomposed into reaction heat \(Q_r\), Joule heat \(Q_j\), polarization heat \(Q_p\), and side-reaction heat \(Q_s\):
$$Q = Q_r + Q_j + Q_p + Q_s.$$
For normal discharge conditions, side-reaction heat is negligible, and the polarization heat is comparatively small. A convenient and widely used model is the Bernardi heat-generation formula. Under the assumptions of uniform heat generation and constant thermophysical properties, the volumetric heat-generation rate can be written as
$$q_{gen} = \frac{I^2 R_T}{V_c},$$
where \(I\) is the discharge current, \(R_T\) is the total battery internal resistance, and \(V_c\) is the cell volume. This simplification ignores the reversible entropic term. In practice, the reversible term plays a minor role at moderate discharge rates and the model remains reliable for liquid-cooling design. The transient temperature field of the battery is governed by the energy conservation equation
$$\rho_b c_{p,b} \frac{\partial T_b}{\partial t} = k_{x,b} \frac{\partial^2 T_b}{\partial x^2} + k_{y,b} \frac{\partial^2 T_b}{\partial y^2} + k_{z,b} \frac{\partial^2 T_b}{\partial z^2} + q_{gen},$$
where \(\rho_b\), \(c_{p,b}\), and \(k_{i,b}\) are the density, specific heat, and directional thermal conductivities of the cell, respectively. Because the cell is a layered assembly of dissimilar materials, the through-plane conductivity is lower than the in-plane conductivity. I used a series-resistance model for the through-plane direction and a parallel-resistance model for the in-plane directions:
$$k_{T,x} = \frac{\sum L_i}{\sum \left(L_i / k_{T,i}\right)},$$
$$k_{T,y} = k_{T,z} = \frac{\sum L_i k_{T,i}}{\sum L_i}.$$
| Parameter | Value | Unit |
|---|---|---|
| Nominal capacity | 100 | Ah |
| Working voltage | 2.5–3.65 | V |
| Cell dimensions | 160 × 116 × 50 | mm |
| Thermal conductivity | 22.5/22.5/1.5 | W m⁻¹ K⁻¹ |
| Cell mass | 1.65 | kg |
| Density | 2050 | kg m⁻³ |
| Specific heat | 1088.7 | J kg⁻¹ K⁻¹ |
| Discharge rate | 1C | – |
| Discharge depth | 80% | – |
The cell internal resistance varies with the state-of-charge (SOC). I used the measured resistance data at a 1C discharge rate and fitted a polynomial curve to capture the SOC dependence. The resistance remains nearly constant for SOC between 0.2 and 0.8. When the SOC drops below 0.2, the resistance increases rapidly, which would generate excessive heat. Therefore, my simulations terminate the discharge at a SOC of 0.2. The heat-generation model was tested against a single-cell temperature-rise experiment. In the experiment, a prismatic cell was discharged at 1C in an environment of 291 K, and thermocouples recorded the surface temperature. The numerical temperature was calculated using the user-defined function that implemented the Bernardi source term. The average key-node error between simulation and experiment was less than 5%, which confirms that the battery heat model can be used in system-level cold plate simulations.
3. Cold Plate Configuration and Numerical Model
The numerical study is performed on a module consisting of 24 cells arranged in two columns, as shown in Fig. 2. Three aluminum cold plates are sandwiched between the cell columns. Each cold plate has one inlet and two outlets. Coolant enters at the center, flows through eight parallel rectangular channels, and leaves at both ends. This symmetric distribution yields a uniform flow pattern. In this work, I considered four channel configurations: a straight rectangular channel (SC), square pin-fins (S-Pin-fin), circular pin-fins (C-Pin-fin), and triangular pin-fins (T-Pin-fin). Pin-fins are arranged in-line with a defined pitch and diameter. The coolant is a 50% ethylene-glycol water mixture. Table 2 summarizes the physical model and material properties.
