Electric vehicles have become an important solution for reducing greenhouse-gas emissions and dependence on fossil fuels. In an electric vehicle, the traction battery is the most critical component because its performance, lifetime and safety directly determine the usability of the vehicle. Among the available chemistries, lithium-ion cells are widely accepted because of their high gravimetric and volumetric energy density, low self-discharge rate and relatively long cycle life. However, the working temperature of an EV battery is a central issue. A lithium-ion battery usually operates best between 25 °C and 40 °C, and the maximum temperature difference inside a pack should be lower than 5 °C. When the cell temperature is too high, side reactions inside the electrode/electrolyte interface are accelerated, causing capacity fade, internal short circuits and thermal runaway. Therefore, an effective battery thermal management system is indispensable for the reliable operation of an EV battery pack.
Four principal cooling methods are used for EV battery systems: air cooling, phase-change-material cooling, heat-pipe cooling and liquid cooling. Air cooling is simple and inexpensive, but its heat transfer coefficient is relatively low. Phase-change materials provide good temperature uniformity but add weight and may suffer from limited heat-dissipation capacity. Heat pipes show excellent thermal conductivity, yet their structure is complex and their cost is high. Liquid cooling, by contrast, uses a working fluid with a large specific heat and high thermal conductivity to remove heat efficiently. As a result, liquid-cooled battery thermal management is considered the most suitable approach for high-rate EV battery operation.
In this study, a prismatic lithium-ion cell was used as the research object. A conventional Tesla valve flow structure, a proposed internal-improvement Tesla valve structure and an external-improvement Tesla valve structure were integrated into liquid cooling plates. The cooling performance was investigated numerically and experimentally, and a multi-objective optimization method was used to balance cell temperature and pumping loss.
1. Battery Cell Characteristics and Thermal Model
The cell studied in this work is a prismatic lithium-ion cell with a nominal voltage of 3.7 V, a rated capacity of 10.15 Ah, and dimensions of 66 mm × 18 mm × 133 mm. The charging cut-off voltage is 4.2 V and the discharging cut-off voltage is 2.75 V. The cell parameters are summarized in Table 1.
| Parameter | Value |
| Dimensions (L×W×H), mm | 66 × 18 × 133 |
| Rated capacity, Ah | 10.15 |
| Nominal voltage, V | 3.7 |
| Charge cut-off voltage, V | 4.2 |
| Discharge cut-off voltage, V | 2.75 |
The heat generated inside a lithium-ion cell originates mainly from polarization heat, ohmic heat, reaction heat and side-reaction heat. The total heat generation rate can be expressed by the Bernardi equation:
$$
q_{\mathrm{gen}} = \frac{1}{V_{\mathrm{cell}}}\left( I^2 R + I T \frac{\partial E}{\partial T} \right),
$$
where I represents the applied current, R is the battery internal resistance, T is temperature, E is the electromotive force, and Vcell is the cell volume. Since the internal resistance depends on the state of charge, a hybrid pulse power characterization experiment was carried out to measure the resistance at different values of SOC. In the experiment, the cell was first fully charged, and then a 1 C pulse discharge of 10 s was applied, followed by a 10 s rest. Each SOC level from 0.9 to 0.1 was tested. The ohmic resistance and polarization resistance were calculated from the instantaneous voltage drops:
$$
R_{\Omega} = \frac{\Delta V_1}{I_f}, \qquad R_{p} = \frac{\Delta V_2}{I_f}, \qquad R = R_{\Omega} + R_p.
$$
The experimentally obtained resistance curve was fitted with a sixth-order polynomial:
$$
\begin{aligned}
R(\mathrm{SOC}) =& 0.0215\, \mathrm{SOC}^6 – 0.66191\, \mathrm{SOC}^5 + 2.25422\, \mathrm{SOC}^4 \\
&- 3.04786\, \mathrm{SOC}^3 + 2.06703\, \mathrm{SOC}^2 \\
&- 0.73309\, \mathrm{SOC} + 0.12956.
\end{aligned}
$$
This expression was compiled as a user-defined function and used in the computational fluid dynamics simulation to reflect the dynamic heat generation behavior of an EV battery cell during discharge.
