Marine vessels are under increasing pressure to decarbonize because international shipping contributes a substantial share of global carbon dioxide emissions. One promising route is the replacement of conventional diesel propulsion with clean electric propulsion based on a large-format marine traction battery. A marine traction battery stores a huge amount of electrical energy and must deliver high discharge currents for long periods, which inevitably produces significant waste heat inside the cells. The electrochemical performance, service lifetime and safety of the traction battery strongly depend on its operating temperature. In general, lithium-ion cells should be kept within 25 °C to 40 °C, and the temperature difference inside a battery module should remain below 5 °C to minimize uneven aging. If the generated heat is not removed effectively, local hot spots appear, internal resistance rises, and thermal runaway may be initiated, eventually causing fire or explosion. Therefore, a robust battery thermal management system has become one of the key technologies in the design of a marine traction battery system.
Many cooling strategies have been investigated for traction battery thermal management, including air cooling, phase-change material cooling, immersion cooling and indirect liquid cooling. Air cooling systems are simple and cheap, but they require a large space and their heat transfer coefficient is limited. Phase-change materials provide passive heat absorption and excellent temperature uniformity, but leakage, low thermal conductivity and high weight hinder their practical application. Immersion cooling can give outstanding performance, but it demands dielectric fluids and careful sealing. Indirect liquid cooling is particularly attractive for marine applications because a liquid-cooled plate can be made very compact, and water or a water-glycol mixture has a high specific heat and a high convective heat-transfer coefficient. The flow channel layout inside a liquid-cooled plate controls both the cooling capacity and the pumping power required to drive the coolant. An optimized channel structure should remove heat effectively, maintain a uniform temperature field and produce a low pressure drop. The present study focuses on the design, optimization and characterization of a bionic liquid-cooled plate for a marine traction battery.
The inspiration for the new cooling plate comes from two natural structures: the hexagonal vein pattern of a leaf and the branching morphology of tree roots. The hexagonal leaf veins transport water and nutrients to every part of the leaf efficiently, while the tree-root network spreads through the soil and distributes fluid through a hierarchy of branches. By transforming these natural features into a liquid-cooled plate, I constructed a bionic flow domain that consists of a hexagonal strengthening region and a root-like arrangement around the hexagonal obstacles. The coolant enters through a narrow inlet, enters the general heat-transfer zone, then flows through the enhanced heat-transfer zone where it is repeatedly split and recombined by the hexagonal elements. This repeated rupture and reconstruction of the thermal boundary layer is intended to enhance convective heat transfer while consuming less pumping energy than a conventional straight-channel liquid-cooled plate. The main objective is to keep the marine traction battery within its favorable temperature window and to minimize the pressure drop so that the coolant pump consumes less energy.
Numerical model and governing equations
I used the finite-volume method to solve the steady and transient heat-transfer and fluid-flow problem. The numerical approach is based on three fundamental conservation equations for an incompressible Newtonian fluid. The mass conservation equation is written as
$$ \frac{\partial \rho}{\partial t} + \frac{\partial(\rho u_x)}{\partial x} + \frac{\partial(\rho u_y)}{\partial y} + \frac{\partial(\rho u_z)}{\partial z} = 0 $$
Because the coolant in this study is treated as incompressible, the density is constant and the continuity equation simplifies to
$$ \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} = 0 $$
The momentum equations are expressed in vector form as
$$ \frac{\partial(\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \mathbf{u}) = -\nabla p + \nabla \cdot (\mu \nabla \mathbf{u}) + \mathbf{S} $$
where \(\mathbf{u}\) is the velocity vector, \(p\) is the static pressure, \(\mu\) is the dynamic viscosity and \(\mathbf{S}\) represents the generalized momentum source terms. The energy equation for the temperature field is
$$ \frac{\partial(\rho T)}{\partial t} + \nabla \cdot (\rho \mathbf{u} T) = \nabla \cdot \left( \frac{\lambda}{c_p} \nabla T \right) + S_T $$
where \(T\) is temperature, \(\lambda\) is thermal conductivity, \(c_p\) is specific heat capacity and \(S_T\) is the volumetric heat source. In the solid aluminium wall of the liquid-cooled plate, the convective term disappears and only heat conduction is considered.