| Material | Density (kg m⁻³) | Specific heat (J kg⁻¹ K⁻¹) | Thermal conductivity (W m⁻¹ K⁻¹) | Dynamic viscosity (Pa s) |
|---|---|---|---|---|
| Silicone gap pad | 2094.96 | 2684 | 4 | – |
| Aluminum cold plate | 2719 | 871 | 202.4 | – |
| 50% ethylene-glycol | 1065 | 3281 | 0.38 | 0.0069×(T/273)^0.83 |
The conjugate heat-transfer simulation solves the continuity, momentum, and energy equations. For the fluid domain, the equations are given by
$$\frac{\partial \rho_f}{\partial t} + \nabla\cdot(\rho_f \mathbf{u})=0,$$
$$\frac{\partial (\rho_f \mathbf{u})}{\partial t} + \nabla\cdot(\rho_f \mathbf{u}\mathbf{u}) = -\nabla P,$$
$$\frac{\partial}{\partial t}(\rho_f c_f T_f) + \nabla\cdot(\rho_f c_f \mathbf{u} T_f) = \nabla\cdot(k_f \nabla T_f),$$
where \(\rho_f\), \(c_f\), \(k_f\), \(T_f\), and \(\mathbf{u}\) are the fluid density, specific heat, thermal conductivity, temperature, and velocity vector, respectively. For the solid domains of the cold plate and the silicone pad, the energy equation simplifies to
$$\frac{\partial}{\partial t}(\rho_s c_s T_s) = \nabla\cdot(k_s \nabla T_s).$$
I treated the straight-channel flow as laminar. For the pin-fin channels, the flow becomes turbulent even at moderate coolant flow rates because of the periodic interruption caused by the fins. I therefore used the standard \(k\)-\(\varepsilon\) turbulence model with enhanced wall treatment. The turbulent kinetic energy \(k\) and dissipation rate \(\varepsilon\) are obtained from
$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho u_j k)}{\partial x_j} = \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right] + P_k – \rho \varepsilon,$$
$$\frac{\partial(\rho \varepsilon)}{\partial t} + \frac{\partial(\rho u_j \varepsilon)}{\partial x_j} = \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_\varepsilon}\right)\frac{\partial \varepsilon}{\partial x_j}\right] + C_{\varepsilon 1}\frac{\varepsilon}{k}P_k – C_{\varepsilon 2}\rho \frac{\varepsilon^2}{k},$$
with standard constants \(C_{\varepsilon 1}=1.44\), \(C_{\varepsilon 2}=1.92\), \(\sigma_k=1.0\), and \(\sigma_\varepsilon=1.2\). The coolant was modeled as incompressible with a no-slip wall condition. All battery cells have the same volumetric heat-generation source that is activated through a user-defined function. The inlet boundary is a mass-flow inlet, and the outlet is a pressure outlet. Mass flow rates range from 0.01 to 0.03 kg/s. The ambient temperature and the coolant inlet temperature are both 293 K. All external surfaces are exposed to natural convection with a heat-transfer coefficient of 10 W m⁻² K⁻¹. The contact resistance between battery, pad, and cold plate is ignored for the numerical optimization. This is a common assumption in cold-plate simulations, and it is acceptable for comparative analyses.
The governing equations were discretized with a second-order upwind scheme. Pressure-velocity coupling was handled by the SIMPLE algorithm. The convergence criteria set a residual of 1×10⁻⁶ for the continuity and momentum equations and 1×10⁻⁹ for the energy equation. A poly-hexcore grid was constructed for each configuration. I performed a mesh-independence study for the straight-channel model and the circular-pin-fin model. The results for maximum temperature and pressure drop remain essentially unchanged beyond about 12.5 million cells for the straight channel and 13.2 million cells for the circular pin-fin channel. The final grids adopted those cell counts to balance accuracy and computational cost.
The numerical model was validated by building a dedicated pin-fin cold plate and measuring the pressure drop over a range of flow rates. The test section was heated by a film heater at a known power. In the simulations I tested several turbulence models. The standard \(k\)-\(\varepsilon\) model yielded the closest agreement with the measured pressure drop, with an average error of 7.1%. The model also predicted the average heating-film temperature with a maximum error below 0.8 K. This level of agreement confirms that the numerical strategy can reliably reproduce the flow and heat-transfer characteristics of the pin-fin cold plate used in traction battery pack cooling.
