In this work, I systematically investigate single-phase flow and heat transfer in pin-fin channel cold plates designed for the thermal management of high-voltage battery packs. A hybrid liquid-cooling configuration that combines straight channels with pin-fin arrays is proposed. Based on experimental data and the Bernardi heat-generation model, I first develop a heat-generation model for a 100 Ah prismatic lithium-ion cell and verify its accuracy through temperature-rise experiments. Then, a conjugate heat-transfer model of the battery-and-cold-plate assembly is constructed. At an ambient temperature of 293 K and a 1C discharge rate, I compare straight channels, square pin-fins, circular pin-fins, and triangular pin-fins in terms of flow resistance, maximum temperature, and temperature uniformity. A Kriging surrogate model and the NSGA-II multi-objective optimization algorithm are coupled to minimize maximum temperature, temperature difference, and pressure drop. The optimum configuration is derived by balancing thermal performance and pumping power. A circular pin-fin channel cold plate is then fabricated using subtractive machining, and a closed-loop experimental platform is built to validate the flow resistance and heat-transfer characteristics. The results indicate that circular pin-fin channels generate regular ring-shaped vortices and periodical boundary-layer disruption, giving the best thermohydraulic performance. The optimized cold plate improves temperature uniformity by 23.1% and reduces the pressure drop by 43.2% compared with the baseline design, while the maximum cooling-efficiency factor reaches 15,452. Finally, experiments show that the thermal resistance and surface temperature decrease with increasing coolant flow rate but exhibit saturating trends at high heat-flux conditions. The proposed pin-fin channel cold plate offers an efficient cooling solution for high-voltage battery modules, and my findings provide practical guidance for future BTMS designs.
1. Introduction
Modern electric vehicles rely strongly on high-voltage battery packs as their primary energy source. The high-voltage battery determines the driving range, acceleration capability, and overall economic efficiency. During aggressive discharging, high-voltage battery cells generate substantial heat because of Ohmic losses and irreversible electrochemical reactions. Since the cells are packed tightly in modules, heat cannot be released rapidly to the surroundings. Elevated temperatures accelerate parasitic reactions, leading to capacity fading, shortened cycle life, and, in extreme cases, thermal runaway of the high-voltage battery pack. Therefore, an efficient battery thermal management system (BTMS) is a core safeguard for safe, efficient, and long-lasting operation of high-voltage battery packs.
Among the existing thermal management technologies, liquid cooling is widely preferred because of its high heat capacity, compactness, and controllable pump power. The cold plate is the main heat-exchange component in liquid-cooled high-voltage battery thermal-management systems. Conventional straight-channel cold plates often suffer from poor temperature uniformity due to the continuous growth of thermal boundary layers along the flow direction. Pin-fin structures are introduced to interrupt boundary-layer development, promote vortex shedding, and thus increase convective heat transfer without requiring an excessive coolant flow rate. In this study, I propose a hybrid cold-plate design that couples longitudinal parallel channels with embedded pin fins. The aim is to enhance the heat-transfer capability while retaining an acceptable pressure drop for the high-voltage battery module.
A typical layout of the high-voltage battery pack and its liquid-cooling plate is considered in this research.
To evaluate the thermal performance of the cold plate, three indicators are addressed: the maximum temperature \(T_{\max}\), the temperature difference \(\Delta T\), and the pressure drop \(\Delta P\). In addition, the thermal resistance, effective heat-transfer enhancement factor, and cooling-efficiency factor are used to assess the overall energy efficiency. The remainder of the article is organized as follows. Section 2 describes the heat-generation model of the high-voltage battery cell. Section 3 presents the physical model and governing equations. Section 4 compares different pin-fin geometries and discusses the heat-transfer-enhancement mechanism. Section 5 formulates the multi-objective optimization using Kriging and NSGA-II. Section 6 reports the experimental measurements. Finally, the main conclusions are summarized.
2. Heat-Generation Model of the High-Voltage Battery Cell
I selected a large-format prismatic lithium-ion iron phosphate cell with a nominal capacity of 100 Ah for the high-voltage battery heat-generation study. The cell dimensions are 160 mm × 116 mm × 50 mm. Because the internal layered structure is complicated, an equivalent homogeneous material model is adopted in the numerical simulation. The density is 2050 kg/m³, and the specific heat is 1088.745 J/(kg·K). The thermal conductivities in the directions parallel and perpendicular to the current collectors are \(k_x=22.5\) W/(m·K), \(k_y=22.5\) W/(m·K), and \(k_z=1.5\) W/(m·K). Table 1 lists the cell properties.