The battery temperature field was determined by solving the three-dimensional heat conduction equation:
$$
\rho_{\mathrm{bat}} c_{\mathrm{bat}} \frac{\partial T}{\partial t}
= \frac{\partial}{\partial x}\left( \lambda_x \frac{\partial T}{\partial x} \right)
+ \frac{\partial}{\partial y}\left( \lambda_y \frac{\partial T}{\partial y} \right)
+ \frac{\partial}{\partial z}\left( \lambda_z \frac{\partial T}{\partial z} \right) + q_{\mathrm{gen}}.
$$
The cell and module surfaces were exposed to an environment at 28 °C with a natural-convection heat transfer coefficient of 3 W/(m²·K). The basic thermal-physical properties of the materials are listed in Table 2.
| Component | Material | ρ, kg/m³ | c, J/(kg·K) | λ, W/(m·K) | μ, Pa·s |
| Liquid cooling plate | Aluminium | 2719 | 871 | 202.4 | — |
| Coolant | 50% aqueous glycol solution | 1068.75 | 3319 | 0.384 | 0.00294 |
| Battery cell | — | 2338 | 1085 | λx=22.8, λy=16.5, λz=5.1 | — |
Temperature-rise experiments at discharge rates of 1 C and 2.5 C were performed in a constant-temperature chamber. Five K-type thermocouples were attached to the positive terminal area, negative terminal area, central area, bottom-positive area and bottom-negative area. At the end of the 2.5 C discharge, the measured temperatures near the terminals exceeded 53 °C, while the bottom area was cooler. The cell model was then used to predict the discharge temperature and compared with the experimental data. The maximum deviation between the simulated and measured temperatures was less than 4%, which confirmed the reliability of the battery thermal model.
2. Tesla Valve Liquid Cooling Plate Design
A conventional Tesla valve is a passive fluidic device containing no moving parts. When fluid passes in one direction, the branched channels promote a smooth merging flow; in the opposite direction, the merging streams oppose each other and increase hydraulic resistance. This feature can enhance heat transfer in an EV battery cooling plate while acting as a flow-control structure.
In this work, the traditional Tesla valve was used as the starting point. Two new configurations were proposed. The first is called the internal-improvement Tesla valve, in which the side branch lies inside the angle formed by the main straight channels. The second is called the external-improvement Tesla valve, in which the side branch lies outside that angle. For each type, the forward direction is denoted by the suffix -D, and the reverse direction by -R. The key geometry parameters are the coolant inlet velocity v, the number of channels per plate S, the valve angle α, and the flow-channel angle β. Four different flow-channel shapes, namely M-shaped, conventional Tesla valve, internal-improvement Tesla valve and external-improvement Tesla valve, were therefore compared.
The EV battery module considered in the simulation consisted of ten cells arranged as a 5×2 array. Each liquid cooling plate was placed between two rows of cells. The total pressure drop, \(P_{\mathrm{all}}\), represented the sum of pressure drops in all channels and was used as a measure of pumping-power consumption.
3. Numerical Model and Meshing
Commercial CFD software was used to solve the conjugate heat transfer problem. The coolant flow was assumed to be incompressible, steady in the fluid domain and laminar or transitional depending on the Reynolds number. The continuity, momentum and energy equations for the coolant are written as
$$
\frac{\partial \rho_{\mathrm{liq}}}{\partial t} + \nabla \cdot (\rho_{\mathrm{liq}} \mathbf{v}) = 0,
$$
$$
\frac{\partial (\rho_{\mathrm{liq}} \mathbf{v})}{\partial t} + \nabla \cdot (\rho_{\mathrm{liq}} \mathbf{v} \mathbf{v}) = -\nabla P + \nabla \cdot \left( \mu_{\mathrm{liq}} \nabla \mathbf{v} \right),
$$
$$
\frac{\partial (\rho_{\mathrm{liq}} c_{\mathrm{liq}} T)}{\partial t} + \nabla \cdot (\rho_{\mathrm{liq}} c_{\mathrm{liq}} \mathbf{v} T)
= \nabla \cdot (\lambda_{\mathrm{liq}} \nabla T).
$$
A poly-hexcore mesh was generated for the computational domain with boundary layers at the fluid-solid interfaces. To ensure mesh independence, seven grid sizes were tested, from 2,196,260 to 5,912,478 elements. When the number of elements exceeded 3,608,307, the maximum temperature difference and total pressure drop changed by less than 0.01 °C and 10 Pa, respectively. Therefore, the mesh with 3,608,307 elements was used for the subsequent simulations.