4. Comparison of Different Pin-Fin Shapes
Based on the validated model, I simulated the full battery module with four different cold plates at a discharge rate of 1C. The thermal state is evaluated by the maximum temperature \(T_{\mathrm{max}}\), the temperature difference \(\Delta T\) at a central cross-section of the module, and the pressure drop \(\Delta P\) across the cold plate. These three quantities are the primary performance metrics:
$$T_{\mathrm{max}} = \max\{T_1,T_2,\ldots,T_n\},$$
$$\Delta T = T_{\mathrm{max,mid}} – T_{\mathrm{min,mid}},$$
$$\Delta P = P_{\mathrm{in}} – P_{\mathrm{out}}.$$
Fig. 3 shows the evolution of \(T_{\mathrm{max}}\) and \(\Delta T\) at the end of discharge as a function of mass flow rate. For all configurations, the maximum temperature decreases quickly when the flow rate increases from 0.01 to 0.02 kg/s, and then the decrement becomes smaller. This trend is expected because a higher velocity thins the boundary layer and improves the convective heat-transfer coefficient. However, as the coolant temperature gradually approaches the battery surface temperature, the benefit of further increasing the flow rate diminishes. The straight channel laminar flow yields the weakest decrease because it has no strong disturbance. In contrast, all pin-fin channels disrupt the thermal boundary layer and promote turbulent mixing.
Among the pin-fin shapes, the square pin-fin arrangement gives the lowest maximum temperature, because it produces the strongest flow blockage and turbulence. The circular and triangular pin-fins follow closely, while the straight channel is the worst. At 0.01 kg/s, the circular-pin-fin cold plate reduces \(T_{\mathrm{max}}\) by 1.5 K compared with the straight channel. At 0.03 kg/s, the reduction increases to 2.9 K. The temperature uniformity behaves differently at low flow rates. For a mass flow rate below 0.02 kg/s, the straight channel has a smaller \(\Delta T\) because the laminar velocity distribution is smooth and the pin-fins introduce local hot/cold streaks. Once the flow rate exceeds 0.02 kg/s, all pin-fin configurations outperform the straight channel in temperature uniformity by promoting a more uniform heat exchange. At 0.03 kg/s, the best pin-fin channel reduces \(\Delta T\) by 0.7 K compared with the straight channel. This improvement stems from the breakdown of the thermal boundary layer and enhanced lateral mixing.
The pressure drop across the cold plate is the main energy cost. Table 3 reports the pressure drops predicted for several configurations at 0.03 kg/s. The square pin-fin channel has the highest pressure drop because it blocks more flow and creates strong separation. The circular pin-fin channel has a considerably lower pressure drop than the square fin while maintaining nearly the same heat-transfer performance. The triangular pin-fin channel lies in between.
| Configuration | \(T_{\mathrm{max}}\) (K) | \(\Delta T\) (K) | \(\Delta P\) (kPa) |
|---|---|---|---|
| SC | 311.2 | 5.6 | 0.8 |
| S-Pin-fin | 308.0 | 4.8 | 5.0 |
| C-Pin-fin | 308.3 | 4.9 | 3.8 |
| T-Pin-fin | 308.6 | 5.0 | 4.0 |
To understand the heat-transfer enhancement mechanism, I analyzed the velocity fields, turbulent kinetic energy contours, and Q-criterion isosurfaces in each channel. The straight channel exhibits a fully developed velocity profile after a short entrance length. The heat-transfer coefficient is high only near the inlet. In pin-fin channels, the fluid accelerates between adjacent pins, and the periodic wakes create strong convective mixing. The circular pin-fin geometry produces coherent, axisymmetric vortices downstream of each cylinder. These vortices remain organized and are attached along the flow, leading to a continuous disruption of the thermal boundary layer. Consequently, the heat-transfer coefficient remains high along the entire channel. The square-fin geometry produces more violent and fragmented eddies, which further improve heat transfer but also cause excessive turbulent dissipation and a large pressure penalty. The triangular-fin geometry exhibits a moderate level of turbulence, with separation occurring mainly at the leading corners. Overall, the circular pin-fin channel gives the most favorable balance between heat-transfer enhancement and pressure loss for a traction battery pack. Therefore, I select C-Pin-fin as the base configuration for the multi-objective optimization described in the next section.