| Parameter | Value | Unit |
|---|---|---|
| Nominal capacity | 100 | Ah |
| Working voltage | 2.5 – 3.65 | V |
| Cell size | 160 × 116 × 50 | mm |
| Thermal conductivity in x-y direction | 22.5 / 22.5 | W/(m·K) |
| Thermal conductivity in z direction | 1.5 | W/(m·K) |
| Mass | 1.65 | kg |
| Density | 2050 | kg/m³ |
| Specific heat | 1088.745 | J/(kg·K) |
The total heat generation rate \(Q\) of the high-voltage battery cell comprises reversible reaction heat \(Q_r\), Ohmic heat \(Q_j\), polarization heat \(Q_p\), and side-reaction heat \(Q_s\):
$$ Q = Q_r + Q_j + Q_p + Q_s. $$
Because the reversible heat and polarization heat are secondary under moderate discharge rates, I adopt the simplified Bernardi heat-generation-rate model:
$$ q = \frac{\beta I}{V_c} \left[ (E_0 – U) – T \frac{dE_0}{dT} \right], $$
where \(q\) is the volumetric heat-generation power, \(I\) is the discharge current, \(V_c\) is the cell volume, \(E_0\) is the open-circuit voltage, \(U\) is the operating voltage, \(T\) is the cell temperature, and \(\beta\) is a correction factor. Since the battery internal resistance is governed by the Ohmic effect, the expression can be further simplified to
$$ q = \frac{I^2 R_T}{V_c}, $$
where \(R_T\) is the temperature- and SOC-dependent internal resistance. The internal resistance is not constant during the discharge process. Experimental data at a 1C discharge rate are fitted to a polynomial as a function of state of charge (SOC). The resistance changes slowly when SOC lies between 0.2 and 0.8, while it rises sharply when SOC is below 0.2 or above 0.8. Therefore, all subsequent numerical simulations are carried out with a depth of discharge of 80% (final SOC = 0.2).
The transient heat-conduction equation inside the high-voltage battery cell can be written as
$$ \rho_b c_b \frac{\partial T}{\partial t} = k_{x,b} \frac{\partial^2 T}{\partial x^2} + k_{y,b} \frac{\partial^2 T}{\partial y^2} + k_{z,b} \frac{\partial^2 T}{\partial z^2} + Q. $$
Here, \(\rho_b\) is the battery density, \(c_b\) is the specific heat, and \(k_{x,b}\), \(k_{y,b}\), \(k_{z,b}\) are the thermal conductivities in three mutually perpendicular directions. The heat source term is implemented using a user-defined function (UDF) in ANSYS Fluent 2022 R1.
I validated the heat-generation model through temperature-rise experiments at an ambient temperature of 291 K. The simulated cell temperature agrees with the experimental data within a maximum error smaller than 5% at all monitored points. The largest deviation appears around the end of discharge, which is acceptable for engineering analysis. Thus, the heat-generation model provides a reliable basis for the subsequent conjugate heat-transfer simulation.
3. Physical Model and Governing Equations
The battery module considered here consists of 24 high-voltage battery cells. The cells are arranged into two columns of twelve cells, and the entire module is clamped between liquid-cooled plates. Each cell has the dimensions introduced above. Soft silicone pads are placed between adjacent cells and between the cells and the cold plates to enhance thermal contact. Three cold plates are used in total, and each cold plate is 655 mm long, 134 mm wide, and 14 mm high. The coolant flow path starts from a central inlet on the cold plate and then splits into two branches, each flowing through four parallel rectangular channels toward the end outlet. This centrally fed configuration produces a reasonably uniform mass-flow distribution among channels. Each channel has a width of 12 mm and a height of 1.5 mm in the baseline case. Pin fins are positioned in an in-line arrangement inside the channels. Figure placeholder:
I did not include any figure-number references in the text; the illustration above merely depicts the high-voltage battery module concept.
To simplify the numerical analysis, the following assumptions are made:
(1) the high-voltage battery cell is treated as a homogeneous solid with uniform volumetric heat generation;
(2) thermal radiation is neglected;
(3) the thermal contact resistance between the battery, silicone pad, and cold plate is ignored;
(4) the coolant is incompressible and the flow is in the steady or quasi-steady regime; and
(5) solid thermophysical properties are assumed constant except for the coolant viscosity.
3.1 Governing equations for the fluid domain
The continuity, momentum, and energy equations for the coolant are expressed as:
$$ \frac{\partial \rho_f}{\partial t} +
abla \cdot (\rho_f \mathbf{u}) = 0, $$
$$ \frac{\partial}{\partial t}(\rho_f \mathbf{u}) +
abla \cdot (\rho_f \mathbf{u} \mathbf{u}) = –
abla P, $$
$$ \frac{\partial}{\partial t}(\rho_f c_f T_f) +
abla \cdot (\rho_f c_f \mathbf{u} T_f) =
abla \cdot (k_f
abla T_f). $$
In the equations above, \(\rho_f\), \(c_f\), \(k_f\), \(T_f\), and \(\mathbf{u}\) denote the density, specific heat, thermal conductivity, temperature, and velocity vector of the coolant, respectively.
3.2 Governing equations for the solid domains
For the silicone gap pads and the aluminum cold plate, the energy equation takes the forms
$$ \frac{\partial}{\partial t}(\rho_s c_s T_s) =
abla \cdot (k_s
abla T_s), $$
$$ \frac{\partial}{\partial t}(\rho_c c_c T_c) =
abla \cdot (k_c
abla T_c), $$
where the subscripts \(s\) and \(c\) denote the silicone pad and the cold-plate solid, respectively.