Heat generation and heat transfer in the traction battery
The thermal load applied to the liquid-cooled plate originates from electrochemical heat generation inside the marine traction battery. The total heat generation rate of a lithium-ion traction battery can be divided into four contributions: electrochemical reaction heat \(Q_R\), polarization heat \(Q_P\), Joule heat \(Q_J\) and side-reaction heat \(Q_S\). Thus,
$$ Q_{\text{total}} = Q_R + Q_P + Q_J + Q_S $$
The electrochemical reaction heat represents the entropy change associated with lithium-ion intercalation and deintercalation. The reaction heat is related to the cell current by
$$ Q_R = \frac{N m Q I}{M F} $$
where \(N\) is the number of cells, \(m\) is the mass of active material, \(Q\) is the heat generated per unit amount of substance, \(I\) is the current, \(M\) is molar mass and \(F\) is Faraday’s constant. The polarization heat is caused by polarization resistance inside the cell and can be written as
$$ Q_P = I^2 R_P $$
where \(R_P\) is the polarization resistance. Joule heat, also called ohmic heat, is generated when the current flows against the ohmic resistance \(R_o\) of the cell:
$$ Q_J = I^2 R_o $$
The side-reaction heat is usually negligible under normal operating conditions; the main contributions are reaction heat, polarization heat and Joule heat. The generated heat is first transported through the cell by conduction, reaches the surface of the cell and then enters the aluminium cooling plate. Inside the liquid-cooled plate, convective heat transfer between the coolant and the heated wall removes heat. Radiation heat transfer is comparatively small in this temperature range and was not included in the numerical simulation.
Geometric parameters of the bionic liquid-cooled plate
The bionic liquid-cooled plate was designed for a prismatic lithium-iron-phosphate module with a nominal capacity of 16.18 kWh. The module heat generation used in the simulation is 250 W, which corresponds to a surface heat flux of 785 W/m² on the heated plane of the cooling plate. The overall dimensions of the liquid-cooled plate are 773 mm by 634 mm by 6 mm. The coolant inlet and outlet are rectangular, with a height of 2 mm and a width of 32 mm. The thickness of the fluid domain is also 2 mm. The aluminium plates above and below the fluid channel are therefore 2 mm thick, which provides good structural rigidity and manufacturing feasibility.
Four structural parameters determine the internal topology of the enhanced heat-transfer region. The parameter \(a\) denotes the size of the hexagonal obstacles, \(b\) denotes the transverse distance between adjacent hexagonal obstacles, \(c\) denotes the longitudinal distance between adjacent hexagonal obstacles and \(d\) denotes the distance between the last row of hexagonal obstacles and the outlet. In the optimization model, these four parameters are mapped to four independent variables \(x_1\), \(x_2\), \(x_3\) and \(x_4\):
$$ x_1 = a, \qquad x_2 = b, \qquad x_3 = c, \qquad x_4 = d $$
The four structural parameters define not only the size of the heat-transfer-enhancing elements but also their arrangement. The hexagonal shape has a streamlined front edge that reduces form drag. The root-like arrangement creates multiple oblique flow paths that disturb the boundary layer without producing excessive blockage. The numerical model therefore captures the interaction between the flow field and the heated wall accurately.
Boundary conditions and numerical setup
I made several simplifying assumptions in the numerical model. The coolant is assumed to be incompressible, steady and Newtonian. The material properties of the fluid and solid are treated as constants because the temperature change is moderate. All outer surfaces of the liquid-cooled plate are adiabatic except the heated upper surface where the traction battery is attached. Gravitational effects and viscous dissipation are neglected, and radiative heat exchange is not considered. The inlet boundary condition is a uniform velocity inlet, while the outlet is a pressure outlet.