4. Comparison of Basic Flow-Channel Types
The M-shaped plate was used as a baseline. The same channel width, plate size and operating conditions were maintained for the traditional Tesla valve, internal-improvement Tesla valve and external-improvement Tesla valve plates. The number of channels per plate was \(S=3\), the valve angle was \(\alpha=28^{\circ}\), the channel angle was \(\beta=90^{\circ}\), and the flow direction was forward. The discharge rate was 2.5 C and the initial temperature was 28 °C. The simulated results at a coolant velocity of \(v=1.00\) m/s are summarized in Table 3.
| Flow-channel type | Tmax, °C | ΔTmax, °C | Pall, Pa |
| M-shaped | 31.97 | 2.70 | 290081.69 |
| Traditional Tesla valve | 31.84 | 2.70 | 298593.26 |
| Internal-improvement Tesla valve | 31.69 | 2.76 | 177845.03 |
| External-improvement Tesla valve | 31.59 | 2.67 | 195060.47 |
It can be observed from Table 3 that all Tesla-valve-based channels reduce the maximum cell temperature compared with the M-shaped channel. More importantly, the conventional Tesla valve significantly increases the total pressure drop, reaching almost 300 kPa. In contrast, the internal-improvement Tesla valve and external-improvement Tesla valve reduce the total pressure drop markedly, while retaining the thermal benefit of the Tesla valve structure. The internal-improvement Tesla valve gives the lowest pressure drop of about 178 kPa in this high-flow-rate case. From the standpoint of minimizing pumping loss, this channel type was preferred for the subsequent work.
The direction of coolant flow was also examined. For an internal-improvement Tesla valve plate with \(S=3\), \(\alpha=30^{\circ}\), \(\beta=120^{\circ}\) and \(v=0.20\) m/s, forward flow and reverse flow were compared. The results are shown in Table 4.
| Flow direction | Tmax, °C | ΔTmax, °C | Pall, Pa |
| IITV-D (forward) | 33.33 | 3.24 | 10268.74 |
| IITV-R (reverse) | 33.38 | 3.98 | 10957.83 |
The maximum temperature difference in the reverse direction is larger, and the reverse direction also gives a higher pressure drop. This behavior is consistent with the Tesla-valve mechanism: forward flow merges smoothly, whereas reverse flow creates opposing streams and greater flow resistance. Therefore, the forward-flow internal-improvement Tesla valve plate, abbreviated IITV-D, was selected as the principal object for further parametric analysis and optimization. This design enables an EV battery pack to be cooled effectively while reducing the parasitic loss of the cooling system.
5. Single-Factor Parametric Analysis
For the IITV-D liquid cooling plate, four parameters were evaluated: coolant velocity \(v\), number of flow channels \(S\), valve angle \(\alpha\) and flow-channel angle \(\beta\). The evaluation metrics were the highest cell temperature \(T_{\mathrm{max}}\), the maximum temperature difference inside the module \(\Delta T_{\mathrm{max}}\), and the total pressure drop \(P_{\mathrm{all}}\). The discharge rate remained 2.5 C and the ambient temperature was 28 °C.
5.1 Effect of Coolant Velocity
With \(S=3\), \(\alpha=30^{\circ}\) and \(\beta=120^{\circ}\), the inlet velocity was increased from 0.2 to 1.0 m/s. As the velocity increased, \(T_{\mathrm{max}}\) decreased from 33.33 °C to 31.91 °C and \(\Delta T_{\mathrm{max}}\) decreased from 3.24 °C to 2.77 °C, but \(P_{\mathrm{all}}\) increased from about 10.27 kPa to 105.76 kPa. The cooling enhancement was more pronounced between 0.2 and 0.4 m/s than between 0.8 and 1.0 m/s. Thus, higher flow velocity improves the thermal performance of the EV battery module but requires more pumping power.