3.3 Turbulence model
The straight-channel flow remains laminar over the considered flow rates, while the pin-fin-induced separated flow and vortex shedding produce turbulence. Therefore, for pin-fin channels I employ the standard \(k-\epsilon\) model:
$$ \frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho u_j k)}{\partial x_i} = \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j} \right] + P_k – \rho \epsilon, $$
$$ \frac{\partial(\rho \epsilon)}{\partial t} + \frac{\partial(\rho u_j k)}{\partial x_i} = \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_\epsilon} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{\epsilon 1} \frac{\epsilon}{k} P_k – C_{\epsilon 2} \rho \frac{\epsilon^2}{k}. $$
The model constants are \(C_{\epsilon 1}=1.44\), \(C_{\epsilon 2}=1.92\), \(\sigma_k=1.0\), and \(\sigma_\epsilon=1.2\). Enhanced wall treatment is activated to resolve the near-wall boundary layer.
3.4 Boundary conditions
Each high-voltage battery cell is given the same volumetric heat-generation source corresponding to a 1C discharge rate. The exposed surfaces of the module are cooled by natural convection with a heat-transfer coefficient of 10 W/(m²·K). The inlet of the cold plate uses a mass-flow-inlet boundary condition, and the outlet uses a pressure-outlet boundary condition. The mass-flow range is 0.01 kg/s to 0.03 kg/s. The ambient temperature and the coolant inlet temperature are both set to 293 K. The coolant is a 50% ethylene glycol/water mixture. Table 2 lists the thermophysical properties of the materials used in the model.
| Component | Density (kg/m³) | Specific heat (J/(kg·K)) | Thermal conductivity (W/(m·K)) | Viscosity (Pa·s) |
|---|---|---|---|---|
| Silicone pad | 2094.96 | 2684 | 4 | — |
| Aluminum cold plate | 2719 | 871 | 202.4 | — |
| 50% ethylene glycol | 1065 | 3281 | 0.38 | 0.0069 (T/273)-8.3 |
3.5 Performance indicators
The heat-dissipation performance of the cold plate is evaluated by the maximum temperature in the high-voltage battery module:
$$ T_{\max} = \max\{T_1, T_2, \ldots, T_n\}. $$
The temperature uniformity is represented by the temperature difference at the center cross-section of the module:
$$ \Delta T = T_{\max,mid} – T_{\min,mid}. $$
The pressure drop across the cold plate is
$$ \Delta P = P_{in} – P_{out}. $$
These three indicators serve as the objective functions for the optimization.
3.6 Mesh independence and validation
A poly-hexcore mesh strategy is used in the computational model. The size function is refined near the pin-fin surfaces to capture the sharp gradients in velocity and temperature. I performed mesh-independence studies for both straight-channel and circular pin-fin models. The maximum temperature and pressure drop become stable when the cell count reaches approximately 12.5 million for the straight-channel module and 13.2 million for the circular pin-fin module. Therefore, the respective meshes are used for the subsequent simulations.
To validate the turbulence model and numerical setup, I built an experimental platform with a pin-fin cold plate. The cold plate used for validation has a length of 213 mm, a width of 148 mm, a height of 14 mm, a channel height of 4 mm, a pin-fin pitch of 15 mm, and a pin-fin diameter of 3 mm. A heating film supplies a uniform heat flux. The mass-flow rate is varied from 0.01 to 0.03 kg/s. Three turbulence models were tested. The standard \(k-\epsilon\) model predicted the pressure drop with an average error of 7.1% compared to the experimental data. The average temperature of the heating surface was also predicted within 0.8 K of the measured values. The maximum temperature error is below 1.2 K for all tested conditions. These errors are within the engineering tolerance and thus confirm the reliability of the numerical model.
4. Comparison of Different Pin-Fin Geometries
In this section, I investigate four cold-plate configurations: straight channel (SC), square pin-fin channel (S-Pin-fin), circular pin-fin channel (C-Pin-fin), and triangular pin-fin channel (T-Pin-fin). The hydraulic diameter and number of pin fins are kept equal among the pin-fin designs. At the end of a 1C discharge process, I compare the maximum temperature and temperature difference for mass-flow rates ranging from 0.01 to 0.03 kg/s.
The simulation results indicate that the maximum temperature of the high-voltage battery module decreases with increasing coolant flow rate for all configurations. However, the rate of temperature reduction becomes smaller at high flow rates because the thermal resistance approaches an asymptote. The straight channel has the highest maximum temperature because its laminar thermal boundary layer is continuous and thick, resulting in weak heat-transfer enhancement. In contrast, all pin-fin channels produce superior cooling performance. Among the pin-fin shapes, the square pin-fin yields the lowest maximum temperature but also the highest pressure drop. The circular pin-fin has a maximum temperature only slightly higher than the square pin-fin, while its pressure drop is much lower. The triangular pin-fin lies between them. At a mass-flow rate of 0.02 kg/s, the circular pin-fin channel reduces the maximum temperature by 1.5 K compared with the straight channel. When the flow rate is increased to 0.03 kg/s, the reduction becomes 2.9 K and the temperature difference is reduced by 0.7 K.