The working coolant is water and the plate material is aluminium. Their thermophysical properties are summarized below.
| Substance | Density | Viscosity | Thermal conductivity | Specific heat |
|---|---|---|---|---|
| Water | 998.2 kg/m³ | 0.001003 kg/(m·s) | 0.6 W/(m·K) | 4182 J/(kg·K) |
| Aluminium | 2719 kg/m³ | — | 155.5 W/(m·K) | 3485 J/(kg·K) |
The inlet velocity is 0.2 m/s and the inlet coolant temperature is 303.15 K. The hydraulic diameter of the rectangular inlet is 3.76 mm, which gives a Reynolds number of about 680:
$$ Re = \frac{\rho V D_h}{\mu} $$
Because the Reynolds number is well below the transitional value for internal flow, the laminar viscous model was selected. The SIMPLE-family pressure-based solver is suitable for such low-speed incompressible flows. In the present work, I used the coupled pressure-velocity scheme with second-order upwind discretization for the convective terms. The temperature and pressure fields were monitored during iteration, and the solution was considered converged when the residuals fell below 10⁻⁵ and the maximum temperature and pressure drop no longer changed.
Mesh generation and grid independence
Because the flow domain contains repeated hexagonal obstacles, the mesh must accurately resolve the boundary layers along the obstacle walls. I generated a structured hexahedral mesh using a blocking technique. The two-dimensional block topology was first created in the plane of the liquid-cooled plate, and the three-dimensional volume mesh was then produced by sweeping the block mesh through the 2 mm fluid-domain thickness. Structural hexahedral cells improve numerical accuracy and reduce the total cell count compared with tetrahedral meshes. In the final mesh, more than 88 percent of the cells have a quality greater than 0.9, indicating excellent cell orthogonality and skewness.
To determine whether the numerical results depend on cell count, I generated five meshes with different numbers of cells. The maximum temperature and pressure drop were monitored for each mesh. When the number of cells increased from about 0.99 million to 9.55 million, the maximum temperature changed gradually and the pressure drop also changed gradually. The difference in maximum temperature between the four-million-cell mesh and the nine-million-cell mesh was less than 0.2 °C, while the pressure-drop difference was negligible. The cell count of 4,740,770 was therefore selected as the compromise between accuracy and computational cost. The table below summarizes the grid-independence test.
| Mesh case | Number of cells | Maximum temperature | Pressure drop |
|---|---|---|---|
| M1 | 987,282 | Slightly under-predicted cooling | Reference pressure drop |
| M2 | 1,846,930 | Intermediate value | Slight increase |
| M3 | 2,934,451 | Intermediate value | Small increase |
| M4 | 4,740,770 | Almost unchanged from M5 | Almost unchanged from M5 |
| M5 | 9,548,515 | Reference maximum temperature | Reference value |
Validation of the numerical procedure
Before applying the numerical model to the bionic liquid-cooled plate, I validated the procedure against a previously published experimental study of a liquid-cooled plate with similar dimensions. In that experiment, the plate thickness was 6 mm, the channel thickness was 2 mm and the plate size was 154 mm by 79 mm. The discharge rate was 5C and the corresponding heat flux on the heated surface was 3412 W/m². The coolant inlet temperature was 25 °C and the coolant mass flow rates were 0.5 g/s, 1 g/s, 1.5 g/s and 2 g/s. My numerical setup reproduced the same geometry and solved for the maximum wall temperature at each flow rate. The simulated maximum temperatures agreed with the measured data within 5 percent for all four flow rates. This level of agreement confirmed that the laminar model and the present boundary condition treatment are adequate for predicting the thermal performance of liquid-cooled plates for traction battery thermal management.
Influence of fluid-domain thickness
After the numerical model was validated, I studied the influence of two external parameters on the temperature and pressure-drop characteristics of the bionic liquid-cooled plate. The first external parameter is the thickness of the fluid domain. Since the cooling plate is mounted inside a tightly packed marine traction battery container, the total thickness of the plate must remain small. I kept the total plate thickness constant while varying only the fluid-domain thickness from 1 mm to 4 mm. The inlet velocity and coolant temperature were kept constant, so changing the fluid-domain thickness also changed the coolant mass flow because the inlet area changed.