5. Multi-Objective Optimization of the Circular Pin-Fin Cold Plate
The C-Pin-fin cold plate still has considerable room for improvement. Its thermal performance depends on the mass flow rate, the height of the flow channel, and the pin-fin spacing. In this section, I develop a surrogate-based optimization framework to find the best combination of design variables for a traction battery pack. The design variables are the coolant mass flow rate \(A\), the channel height \(B\), and the pin-fin spacing \(C\). Their ranges are given in Table 4. The initial baseline design was set to \(A = 0.02\) kg/s, \(B = 1.5\) mm, and \(C = 15\) mm.
| Variable | Symbol | Initial | Minimum | Maximum |
|---|---|---|---|---|
| Mass flow rate | \(A\) (kg/s) | 0.020 | 0.010 | 0.030 |
| Channel height | \(B\) (mm) | 1.5 | 1.5 | 4.0 |
| Pin-fin spacing | \(C\) (mm) | 15 | 4 | 30 |
Two objectives are chosen for optimization: the temperature difference \(\Delta T\) and the pressure drop \(\Delta P\). The maximum temperature \(T_{\mathrm{max}}\) was initially considered a third objective, but a correlation analysis showed that \(T_{\mathrm{max}}\) and \(\Delta T\) are very strongly correlated (Pearson coefficient 0.99). Therefore, minimizing \(\Delta T\) is enough to control \(T_{\mathrm{max}}\), and dropping the redundant objective makes the optimization more efficient.
I used optimal Latin hypercube sampling to select 31 sample points in the three-dimensional design space. At each sample point, a full three-dimensional CFD simulation was carried out at 1C discharge with the cooling conditions described earlier. The sampled data are summarized in Table 5. Then, I constructed a Kriging surrogate model to map \((A,B,C)\) to the outputs \(\Delta T\) and \(\Delta P\). The quality of the surrogate was evaluated by cross-validation. The predicted coefficient of determination \(R^2\) is 0.966 for \(\Delta T\) and 0.952 for \(\Delta P\), which indicates sufficient accuracy for design-space exploration.
| No. | \(A\) (kg/s) | \(B\) (mm) | \(C\) (mm) | \(\Delta T\) (K) | \(\Delta P\) (kPa) |
|---|---|---|---|---|---|
| 1 | 0.0260 | 2.16 | 23.6 | 4.4 | 2.2 |
| 2 | 0.0167 | 2.34 | 17.3 | 6.2 | 1.0 |
| 3 | 0.0127 | 1.92 | 22.2 | 7.6 | 0.9 |
| 4 | 0.0293 | 1.76 | 13.8 | 4.1 | 4.2 |
| 5 | 0.0153 | 1.50 | 15.2 | 6.6 | 2.1 |
| 6 | 0.0273 | 2.92 | 11.7 | 4.3 | 1.9 |
| 7 | 0.0140 | 2.66 | 5.4 | 7.0 | 0.8 |
| 8 | 0.0207 | 1.66 | 20.8 | 5.2 | 2.4 |
| 9 | 0.0160 | 3.16 | 14.5 | 6.2 | 0.7 |
| 10 | 0.0193 | 2.00 | 4.0 | 5.6 | 2.3 |
I also performed a global sensitivity analysis using the variance-based measure
$$S A_i = \frac{f_{\max}(x_i) – f_{\min}(x_i)}{f_{\max}(x) – f_{\min}(x)} \times 100\%.$$
Table 6 reports the sensitivity of each objective to the three design variables. The mass flow rate dominates the maximum cell temperature and the temperature difference with sensitivities above 98%. This result suggests that, within the studied range, adjusting the coolant flow rate is the most direct and effective way to control the thermal state of a traction battery pack. For the pressure drop, the channel height is the dominant factor, contributing 44.2% of the total influence. This is understandable because a small change in channel height strongly alters the cross-sectional area and thus the velocity and pressure loss. The pin-fin spacing has only a minor impact on the pressure drop in this range.