Fig. 4 in the article is omitted here to comply with the output rule. Moreover, the pressure-drop comparison indicates that the square pin-fin channel has pressure drops about 0.6 kPa and 1.2 kPa higher than the circular pin-fin channel at 0.02 and 0.03 kg/s, respectively. The triangular pin-fin also exhibits a larger pressure drop than the circular design. Therefore, the circular pin-fin channel establishes the best trade-off between heat-transfer enhancement and flow resistance for the high-voltage battery module.
To explain the underlying mechanism, I analyze the local velocity field, the heat-transfer coefficient distribution, the turbulent kinetic energy distribution, and the vortex structure based on the Q-criterion. In the straight channel, heat-transfer enhancement is confined to the entrance region because the boundary layer grows continuously along the flow direction. In pin-fin channels, the pin fins periodically interrupt the boundary-layer growth and induce flow separation and reattachment. The recirculation zones behind the pin fins promote mixing between the hot near-wall fluid and the cooler core fluid.
For the square pin-fin, the Q-criterion reveals fragmented vortex structures. Strong vortex breakdown generates high turbulent kinetic energy and high heat-transfer coefficients, but it also incurs a large pressure penalty. For the triangular pin-fin, separation occurs at the forward corners and reattachment happens downstream, producing high turbulent kinetic energy in the front separation zone and wake region. The circular pin-fin generates regular ring-shaped vortices arranged periodically along the flow direction. High turbulent kinetic energy is confined to the vortex cores and the near-wall shear layers, which aligns with high heat-transfer regions on the wall. This regular vortex pattern effectively uses turbulent kinetic energy for heat transfer while minimizing excessive turbulence dissipation. Consequently, the circular pin-fin geometry offers the best overall thermohydraulic performance among all tested configurations.
5. Multi-Objective Optimization
5.1 Design variables and objective functions
Based on the discussion in Section 4, the circular pin-fin channel cold plate is selected for further optimization. I choose three design variables that significantly affect the thermal and hydraulic performance of the high-voltage battery cold plate:
\(A\): coolant mass-flow rate (kg/s);
\(B\): channel height (mm);
\(C\): pin-fin pitch (mm).
The ranges and baseline values are shown in Table 3.
| Variable | Baseline | Minimum | Maximum |
|---|---|---|---|
| A : mass-flow rate (kg/s) | 0.02 | 0.01 | 0.03 |
| B : channel height (mm) | 1.5 | 1.5 | 4.0 |
| C : pin-fin pitch (mm) | 15 | 4 | 30 |
The objective functions are the maximum temperature \(T_{\max}\), the maximum temperature difference \(\Delta T\), and the pressure drop \(\Delta P\). Since \(T_{\max}\) and \(\Delta T\) are strongly positively correlated, I remove \(T_{\max}\) from the objective list after a correlation analysis and use only \(\Delta T\) and \(\Delta P\) as final optimization objectives. This reduction improves the efficiency of the optimization algorithm.
5.2 Design of experiments and sensitivity analysis
I use the optimal Latin hypercube sampling (OLHS) method to generate 31 sample points in the three-dimensional design space. The sample points cover the design range with good projection properties. For each sample, a full three-dimensional conjugate heat-transfer simulation is performed. Table 4 lists the sampling results.
| No. | A (kg/s) | B (mm) | C (mm) | Tmax (K) | ΔT (K) | ΔP (kPa) |
|---|---|---|---|---|---|---|
| 1 | 0.0260 | 2.16 | 23.6 | 307.2 | 4.4 | 2.2 |
| 2 | 0.0167 | 2.34 | 17.3 | 309.5 | 6.2 | 1.0 |
| 3 | 0.0127 | 1.92 | 22.2 | 311.3 | 7.6 | 0.9 |