The results show that the pressure drop decreases monotonically as the fluid-domain thickness increases. The reason is that a thicker channel gives a larger hydraulic diameter and therefore lowers the velocity for a fixed inlet velocity and reduces flow resistance. The maximum temperature, however, first decreases and then increases with increasing fluid-domain thickness. The initial decrease is caused by the larger coolant mass flow and the larger heat-capacity flow rate. The subsequent increase is caused by the thickening of the aluminium wall between the heated surface and the coolant. The extra aluminium acts as a thermal resistance, so the heat path from the traction battery to the coolant becomes less effective. When the fluid-domain thickness is set to 2 mm, the maximum temperature reaches its lowest value, while the pressure drop remains at an acceptable level. The 3 mm and 4 mm cases reduce the pressure drop further, but the temperature penalty becomes significant. A fluid-domain thickness of 2 mm was therefore selected for the remainder of the study.
Effect of coolant velocity
The second external parameter investigated in this section is the inlet coolant velocity. I compared the bionic liquid-cooled plate, the hexagonal liquid-cooled plate and a conventional straight-channel liquid-cooled plate under the same inlet temperature and heat-flux conditions. The inlet velocity was changed from 0.1 m/s to 0.3 m/s in steps of 0.05 m/s. For all three liquid-cooled plates, the maximum temperature decreases as the inlet velocity increases. A larger coolant velocity increases the mass flow rate and intensifies convective heat transfer, so more heat can be extracted from the heated surface. However, the benefit of increasing the velocity diminishes above 0.2 m/s. Because the Reynolds number remains in the laminar regime, the thermal boundary layer becomes thinner but still limits the heat transfer. The reduction in maximum temperature from 0.2 m/s to 0.3 m/s is much smaller than the reduction from 0.1 m/s to 0.2 m/s.
From the viewpoint of energy consumption, pumping power increases rapidly with increasing coolant velocity. Since the pressure drop and mass flow both increase at higher velocities, the additional energy required to drive the coolant is not justified if the temperature reduction is small. Therefore, the inlet velocity of 0.2 m/s was chosen as a balanced operating point for the comparative study.
Orthogonal experimental design
To evaluate the combined influence of the four structural parameters on the thermal and hydraulic performance of the bionic liquid-cooled plate, I used an orthogonal design with four factors and four levels. This method is efficient because it reduces a full factorial design of \(4^4 = 256\) cases to 16 representative cases. The four parameters and their levels are listed in the table below.
| Parameter | Unit | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|---|
| a | mm | 20 | 26 | 32 | 38 |
| b | mm | 8 | 12 | 16 | 20 |
| c | mm | 30 | 40 | 50 | 60 |
| d | mm | 80 | 90 | 100 | 110 |
I constructed the physical model for each of the 16 cases, generated the structured mesh, and performed a numerical simulation under identical inlet conditions: coolant temperature 30 °C, coolant velocity 0.2 m/s and heat flux 785 W/m². For each case I recorded the maximum temperature and the pressure drop. The results are reported below.