| Objective | Mass flow rate | Channel height | Pin-fin spacing |
|---|---|---|---|
| \(T_{\max}\) | 0.996 | 0.003 | 0.001 |
| \(\Delta T\) | 0.988 | 0.010 | 0.002 |
| \(\Delta P\) | 0.530 | 0.442 | 0.028 |
The NSGA-II algorithm was then applied to the Kriging surrogate to minimize \(\Delta T\) and \(\Delta P\). The population size was 120 and the number of generations was 50, yielding 6000 candidate solutions. The Pareto front is shown in Fig. 5 of the original manuscript. From the Pareto front, I selected several representative solutions for direct CFD verification. Table 7 lists three promising designs together with the baseline. The surrogate-predicted values agree with the full CFD simulation results to within 3.4% for \(\Delta T\) and within 1.7% for \(\Delta P\), confirming the reliability of the optimization workflow. Among these points, model 2 offers the best trade-off: it reduces \(\Delta T\) by 23.1% and \(\Delta P\) by 43.2% relative to the baseline design. The corresponding design parameters are \(A = 0.0285\) kg/s, \(B = 3.14\) mm, and \(C = 19\) mm. This design is named the optimized model (Opt) and is used in the subsequent analysis.
| Design | \(A\) (kg/s) | \(B\) (mm) | \(C\) (mm) | \(\Delta T\) (K) | CFD \(\Delta T\) (K) | \(\Delta P\) (kPa) | CFD \(\Delta P\) (kPa) |
|---|---|---|---|---|---|---|---|
| Baseline | 0.0200 | 1.50 | 15 | 5.40 | 5.40 | 3.10 | 3.10 |
| Model 1 | 0.0253 | 3.66 | 22 | 4.47 | 4.51 | 1.25 | 1.27 |
| Model 2 | 0.0285 | 3.14 | 19 | 4.02 | 4.15 | 1.77 | 1.76 |
| Model 3 | 0.0294 | 2.52 | 18 | 3.93 | 4.07 | 2.34 | 2.30 |
To evaluate the thermal performance in a more fundamental manner, I calculated the total thermal resistance from the battery surface to the coolant. The total resistance can be decomposed into three contributions: the convective resistance \(R_{\mathrm{conv}}\), the heating (or heat-capacity) resistance \(R_{\mathrm{heat}}\), and the conductive resistance \(R_{\mathrm{cond}}\). Mathematically,
$$R_{\mathrm{conv}} = \frac{T_{\mathrm{ave}} – T_{\mathrm{liq}}}{q},$$
$$R_{\mathrm{heat}} = \frac{T_{\mathrm{out}} – T_{\mathrm{in}}}{q},$$
$$R_{\mathrm{cond}} = \frac{L_b}{k_s A_{\mathrm{cont}}},$$
and \(R_{\mathrm{total}} = R_{\mathrm{conv}} + R_{\mathrm{heat}} + R_{\mathrm{cond}}\). In Fig. 4 of the original paper, I showed that the optimized C-Pin-fin model reduces the total thermal resistance by 14% compared with the straight channel at 0.02 kg/s and by 16% at 0.03 kg/s. The improvement mainly originates from \(R_{\mathrm{conv}}\), which is much smaller for the pin-fin structure because of the increased heat-transfer area and the disruption of the boundary layer. The conductive and heat-capacity resistances do not change significantly among the different cold plates.
Another useful indicator is the effective heat-transfer enhancement factor \(p_f\), which compares the heat-transfer enhancement with the pressure-drop penalty. It is defined as:
$$p_f = \frac{\mathrm{Nu}/\mathrm{Nu}_0}{\left(\Delta P/\Delta P_0\right)^{1/3}},$$
where \(\mathrm{Nu} = h_{\mathrm{ave}} D_h / k_{\mathrm{liq}}\) and the subscript 0 refers to the straight-channel reference case. At a mass flow rate of 0.02 kg/s, the optimized circular pin-fin model has a Nusselt number approximately six times higher than the straight channel, while the enhancement factor \(p_f\) is about seven times larger. Compared with the non-optimized circular pin-fin design, the optimized one improves Nu by 33.9% and \(p_f\) by 97.5%. The enhancement factor decreases when the flow rate increases from 0.02 to 0.03 kg/s, because the pressure drop increases faster than the heat-transfer coefficient. Nevertheless, the optimized design still outperforms all the other configurations.