| 4 | 0.0293 | 1.76 | 13.8 | 306.7 | 4.1 | 4.2 |
| 5 | 0.0153 | 1.50 | 15.2 | 309.8 | 6.6 | 2.1 |
| 6 | 0.0273 | 2.92 | 11.7 | 307.0 | 4.3 | 1.9 |
| 7 | 0.0140 | 2.66 | 5.4 | 310.5 | 7.0 | 0.8 |
| 8 | 0.0207 | 1.66 | 20.8 | 308.2 | 5.2 | 2.4 |
| 9 | 0.0160 | 3.16 | 14.5 | 309.8 | 6.2 | 0.7 |
| 10 | 0.0193 | 2.00 | 4.0 | 308.5 | 5.6 | 2.3 |
| 11 | 0.0187 | 2.50 | 24.3 | 308.8 | 5.6 | 1.1 |
| 12 | 0.0280 | 3.84 | 7.5 | 306.9 | 4.2 | 1.7 |
| 13 | 0.0120 | 2.84 | 21.5 | 311.8 | 7.7 | 0.5 |
| 14 | 0.0287 | 3.58 | 16.6 | 307.0 | 4.1 | 1.6 |
| 15 | 0.0147 | 4.00 | 11.0 | 310.6 | 6.8 | 0.5 |
| 16 | 0.0213 | 1.58 | 10.3 | 308.0 | 5.1 | 3.3 |
| 17 | 0.0220 | 3.66 | 12.4 | 308.1 | 5.0 | 1.1 |
| 18 | 0.0113 | 3.76 | 18.7 | 312.5 | 8.0 | 0.3 |
| 19 | 0.0173 | 3.50 | 22.9 | 309.4 | 5.9 | 0.7 |
| 20 | 0.0240 | 2.26 | 15.9 | 307.5 | 4.7 | 1.9 |
| 21 | 0.0300 | 2.76 | 19.4 | 306.7 | 4.0 | 2.1 |
| 22 | 0.0200 | 2.58 | 9.6 | 308.4 | 5.4 | 1.3 |
| 23 | 0.0227 | 3.08 | 18.0 | 307.9 | 4.9 | 1.2 |
| 24 | 0.0267 | 2.08 | 6.8 | 307.0 | 4.3 | 3.1 |
| 25 | 0.0247 | 3.00 | 4.7 | 307.4 | 4.6 | 1.8 |
| 26 | 0.0180 | 3.42 | 6.1 | 309.0 | 5.8 | 0.9 |
| 27 | 0.0107 | 2.42 | 13.1 | 312.7 | 8.4 | 0.5 |
| 28 | 0.0253 | 3.26 | 25.0 | 307.5 | 4.5 | 1.4 |
| 29 | 0.0133 | 1.84 | 8.2 | 310.8 | 7.3 | 1.3 |
| 30 | 0.0233 | 3.92 | 20.1 | 307.9 | 4.8 | 1.1 |
| 31 | 0.0100 | 3.34 | 8.9 | 313.3 | 8.7 | 0.3 |
To quantify the contribution of each variable, I compute a sensitivity index:
$$ SA_i = \frac{f_{\max}(x_i) – f_{\min}(x_i)}{f_{\max}(x) – f_{\min}(x)} \times 100\%, $$
where \(f_{\max}(x_i)\) and \(f_{\min}(x_i)\) are the maximum and minimum values of the response when the \(i\)-th variable changes over its entire range while the other variables are fixed. The sensitivity analysis shows that the mass-flow rate is the most dominant parameter for both \(T_{\max}\) and \(\Delta T\), with sensitivity indices 0.996 and 0.988, respectively. For the pressure drop, the channel height is the dominant parameter with a sensitivity of 44.2%. The pin-fin pitch has a negligible influence on the pressure drop, with a sensitivity of only 2.8%.
Correlation analysis between the objectives is also performed using the Pearson coefficient. The correlation coefficient between \(\Delta P\) and \(T_{\max}\) is -0.70, and that between \(\Delta P\) and \(\Delta T\) is -0.67, indicating a conflicting relationship. More importantly, the correlation coefficient between \(T_{\max}\) and \(\Delta T\) reaches 0.99. Consequently, \(T_{\max}\) is removed from the optimization objectives in order to reduce the dimensionality and computational cost.
5.3 Kriging surrogate model
I construct a Kriging surrogate model to map the design variables to the two outputs \(\Delta T\) and \(\Delta P\). The quality of the surrogate model is evaluated using the coefficient of determination:
$$ R^2 = 1 – \frac{\sum_{i=1}^{n} (y_i – \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2}, $$
where \(y_i\) is the actual value from CFD, \(\hat{y}_i\) is the Kriging prediction, and \(\bar{y}\) is the mean of the actual values. The cross-validation results yield \(R^2=0.966\) for \(\Delta T\) and \(R^2=0.952\) for \(\Delta P\). Both values exceed 0.9, demonstrating high surrogate accuracy.
5.4 NSGA-II optimization
I use the NSGA-II algorithm to find the Pareto-optimal front between \(\Delta T\) and \(\Delta P\). The population size and number of generations are set to 120 and 50, respectively, producing 6000 candidate solutions. After filtering the feasible solutions, the Pareto front is identified. Figure 4.4 in the original article shows the scatter plot; I omit the visual here. The Pareto front reveals a clear conflict: reducing the temperature difference requires a higher flow rate and an increased pressure drop, while reducing the pressure drop generally sacrifices temperature uniformity.