| Case | a (mm) | b (mm) | c (mm) | d (mm) | Tmax (°C) | ΔP (Pa) |
|---|---|---|---|---|---|---|
| 1 | 20 | 8 | 30 | 80 | 35.43 | 173.47 |
| 2 | 20 | 12 | 40 | 90 | 36.08 | 158.21 |
| 3 | 20 | 16 | 50 | 100 | 36.14 | 157.52 |
| 4 | 20 | 20 | 60 | 110 | 36.33 | 153.46 |
| 5 | 26 | 8 | 50 | 90 | 35.46 | 171.68 |
| 6 | 26 | 12 | 60 | 80 | 35.88 | 158.92 |
| 7 | 26 | 16 | 30 | 110 | 36.06 | 157.39 |
| 8 | 26 | 20 | 40 | 100 | 36.35 | 157.34 |
| 9 | 32 | 8 | 60 | 100 | 35.45 | 167.94 |
| 10 | 32 | 12 | 50 | 110 | 35.84 | 165.11 |
| 11 | 32 | 16 | 40 | 80 | 36.16 | 170.11 |
| 12 | 32 | 20 | 30 | 90 | 36.13 | 154.75 |
| 13 | 38 | 8 | 40 | 110 | 35.47 | 180.66 |
| 14 | 38 | 12 | 30 | 100 | 35.81 | 158.73 |
| 15 | 38 | 16 | 60 | 90 | 35.91 | 158.69 |
| 16 | 38 | 20 | 50 | 80 | 35.96 | 158.38 |
Examination of the temperature field of the sixteen cases reveals a common pattern. The highest temperature consistently appears at the two corners of the liquid-cooled plate near the outlet. These corners are the last regions to be reached by the coolant and therefore receive the least effective cooling. Besides the numerical difference in maximum temperature, the spatial temperature distribution changes significantly with the structural parameters. A suitable arrangement of hexagonal obstacles creates a more uniform temperature contour and avoids large hot areas at the downstream corners. Therefore, the selection of the structural parameters is essential for achieving a high cooling performance in the marine traction battery system.
Range analysis of the orthogonal results
I analyzed the orthogonal-results table by range analysis, also called the extreme-difference method. For each parameter and each level, I calculated the total value \(k\) and then the average value \(K\). The range \(R\) is the difference between the maximum and minimum average values for that parameter. A larger range means the parameter has a stronger influence on the corresponding objective. The range results are summarized below.
| Response | a | b | c | d | Significance order |
|---|---|---|---|---|---|
| Tmax range | 0.21 | 0.74 | 0.17 | 0.08 | b > a > c > d |
| Pressure-drop range | 3.81 | 17.46 | 6.83 | 4.84 | b > c > a > d |
The transverse spacing \(b\) is the most influential parameter for both the maximum temperature and the pressure drop. The longitudinal spacing \(c\) has a much stronger effect on pressure drop than on temperature, while the hexagonal size \(a\) primarily affects temperature. The distance \(d\) between the hexagonal zone and the outlet has the weakest influence on both responses. This range analysis provides useful insight for structural optimization. A larger transverse spacing tends to give a higher maximum temperature but a lower pressure drop. In other words, there is a clear trade-off between cooling capacity and pumping energy. Because the inlet and outlet geometry remains unchanged and the inlet velocity is constant, the coolant mass flow is the same for all 16 cases. Therefore, the pressure drop can be directly translated into pumping-power consumption, and reducing the pressure drop is equivalent to reducing the energy required to drive the coolant through the plate.
Multi-objective optimization of the structural parameters
To find a structural arrangement that gives both a low maximum temperature and a low pressure drop, a multi-objective optimization must be solved. The two objectives conflict with each other, so there is no unique minimum that simultaneously satisfies both objectives. Instead, the solution is a Pareto front containing a set of non-dominated solutions. I used the NSGA-II algorithm, which is well suited to this problem because it preserves population diversity and converges quickly. The algorithm first requires a mathematical surrogate that relates the design variables to the two objectives. I fitted the results of the 16 orthogonal experiments with a linear regression model. The regression functions for the maximum temperature and the pressure drop are
$$ y_P = 0.224917 x_1 – 1.292 x_2 – 0.07405 x_3 – 0.03645 x_4 + 181.0079 $$
$$ y_T = -0.01108 x_1 + 0.059625 x_2 – 0.0006 x_3 + 0.00245 x_4 + 35.18467 $$
The determination coefficient of the temperature model is about 0.81 and that of the pressure-drop model is about 0.63. Although the pressure-drop regression is less accurate, it still captures the basic trend and is adequate for the preliminary optimization. The constraints of the design variables are
$$ 20 \le x_1 \le 38, \qquad 8 \le x_2 \le 20, \qquad 30 \le x_3 \le 60, \qquad 80 \le x_4 \le 110 $$
Therefore, the multi-objective optimization problem is formulated as
$$ \min F(x_1,x_2,x_3,x_4) = \begin{cases} \min y_T(x_1,x_2,x_3,x_4) \\[2mm] \min y_P(x_1,x_2,x_3,x_4) \end{cases} $$
subject to the above inequality constraints. The NSGA-II parameters were set to a population size of 50, a maximum generation number of 150 and a fitness tolerance of 0.0001. The algorithm produced a clear Pareto front. As expected, the maximum temperature tends to rise when the pressure drop is reduced, and the pressure drop tends to increase when the maximum temperature is reduced. I selected a balanced solution from the middle portion of the Pareto front because this point provides a good trade-off between the cooling requirement and the pumping-energy requirement of the marine traction battery.