5.2 Effect of Channel Number
With \(v=0.2\) m/s, \(\alpha=30^{\circ}\) and \(\beta=120^{\circ}\), the number of channels was changed from 2 to 6. When more channels were used, the heat transfer area increased and the maximum temperature decreased from 34.73 °C to 31.93 °C. The maximum temperature difference decreased from 3.68 °C to 2.76 °C. The total pressure drop increased almost linearly with the number of channels, from about 6.85 kPa to 20.56 kPa. Channel number is therefore a major geometric factor that affects both cooling performance and system resistance.
5.3 Effect of Valve Angle
The valve angle \(\alpha\) was changed from 26° to 34°. The maximum cell temperature varied only from 33.31 °C to 33.35 °C, and the maximum temperature difference remained near 3.24 °C. The pressure drop, however, increased from 9216 Pa to 10903 Pa as \(\alpha\) increased. The results indicate that \(\alpha\) has a minor effect on cell temperature but a visible effect on pressure loss.
5.4 Effect of Channel Angle
The channel angle \(\beta\) was changed from 90° to 120°. As \(\beta\) increased, \(T_{\mathrm{max}}\) increased slightly from 33.06 °C to 33.33 °C, whereas \(P_{\mathrm{all}}\) decreased from 16407 Pa to 10268 Pa. A larger channel angle makes the channel more straight and reduces the resistance to flow. Therefore, \(\beta\) mainly affects hydraulic performance rather than thermal performance.
These single-factor trends are summarized in Table 5.
| Parameter | Range | Tmax change, °C | ΔTmax change, °C | Pall change |
| v | 0.2 – 1.0 m/s | 33.33 to 31.91 | 3.24 to 2.77 | 10.3 to 105.8 kPa |
| S | 2 – 6 | 34.73 to 31.93 | 3.68 to 2.76 | 6.9 to 20.6 kPa |
| α | 26° – 34° | 33.31 to 33.35 | 3.23 to 3.24 | 9.2 to 10.9 kPa |
| β | 90° – 120° | 33.06 to 33.33 | 3.31 to 3.24 | 16.4 to 10.3 kPa |
In general, increasing \(v\) and \(S\) improves the cooling performance of the EV battery pack but also increases the pressure drop. The angles \(\alpha\) and \(\beta\) have comparatively small thermal effects and can be used to control pressure loss without significantly sacrificing temperature uniformity.
6. Orthogonal Experiments and Range Analysis
In order to distinguish the significance of each design factor, four-factor and four-level orthogonal experiments were arranged. The factor levels are listed in Table 6.
| Level | v, m/s | S | α, ° | β, ° |
| 1 | 0.2 | 2 | 28 | 90 |
| 2 | 0.4 | 3 | 30 | 100 |
| 3 | 0.6 | 4 | 32 | 110 |
| 4 | 0.8 | 5 | 34 | 120 |
The \(L_{16}\) orthogonal array and the corresponding simulation results are shown in Table 7.
| No. | v | S | α | β | Tmax, °C | ΔTmax, °C | Pall, Pa |
| 1 | 1 | 1 | 1 | 1 | 34.26 | 3.74 | 10636.28 |
| 2 | 1 | 2 | 2 | 2 | 33.16 | 3.28 | 13497.30 |
| 3 | 1 | 3 | 3 | 3 | 32.61 | 3.00 | 15445.82 |
| 4 | 1 | 4 | 4 | 4 | 32.21 | 2.83 | 18175.40 |
| 5 | 2 | 1 | 2 | 3 | 33.42 | 3.34 | 20116.39 |
| 6 | 2 | 2 | 1 | 4 | 32.41 | 2.95 | 26114.67 |
| 7 | 2 | 3 | 4 | 1 | 31.83 | 2.82 | 61898.36 |
| 8 | 2 | 4 | 3 | 2 | 31.54 | 2.66 | 63528.00 |
| 9 | 3 | 1 | 3 | 4 | 33.03 | 3.18 | 34525.98 |
| 10 | 3 | 2 | 4 | 3 | 32.13 | 2.90 | 57657.94 |
| 11 | 3 | 3 | 1 | 2 | 31.61 | 2.70 | 88159.44 |
| 12 | 3 | 4 | 2 | 1 | 31.34 | 2.62 | 137659.87 |
| 13 | 4 | 1 | 4 | 2 | 32.79 | 3.20 | 73520.17 |
| 14 | 4 | 2 | 3 | 1 | 31.92 | 2.92 | 131122.20 |
| 15 | 4 | 3 | 2 | 4 | 31.55 | 2.64 | 102598.78 |
| 16 | 4 | 4 | 1 | 3 | 31.25 | 2.52 | 136150.14 |
Range analysis was carried out for each response. The range values are listed in Table 8.