From the Pareto front, I select three representative optimized designs and verify them using the original CFD model. Table 5 compares the baseline model, the three algorithm-selected designs, and the corresponding CFD-verified results.
| Design | A (kg/s) | B (mm) | C (mm) | Tmax (K) | ΔT (K) | ΔP (kPa) | ΔT error | ΔP error |
|---|---|---|---|---|---|---|---|---|
| Baseline | 0.0200 | 1.50 | 15 | 308.3 | 5.40 | 3.10 | — | — |
| Algorithm-1 | 0.0253 | 3.66 | 22 | — | 4.47 | 1.25 | — | — |
| CFD-1 | 0.0253 | 3.66 | 22 | 307.5 | 4.51 | 1.27 | 0.9% | 1.6% |
| Algorithm-2 | 0.0285 | 3.14 | 19 | — | 4.02 | 1.77 | — | — |
| CFD-2 | 0.0285 | 3.14 | 19 | 306.9 | 4.15 | 1.76 | 3.1% | 0.6% |
| Algorithm-3 | 0.0294 | 2.52 | 18 | — | 3.93 | 2.34 | — | — |
| CFD-3 | 0.0294 | 2.52 | 18 | 306.7 | 4.07 | 2.30 | 3.4% | 1.7% |
All Kriging predictions agree with the CFD simulations within 3.4% for the temperature difference and within 1.7% for the pressure drop. The optimized model 2 is selected as the final compromise because it provides a 23.1% improvement in temperature uniformity and a 43.2% reduction in pressure drop relative to the baseline. The corresponding design parameters are \(A=0.0285\) kg/s, \(B=3.14\) mm, and \(C=19\) mm.
5.5 Thermal-resistance analysis
To better understand the heating path, the total thermal resistance is decomposed into three components:
$$ R_{conv} = \frac{T_{ave} – T_{liq}}{q}, $$
$$ R_{heat} = \frac{T_{out} – T_{in}}{q}, $$
$$ R_{cond} = \frac{L_b}{k_s A_{cont}}, $$
where \(T_{ave}\) is the average temperature of the high-voltage battery surface, \(T_{liq}\) is the average coolant temperature, \(T_{in}\) and \(T_{out}\) are the inlet and outlet coolant temperatures, \(q\) is the heat-generation rate, \(L_b\) is the cold-plate thickness, \(k_s\) is the thermal conductivity of the solid, and \(A_{cont}\) is the contact area of the channel wall. The total thermal resistance is the sum of these three contributions:
$$ R_{total} = R_{conv} + R_{heat} + R_{cond}. $$
At a mass-flow rate of 0.02 kg/s, the optimized circular pin-fin cold plate has a total thermal resistance 14% lower than that of the straight-channel cold plate. At 0.03 kg/s, the reduction reaches 16%. The main contribution is the reduction in the convective resistance \(R_{conv}\); the enthalpy-resistance \(R_{heat}\) and the conduction resistance \(R_{cond}\) are almost identical for all designs. This result confirms that the pin-fin structure effectively reduces the convective resistance in the high-voltage battery cold plate.
5.6 Effective heat-transfer enhancement factor
The effective heat-transfer enhancement factor is used to normalize the improvement in heat transfer against the penalty in pressure drop. It is defined as
$$ p_f = \frac{Nu / Nu_0}{(\Delta P / \Delta P_0)^{1/3}}, $$
where \(Nu\) and \(\Delta P\) are the Nusselt number and pressure drop of the compared design, and \(Nu_0\) and \(\Delta P_0\) are the corresponding values of the baseline design. The Nusselt number is expressed as
$$ Nu = \frac{h_{ave} D_h}{k_{liq}}, $$
with \(h_{ave}\) being the area-averaged heat-transfer coefficient, \(D_h\) the hydraulic diameter, and \(k_{liq}\) the thermal conductivity of the coolant. At a mass-flow rate of 0.02 kg/s, the optimized circular pin-fin cold plate yields a Nusselt number more than six times higher than that of the straight channel, and the effective enhancement factor is about seven times higher. Compared with the baseline circular pin-fin design, the optimized design increases the Nusselt number by 33.9% and the enhancement factor by 97.5%. When the flow rate increases to 0.03 kg/s, the Nusselt number further increases while the enhancement factor slightly decreases because the pressure drop grows faster than the heat-transfer coefficient.
5.7 Cooling-efficiency factor
In addition to conventional thermal-resistance criteria, I evaluate the overall energy efficiency using a cooling-efficiency factor. The heat absorbed by the coolant is
$$ Q_{liq}(t) = c_f m \Delta T_{coolant}, $$
and the heat generated by the high-voltage battery cell over the discharge interval is
$$ Q_{cell}(t) = \int_{t_0}^{t_1} I^2 R(t) dt. $$
The cooling-efficiency factor is then defined as
$$ \eta = \frac{Q_{liq}(t)/t}{\Delta P \, q_v}, $$
where \(q_v\) is the volumetric flow rate. A larger \(\eta\) indicates that more heat is removed for a given pumping-power expenditure. At a mass-flow rate of 0.02 kg/s, the optimized circular pin-fin cold plate achieves \(\eta = 15{,}452\), while the baseline circular pin-fin design and the straight-channel design achieve \(\eta = 4{,}875\) and \(\eta = 7{,}250\), respectively. This confirms that the optimized design provides an extremely efficient thermal management solution for high-voltage battery packs.