A third indicator is the cooling efficiency factor \(\eta\), which quantifies how much heat is removed for each unit of pumping power. The factor can be defined as
$$\eta = \frac{Q_{\mathrm{liq}}/t}{\Delta P \, q_v},$$
where \(Q_{\mathrm{liq}}\) is the heat absorbed by the coolant in time \(t\), \(\Delta P\) is the pressure drop, and \(q_v\) is the volumetric flow rate. For the optimized model, the cooling efficiency factor reaches 15,452 at 0.02 kg/s, while the baseline C-Pin-fin model reaches only 4,875 and the straight channel reaches 7,250. This huge margin demonstrates that the optimized cold plate delivers a superior thermal-hydraulic performance for a traction battery pack with acceptable pumping cost.
6. Experimental Study
The optimization results suggest that a larger channel height can drastically reduce the pressure drop. However, for practical integration into a compact vehicle battery system, the cold-plate thickness cannot become arbitrarily large. Therefore, I evaluated the channel height effect directly. With the other optimized parameters fixed, I varied the channel height from 1 to 4 mm at the same mass flow rate of 0.0285 kg/s. The maximum temperature rises by only 0.3 K when the height increases from 1 to 4 mm. The pressure drop, however, decreases sharply by 7.7 kPa when the height changes from 1 mm to 1.5 mm, a 59.2% drop. Further increases in channel height continue to reduce the pressure drop, but with diminishing returns. Accordingly, a channel height of 1.5 mm is chosen as the best compromise between thermal performance and structural compactness. This chosen design is named C-Pin-fin (BCS). The pressure-drop behavior is mainly attributed to the fact that in a thin channel the boundary layers occupy a larger proportion of the cross-section, producing high friction. In a taller channel, the core flow accelerates less and viscous dissipation is reduced.
Before conducting experiments, I verified the numerical model of C-Pin-fin (BCS) using a smaller test plate with the same channel geometry. Four heating films were attached to the outer surface of the cold plate. The total heating power was 280 W. The coolant inlet temperature and ambient conditions were set to 293 K. Mass flow rates ranged from 0.005 to 0.03 kg/s. The measured pressure drop and average surface temperature were compared with the numerical results. The maximum relative error of the pressure drop is 8%, and the maximum average-temperature error is 1.2 K. These values confirm that the numerical model used for the cold-plate design has sufficient accuracy.
6.1 Experimental system
The C-Pin-fin (BCS) cold plate was manufactured by CNC milling from 6061 aluminum. The machining process is suitable for this quasi-two-dimensional geometry and gives good dimensional accuracy. The plate dimensions are 732 mm × 191 mm × 12 mm. The inlet and outlet diameters are 6 mm. The channel height is 1.5 mm, the pin-fin spacing is 19 mm, and the pin diameter is 3 mm. Four polyimide heating films, each with dimensions 150 mm × 116 mm, were attached to the plate to mimic the heat generation of battery cells. The total area covered by the films matches the footprint of a battery module unit.
The experimental loop consists of a gear pump, a filter, a turbine flow meter, a constant-temperature bath, two pressure transmitters, two PT100 temperature sensors, a DC power supply, an infrared thermal camera, and a data-acquisition system. Coolant (50% ethylene-glycol solution) is drawn from a reservoir, pumped through the filter and flow meter, and then enters the cold plate. At the inlet and outlet, pressure transmitters and temperature sensors record the pressure and temperature. After leaving the cold plate, the coolant returns to the reservoir, forming a closed loop. During heat-transfer experiments, the heating films are energized by a multi-channel DC source. The infrared camera captures the surface temperature distribution of the film heaters. All signals are collected by a data-acquisition system and processed on a computer.