| Response | v | S | α | β |
| Tmax, °C | 1.18 | 1.79 | 0.15 | 0.08 |
| ΔTmax, °C | 0.39 | 0.71 | 0.04 | 0.13 |
| Pall, Pa | 96409.12 | 54178.65 | 15655.11 | 39975.47 |
According to the range analysis, the order of significance for the maximum temperature is \(S > v > \alpha > \beta\). For the maximum temperature difference, the order is \(S > v > \beta > \alpha\). For total pressure drop, the order is \(v > S > \beta > \alpha\). Thus, coolant velocity and flow-channel number are the dominant factors controlling both thermal performance and hydraulic loss. The angles are secondary influences.
7. Multi-Objective Optimization with NSGA-III
To obtain an optimal trade-off among conflicting objectives, the NSGA-III algorithm was applied. This algorithm uses non-dominated sorting and a set of reference points to maintain population diversity. The optimization problem can be stated as
$$
\min \left\{ T_{\mathrm{max}}(v,S,\alpha,\beta),\; \Delta T_{\mathrm{max}}(v,S,\alpha,\beta),\; P_{\mathrm{all}}(v,S,\alpha,\beta) \right\},
$$
subject to the design ranges used in the orthogonal experiments:
$$
0.2 \le v \le 0.8, \quad 2 \le S \le 5, \quad 28^{\circ} \le \alpha \le 34^{\circ}, \quad 90^{\circ} \le \beta \le 120^{\circ}.
$$
Because CFD simulation is too expensive for direct optimization, a quadratic response-surface model was built from the 16 orthogonal cases. The general expression for each objective is
$$
\hat{y} = b_0 + \sum_{i=1}^{4} b_i x_i + \sum_{i=1}^{4} b_{ii} x_i^2 + \sum_{i<j} $$=""
where \(x_i \) represents the normalized design variables \(v\), \(S\), \(\alpha\) and \(\beta\). The coefficients \(b\) were estimated by least-squares regression. The coefficient of determination \(r^2\) is listed in Table 9.
| Objective | Tmax | ΔTmax | Pall |
| r² | 0.9988 | 0.9997 | 0.9999 |
The high \(r^2\) values indicate that the surrogate models represent the CFD data accurately. In the NSGA-III run, the population size was 30, the number of generations was 500, the crossover probability was 0.5 and the mutation probability was 0.05. The Pareto optimal set was obtained. From the point on the Pareto front with the smallest total pressure drop, the selected optimum is listed in Table 10.
| Parameter | NSGA-III optimum | CFD verification |
| v, m/s | 0.20 | 0.20 |
| S | 3 | 3 |
| α, ° | 33.62 | 33.62 |
| β, ° | 105.82 | 105.82 |
| Tmax, °C | 33.21 | 33.22 |
| ΔTmax, °C | 3.25 | 3.26 |
| Pall, Pa | 12486.85 | 13088.59 |
The maximum discrepancies between the response-surface model and the CFD simulation are only 0.04% for \(T_{\mathrm{max}}\), 0.40% for \(\Delta T_{\mathrm{max}}\) and 4.82% for \(P_{\mathrm{all}}\). This confirms the usefulness of the surrogate-based optimization approach for EV battery cooling-plate design.