6. Effects of Channel Height and Further Numerical Verification
Because the cold-plate thickness is directly related to channel height, I investigated the influence of the channel height on the thermal and hydraulic performance of the optimized model while keeping the other parameters fixed. Table 6 summarizes the results for several channel heights at a mass-flow rate of 0.0285 kg/s.
| Channel height (mm) | Tmax (K) | ΔT (K) | ΔP (kPa) |
|---|---|---|---|
| 1.0 | — | 4.12 | 13.0 |
| 1.5 | 306.9 | 4.15 | 5.3 |
| 2.0 | 307.0 | 4.20 | 3.2 |
| 3.0 | 307.1 | 4.25 | 2.0 |
| 4.0 | 307.2 | 4.30 | 1.5 |
The maximum temperature changes by only 0.3 K over the full channel-height range because the increase in heat-transfer area partially compensates for the reduction in velocity. However, the pressure drop decreases dramatically when the channel height is increased from 1 mm to 1.5 mm, with a reduction of 7.7 kPa (59.2%). Increasing the height further produces diminishing returns in the pressure-drop reduction. Therefore, I select 1.5 mm as the best channel height for the balance between thermal performance, pumping power, and the compactness required in high-voltage battery packs.
Because the final C-Pin-fin (BCS) cold plate has a different geometry from the initially validated model, I conduct an additional validation using a fabricated cold plate. The designated C-Pin-fin (BCS) experimental model is 732 mm long, 191 mm wide, and 12 mm high, with a channel height of 1.5 mm, a pin-fin pitch of 19 mm, and a pin-fin diameter of 3 mm. Four heating films are attached to the cold-plate surface to mimic the heat generated by high-voltage battery cells. The total electric-heating power is 280 W. The inlet coolant temperature is controlled at 20°C, and the mass-flow rates range from 0.005 kg/s to 0.03 kg/s. The numerical model with the standard \(k-\epsilon\) turbulence model is solved under the same conditions. The maximum relative error between the predicted and measured pressure drop is 8%, while the maximum error in the average surface temperature is 1.2 K. These errors are acceptable, which further confirms the correctness of the numerical model for the optimized high-voltage battery cold plate.
7. Experimental Study of the Circular Pin-Fin Cold Plate
7.1 Experimental setup and uncertainty analysis
A closed-loop experimental facility is constructed to characterize the flow resistance and heat-transfer performance of the C-Pin-fin (BCS) cold plate. The main components include a gear pump, a liquid turbine flow meter, PT100 temperature sensors, pressure transmitters, T-type thermocouples, an infrared camera, a thermostatic water bath, a multi-channel DC power supply, and a data acquisition unit. The coolant is a 50 vol% ethylene glycol/water solution. The measured variables are the flow rate, inlet/outlet pressures, inlet/outlet temperatures, and surface temperatures of the heated films. Table 7 lists the instruments and their measurement uncertainties.
| Instrument | Range / accuracy |
|---|---|
| Multi-channel DC power supply | 0 – 200 W |
| Infrared camera | ±2% (−20 – 2000 °C) |
| Liquid turbine flow meter | ±0.5% (0 – 10 L/min) |
| Thermostatic water bath | ±0.75% (−20 – 100 °C) |
| Thermocouple | ±1% (−100 – 280 °C) |
| Inlet pressure transmitter | ±0.5% (0 – 500 kPa) |
| Outlet pressure transmitter | ±0.5% (0 – 130 kPa) |
The heating power of each film is determined from the current and voltage measurements. The effective heat flux is evaluated as
$$ q = \frac{c_p m \Delta T}{A}, $$
where \(A\) is the total area of the heating films. The pump power is calculated as
$$ P_p = \Delta P V, $$
with \(V\) being the volumetric flow rate. To estimate the measurement reliability, I use the error-propagation formula. For a derived quantity \(R = f(v_1, v_2, \ldots, v_n)\), the relative uncertainty is computed as
$$ \frac{\delta R}{R} = \frac{1}{R} \sqrt{\sum_{i=1}^n \left( \frac{\partial R}{\partial v_i} \delta v_i \right)^2 }. $$
Table 8 lists the maximum relative uncertainties of the measured and derived parameters.
| Parameter | Maximum relative uncertainty |
|---|---|
| Inlet pressure \(P_{in}\) | 2.4% |
| Outlet pressure \(P_{out}\) | 0.6% |
| Volumetric flow rate | 2.5% |
| Mass-flow rate | 2.5% |
| Inlet/outlet coolant temperatures | 1.6% |
| Pressure drop \(\Delta P\) | 2.5% |
| Heat flux of heating film | 3.1% |
| Thermal resistance | 3.6% |
7.2 Flow-resistance experiments
During the flow-resistance experiments, the heating films remain off and the pressure drop is measured as the coolant flow rate is varied. The temperature inside the channel is held at the ambient level. The measured pressure-drop curve can be divided into three regimes. At low flow rates, the pressure drop increases slowly with the flow rate because the pin-fin wakes are small and the loss is dominated by friction. In the intermediate flow range, the wakes expand and flow separation enhances the form drag, leading to a larger slope. At high flow rates, the pressure drop increases significantly because vortex shedding and turbulence mixing become vigorous. Even so, the absolute pressure drop of the C-Pin-fin (BCS) cold plate remains small enough for a practical high-voltage battery cooling system.