The selected optimum corresponds to the following structural parameters: \(a = 30\) mm, \(b = 8\) mm, \(c = 50\) mm and \(d = 90\) mm. The surrogate model predicted a maximum temperature of 35.49 °C and a pressure drop of 170.08 Pa for this geometry. I then reconstructed the geometry with these parameters and performed a full CFD simulation. The simulated maximum temperature is 35.51 °C and the simulated pressure drop is 163.79 Pa. The difference between the surrogate prediction and the CFD result is very small, which confirms the reliability of the optimization procedure. The table below compares the predicted and simulated values.
| Result quantity | Algorithm prediction | CFD simulation | Error |
|---|---|---|---|
| Maximum temperature (°C) | 35.49 | 35.51 | 0.06% |
| Pressure drop (Pa) | 170.08 | 163.79 | 3.7% |
Comparison with conventional liquid-cooled plates
To evaluate the practical advantage of the optimized bionic liquid-cooled plate, I compared it with two reference geometries: a conventional liquid-cooled plate and a hexagonal liquid-cooled plate. The conventional liquid-cooled plate contains straight rectangular flow channels arranged parallel to the flow direction. The hexagonal liquid-cooled plate contains the same hexagonal obstacles as the bionic plate but without the root-like transverse arrangement. The external dimensions, inlet size, outlet size, coolant inlet temperature, coolant velocity and heat flux are identical for all three geometries. Therefore, any difference in temperature or pressure drop can be attributed solely to the internal flow-channel topology.
Pressure-drop comparison
The pressure-drop comparison was performed at a coolant velocity of 0.2 m/s and a coolant temperature of 30 °C. The conventional liquid-cooled plate produces a pressure drop of 190.91 Pa. The hexagonal plate produces a pressure drop of 165.66 Pa, while the optimized bionic liquid-cooled plate produces a pressure drop of 163.79 Pa. The bionic plate therefore reduces the pressure drop by 14.2 percent compared with the conventional plate. The reduction is caused by the hexagonal obstacle shape and by the root-like arrangement, which guides the flow gradually around the obstacles instead of forcing it through long straight rectangular channels. The small difference between the hexagonal plate and the bionic plate shows that the root-like arrangement does not create additional form drag; on the contrary, it slightly reduces the overall resistance. Since the inlet area and velocity are fixed, the mass flow rate is the same for all three configurations. A lower pressure drop therefore directly lowers the pumping power and reduces the parasitic energy loss of the traction battery cooling loop.
Temperature comparison
Under the same boundary conditions, the maximum temperatures of the conventional plate, the hexagonal plate and the optimized bionic plate are 36.20 °C, 36.01 °C and 35.51 °C, respectively. The optimized bionic plate is therefore 0.69 °C cooler than the conventional plate and about 0.5 °C cooler than the simple hexagonal plate. Although the absolute temperature reduction is not extremely large, it is achieved without increasing pressure drop and without changing mass flow. In fact, the bionic plate is both cooler and less resistant than the conventional plate, which demonstrates that the geometric design successfully improves the synergy between the velocity field and the temperature field.