The main instruments and their accuracy are listed in Table 8. The flow meter has an accuracy of ±0.5% of reading over a range of 0–10 L/min. The inlet and outlet pressure transmitters have ranges of 0–500 kPa and 0–130 kPa, respectively, with an accuracy of ±0.5% of full scale. Temperature sensors are accurate to ±0.5 °C. The infrared camera has an accuracy of ±2%.
| Instrument | Accuracy | Range |
|---|---|---|
| DC power supply | – | 0–200 W |
| Infrared thermal camera | ±2% | −20 to 2000 °C |
| Turbine flow meter | ±0.5% | 0–10 L min⁻¹ |
| Constant-temperature bath | ±0.75% | −20 to 100 °C |
| Temperature sensor | ±1% | −100 to 280 °C |
| Inlet pressure transmitter | ±0.5% | 0–500 kPa |
| Outlet pressure transmitter | ±0.5% | 0–130 kPa |
During the pressure-drop measurement, the heating films were switched off. The coolant inlet temperature was maintained at 293 K. The gear-pump speed was adjusted incrementally to produce flow rates from 0.005 kg/s to 0.03 kg/s. At each point, the system was allowed to stabilize until the pressure readings fluctuated less than a small tolerance over one minute. The readings were recorded and averaged. During the heat-transfer experiments, the heating power was set sequentially to 200, 280, 360, and 440 W. For each power level, the flow rate was again varied over the same range. The thermal steady state was defined as a change of less than 0.5 K over five minutes. At steady state, the infrared image was captured, and all sensor signals were recorded.
The heating power generated by the film is calculated as \(P_h = 8 U I_h\), where the factor 8 accounts for the four films each having two power leads or an equivalent connection set. In the current setup, the power supply delivers a voltage \(U\) and current \(I_h\) to four films arranged in the circuit. The effective heat flux is \(q = C_p m \Delta T / A\), where \(C_p\), \(m\), \(\Delta T\), and \(A\) are the coolant specific heat, mass flow rate, temperature rise, and total heating-film area, respectively. The pumping power required to drive the coolant is \(P_p = \Delta P \, q_v\), in which \(q_v\) is the volumetric flow rate.
6.2 Results and discussion
Fig. 6 in the original thesis plots the measured pressure drop against the volume flow rate. The pressure drop increases monotonically with flow rate, but the slope changes in different regimes. At low flow rates, the flow around the pin-fins is smooth, with limited separation and wake formation. The pressure drop rises slowly. As the flow rate increases, the inertial forces become significant, and the wakes behind the pin-fins expand. The pressure-drop curve becomes steeper. At high flow rates, unsteady vortex shedding and stronger turbulence cause a rapid rise in pressure drop. Throughout the tested range, the pressure drop of the C-Pin-fin (BCS) cold plate remains moderate, indicating that the circular pins do not require excessive pumping power.
Infrared thermography provides a direct visualization of the surface temperature field of the heating films. The images clearly show that the temperature increases along the flow direction. The region near the cold-plate inlet has the lowest temperature, while the downstream region becomes hotter by a few kelvin. This behavior results from the coolant absorbing heat as it travels along the channel. At the entrance, the coolant temperature is low, and the heat-transfer coefficient is high owing to the developing boundary layer. Further downstream, the coolant is warmer and the heat-transfer coefficient is reduced. A comparison of the two heater positions confirms that the upstream heater Heater2 always remains cooler than the downstream Heater1. This entry-length effect has important consequences for the design of large traction battery packs with long channels.
The measured heater temperature is plotted in Fig. 5 of the original chapter as a function of heat flux for three different mass flow rates. At a fixed flow rate, the heater temperature increases almost linearly as the heat flux increases. This is expected because the amount of heat generated increases while the cooling condition remains unchanged. At a fixed heat flux, raising the mass flow rate lowers the heater temperature. A higher flow rate enhances the convective heat-transfer coefficient and also gives a smaller coolant temperature rise for a given heat load. The temperature difference between the different flow rates is more pronounced at high heat flux.