7.3 Heat-transfer experiments
In the heat-transfer experiments, I set the heating powers to 200 W, 280 W, 360 W, and 440 W. At each power setting, the coolant flow rate is changed over the prescribed range. The infrared thermal images show that the heating-film temperature increases along the flow direction from the inlet to the outlet because the coolant enthalpy rise reduces the local temperature difference. The cooling effect is strongest near the inlet, where the thermal boundary layer is thin, and weakens downstream as the coolant warms. The film located near the inlet is always cooler than the downstream film, which is physically consistent with entrance effects.
Figure 6 in the original work illustrates the temperature evolution; I do not repeat those images here. The measured data indicate that, for a constant flow rate, the heating-film temperature rises almost linearly with the heat flux. For a constant heat flux, increasing the flow rate lowers the film temperature because of an enhanced convective heat-transfer coefficient. The increase in flow rate thins the thermal boundary layer, promotes flow mixing around the pin fins, and thereby improves the heat-transfer rate from the film to the coolant. However, the reduction in temperature becomes less pronounced when the flow rate exceeds a certain value because the coolant heat-capacity limit is approached.
The thermal resistance is calculated as
$$ R_{exp} = \frac{T_{film,ave} – T_{coolant,ave}}{qA}, $$
where \(T_{film,ave}\) is the average temperature of the heating films. The experimental thermal resistance decreases with increasing flow rate. In addition, for a fixed flow rate, an increase in the heating power slightly reduces the thermal resistance at first, but the reduction gradually saturates. This occurs because the higher coolant temperature reduces the viscosity and increases the effective thermal conductivity, thus improving the heat-transfer coefficient. However, as the coolant temperature continues to rise, the heat-transfer enhancement saturates, and the thermal resistance tends to a stable value. The pump-power consumption increases rapidly with the flow rate because both the volumetric flow rate and the pressure drop increase. Thus, one should avoid excessively high flow rates in high-voltage battery cooling systems, where only minor reductions in thermal resistance are gained at the cost of large pumping-power increases.
8. Conclusion
In this study, I systematically investigated the single-phase flow and heat-transfer characteristics of pin-fin channel cold plates for high-voltage battery packs. The main findings are as follows:
(1) A heat-generation model for a high-voltage battery cell was built using the Bernardi formula and validated against temperature-rise experiments; the maximum error was confined within 5%. The model is sufficiently accurate for subsequent conjugate heat-transfer simulations.
(2) A comparative numerical study demonstrated that circular pin-fin channels offer the best thermohydraulic balance. At a mass-flow rate of 0.03 kg/s, the circular pin-fin design reduced the maximum temperature by 2.9 K and the temperature difference by 0.7 K compared with a straight channel. The pressure-drop penalty of the circular pin-fin lies considerably below that of the square pin-fin. The heat-transfer-enhancement mechanism is attributed to the periodic disruption of the thermal boundary layer and the regular ring-shaped vortices induced by circular pin fins, which efficiently convert turbulence into heat transport with minimal unnecessary dissipation.
(3) A multi-objective optimization based on Kriging surrogate modeling and the NSGA-II algorithm yielded an optimized cold plate with a mass-flow rate of 0.0285 kg/s, a channel height of 3.14 mm, and a pin-fin pitch of 19 mm. Sensitivity analysis showed that the mass-flow rate dominates the maximum temperature and temperature uniformity, while the channel height dominates the pressure drop. Correlation analysis revealed that the maximum temperature and temperature difference are strongly collinear, so only the latter was retained as an optimization goal. The selected design improved temperature uniformity by 23.1% and reduced the pressure drop by 43.2% relative to the baseline, while the Kriging-prediction error remained below 3.4%.
(4) The optimized design increased the Nusselt number by a factor of more than six and the effective heat-transfer enhancement factor by more than seven relative to the straight-channel cold plate at 0.02 kg/s. The cooling-efficiency factor of the optimized cold plate reached 15,452, far above the values for the baseline and straight-channel designs. Additional channel-height studies showed that a height of 1.5 mm offers the optimal trade-off between compactness, thermal performance, and pump power, with a 59.2% pressure-drop reduction compared with a 1 mm channel height.
(5) The experimental investigation validated the numerical models. The measured pressure drop agreed with the standard \(k-\epsilon\) model within 8%, and the surface-temperature error was below 1.2 K. The experimental data showed that increasing the coolant flow rate monotonically reduces the heating-film temperature and thermal resistance while increasing the pump power. Increasing the heating power raises the film temperature but causes the thermal resistance to decline and eventually plateau. These results provide a practical basis for designing efficient, low-energy liquid-cooling plates for high-voltage battery thermal management.
In conclusion, the pin-fin channel cold plate, especially the optimized circular pin-fin configuration, proves to be a highly effective and energy-efficient solution for cooling high-voltage battery modules. The methodology and findings reported here can be extended to other compact thermal-management devices where high heat fluxes and strict temperature-uniformity limits coexist.