Compared with the initial IITV-D design, the optimized IITV-D design increases \(T_{\mathrm{max}}\) by about 0.8 °C and \(\Delta T_{\mathrm{max}}\) by about 0.3 °C, but reduces the total pressure drop by approximately 52.18%. The resulting cell temperature remains within the desired range for a lithium-ion EV battery, while the pumping power consumption is substantially reduced. The optimized IITV-D plate therefore improves the balance between thermal performance and system hydraulic loss.
8. Experimental Validation
An experimental validation was designed for the optimized IITV-D liquid cooling plate. Because the available charging/discharging test equipment had only two independent channels, a battery cooling unit containing two cells and one IITV-D plate was used instead of a full ten-cell module. The cold plate was manufactured with 3D printing to ensure dimensional accuracy. Its outer dimensions were 132 mm × 4 mm × 133 mm and the channel width was 2 mm. The two cells were placed on one side of the cold plate, and five K-type thermocouples were installed at the top-positive area, top-negative area, geometric center, bottom-positive area and bottom-negative area.
The experimental platform included a high-performance battery testing system, a constant-temperature humidity chamber, K-type thermocouples, silicone tubing, and a liquid-cooling temperature-control unit. The coolant flow rate was set to 0.048 L/min, corresponding to an inlet velocity of 0.2 m/s. The environment temperature was 28 °C. Both cells were charged at 1 C to the nominal capacity and then discharged at 2.5 C. The measured temperature histories were recorded and compared with the CFD predictions for the same geometry and boundary conditions.
Table 11 compares the predicted and measured temperatures at the end of discharge.
| Thermocouple position | Simulation, °C | Experiment, °C | Error, % |
| T1: top-positive | 30.79 | 30.94 | 0.49 |
| T2: top-negative | 31.42 | 31.58 | 0.51 |
| T3: geometric center | 30.58 | 30.73 | 0.49 |
| T4: bottom-positive | 30.99 | 31.17 | 0.58 |
| T5: bottom-negative | 31.59 | 31.78 | 0.60 |
The temperature order, \(T_5 > T_2 > T_4 > T_1 > T_3\), is identical in the numerical and experimental results. The largest relative error is less than 1%, and all discrepancies are far below the target value of 5%. The small deviations can be attributed to manufacturing tolerances inside the tiny 2-mm-wide channel, which may increase the actual flow resistance and slightly reduce the heat removal rate. Nevertheless, the good agreement confirms the accuracy of the simulation and demonstrates that the optimized IITV-D liquid cooling plate provides reliable thermal control for an EV battery cell.
The successful validation of the two-cell cooling unit implies that the same IITV-D plate is suitable for more complex EV battery modules. Because the cooling principle and governing equations remain the same, extending the plate into a larger pack should preserve its beneficial performance. The main contribution of the optimized design is a large reduction in total pressure drop while still meeting the thermal requirements of a lithium-ion battery.
9. Conclusion
This study focused on the design and optimization of Tesla-valve-based liquid cooling channels for EV battery thermal management. A prismatic lithium-ion cell was characterized experimentally, and a numerical thermal model was validated within an error of 4%. Two improved Tesla-valve structures were proposed, and the forward-flow internal-improvement Tesla valve plate was selected because it offers the largest reduction in pressure drop while maintaining good thermal performance.
The single-factor and orthogonal analyses showed that coolant velocity and channel number are the most important parameters. The NSGA-III multi-objective optimization yielded a practical optimum with a coolant velocity of 0.20 m/s, three flow channels, a valve angle of 33.62° and a channel angle of 105.82°. At this optimum, the cell maximum temperature remains near 33.22 °C, the maximum temperature difference is approximately 3.26 °C, and the total pressure drop is about 13.1 kPa. Compared with the initial IITV-D design, the pressure drop is reduced by 52.18% with only a moderate increase in temperature. An experimental cooling-unit test confirmed that the simulation error is smaller than 5%, thereby proving the reliability of the proposed design and the optimization methodology.
The Tesla-valve liquid cooling plate described here provides an attractive solution for high-rate EV battery applications where thermal safety, temperature uniformity and low parasitic power must be balanced simultaneously.