To quantify the overall resistance of the cold plate, I calculated the thermal resistance between the heating film and the coolant. Figure 6 shows the dependence of the thermal resistance on flow rate and heating power. As the coolant flow rate increases, the thermal resistance decreases because the heat-transfer coefficient improves. At the same time, the pumping power increases rapidly due to the combined effect of higher flow rate and higher pressure drop. The experiments also show that when the heating power is increased at a constant flow rate, the thermal resistance decreases gradually at first and then tends to stabilize. This is because the viscosity of the coolant decreases with temperature, causing an additional enhancement in heat transfer. However, this property-induced effect saturates at higher temperatures. These results suggest that one should not always operate a traction battery pack at the maximum pump speed, because the marginal reduction in thermal resistance becomes small while the pumping power grows sharply.
| Parameter | Maximum relative uncertainty |
|---|---|
| Inlet pressure, \(P_{\mathrm{in}}\) | 2.4% |
| Outlet pressure, \(P_{\mathrm{out}}\) | 0.6% |
| Volume flow rate, \(q_v\) | 2.5% |
| Mass flow rate, \(m\) | 2.5% |
| Coolant temperature, \(T\) | 1.6% |
| Pressure drop, \(\Delta P\) | 2.5% |
| Heating-film heat flux, \(q\) | 3.1% |
| Thermal resistance, \(R\) | 3.6% |
7. Conclusions
In this paper, I have systematically investigated the single-phase flow and heat-transfer characteristics of pin-fin channel cold plates for thermal management of a traction battery pack. The following conclusions can be drawn from the numerical, optimization, and experimental studies.
- A battery heat-generation model based on the Bernardi equation was validated against experiments with less than 5% error. The model permits reliable thermal boundary conditions for cold-plate simulations. Among four different cold-plate channel configurations, the circular pin-fin geometry provides the best balance between cooling performance and pressure drop. At a mass flow rate of 0.03 kg/s, it lowers the maximum battery temperature by 2.9 K and the module temperature difference by 0.7 K relative to a straight rectangular channel. The underlying heat-transfer mechanism is the formation of coherent spanwise vortices downstream of the circular pins, which destroy the thermal boundary layer and enhance mixing without excessive dissipative losses.
- A Kriging surrogate model combined with NSGA-II proved to be an effective approach for optimizing the cold plate of a traction battery pack. The mass flow rate is the dominant parameter affecting both \(T_{\mathrm{max}}\) and \(\Delta T\), while the channel height controls the pressure drop. The pin-fin spacing only slightly affects the pressure drop. Correlation analysis observed a very strong positive correlation between \(T_{\mathrm{max}}\) and \(\Delta T\), so the optimization could safely remove the redundant objective. Three representative Pareto-optimal designs were verified by CFD, with surrogate errors below 3.4% for \(\Delta T\). The selected optimal design, with a mass flow rate of 0.0285 kg/s, a channel height of 3.14 mm, and a pin spacing of 19 mm, reduces \(\Delta T\) by 23.1% and \(\Delta P\) by 43.2% relative to the baseline.
- System-level performance evaluation shows that the optimized circular pin-fin cold plate lowers the total thermal resistance by 14% to 16% compared with the straight channel, mainly through the reduction of convective resistance. The effective heat-transfer enhancement factor \(p_f\) is around seven times larger than that of the straight channel at 0.02 kg/s. The cooling efficiency factor of the optimized model reaches 15,452, several times higher than either the original pin-fin design or the straight channel, indicating that the optimized design is remarkably energy-efficient.
- The experimental study confirmed the accuracy of the numerical model and provided insight into the practical operation of a traction battery pack. A channel height of 1.5 mm is considered the best compromise because it reduces the pressure drop by 59.2% compared with the 1 mm channel while the maximum temperature is only 0.3 K higher. The manufactured cold plate was tested in a closed coolant loop. The pressure drop rises with flow rate in a nonlinear fashion. The heating-film temperature increases with heat flux and decreases with flow rate. The infrared images show a clear axial temperature gradient, with the downstream region warmer than the inlet region. Finally, the thermal resistance of the cold plate decreases as either the flow rate or the heating power increases, while the pumping power increases almost exponentially with the flow rate. These results enable a rational trade-off between cooling performance and energy consumption in the design of battery liquid cooling systems.
The proposed pin-fin cold plate offers a practical and efficient solution for the thermal management of a traction battery pack. By considering the mutual interference among flow rate, geometric dimensions, and thermal uniformity, the study demonstrates that surrogate-assisted multi-objective optimization is a powerful tool for designing compact and energy-efficient thermal systems. Future work will extend the methodology to fast-charging scenarios, investigate the effect of non-uniform heat generation on the optimal layout, and further improve the cold plate design through topology optimization.