The temperature uniformity also improves. In the conventional liquid-cooled plate, the downstream isotherms bulge strongly in the central region, causing relatively large temperature gradients in the transverse direction. In the optimized bionic plate, the isotherms are more evenly distributed and the high-temperature zone is confined to a smaller region near the two downstream corners. I also extracted the average temperature on several transverse lines normal to the main flow direction. Near the inlet, the conventional plate has a slightly lower average temperature because the conventional straight channels cool the entire width immediately after the entrance. The bionic plate contains a general heat-transfer zone before the enhanced heat-transfer zone, so its inlet section is less effective. However, from the middle section to the outlet, the bionic plate consistently has a lower mean temperature than the conventional plate. This result suggests that the division of the plate into an ordinary heat-transfer zone near the inlet and an enhanced heat-transfer zone in the middle is a useful design strategy for a large-area traction battery liquid-cooled plate.
Velocity-field comparison
The pressure-drop and temperature differences are caused by the different flow patterns. In the conventional liquid-cooled plate, the coolant flows through five parallel straight channels separated by long rectangular ribs. The core velocity in these channels is relatively high and remains almost constant along the whole flow path. The flow direction is always parallel to the inlet velocity. There is no recirculation or secondary flow except in the four corners of the manifold, so heat transfer is governed only by the development of the thermal boundary layer along the straight channel walls.
In the optimized bionic liquid-cooled plate, the flow field is divided into two main regions. Outside the enhanced heat-transfer zone, the flow spreads through the distribution chamber at a moderate velocity. Inside the enhanced heat-transfer zone, the coolant repeatedly encounters hexagonal obstacles and is split into two streams around each obstacle. The local velocity near the obstacles is lower than the core velocity in the straight channel, but the velocity direction changes frequently. The fluid is forced to accelerate on the sides of the hexagonal obstacles and decelerate in the wake behind them. This periodic acceleration and deceleration promotes mixing and periodically interrupts the thermal boundary layer. The velocity contours therefore show a complex but regular network, with the highest velocity near the outlet and near the narrow gaps between adjacent hexagonal obstacles. The alternating high- and low-velocity regions contribute to a more uniform temperature distribution on the heated surface.
Synergy-angle analysis
To explain the improved convective heat transfer more clearly, I used the field-synergy principle. Convective heat transfer is not determined only by the magnitude of velocity and temperature gradient; the included angle between the velocity vector and the temperature-gradient vector also plays an important role. The field synergy angle is defined as
$$ \beta = \cos^{-1}\left[ \frac{\mathbf{U} \cdot \nabla T}{|\mathbf{U}| \, |\nabla T|} \right] $$
Heat-transfer enhancement is favored when the synergy angle is small, which means that the velocity vector and the temperature-gradient vector are nearly parallel or anti-parallel. In addition, a high local velocity should coincide with a high temperature gradient and a small synergy angle. The velocity and temperature profiles should be as uniform as possible across the cross-section to ensure that the whole surface contributes effectively to heat transfer. I calculated the synergy angle in CFD-POST and produced synergy-angle contours for the conventional and bionic plates. In the conventional plate, a synergy angle smaller than 90° appears only at the inlet of each rectangular channel and in the small recirculation zones behind the rectangular ribs. In the bionic plate, the region with synergy angle smaller than 90° is much larger because the coolant encounters many more obstacle leading edges. The root-like arrangement ensures that the boundary layer is repeatedly destroyed and reconstructed after every hexagonal obstacle. Each redevelopment of the boundary layer is accompanied by a fresh thermal gradient and a favorable velocity direction. The synergy-angle analysis therefore provides theoretical support for the observed temperature reduction.
Summary comparison table
The following table compares the three liquid-cooled plates at the common design condition of 30 °C coolant temperature and 0.2 m/s inlet velocity.
| Configuration | Maximum temperature (°C) | Pressure drop (Pa) | Temperature uniformity |
|---|---|---|---|
| Conventional liquid-cooled plate | 36.20 | 190.91 | Baseline |
| Hexagonal liquid-cooled plate | 36.01 | 165.66 | Improved |
| Optimized bionic liquid-cooled plate | 35.51 | 163.79 | Best |
Influence of coolant temperature
For a practical marine traction battery system, the operating condition of the cooling loop changes with the ambient environment and the power demand. I therefore studied the effect of coolant inlet temperature on the bionic liquid-cooled plate. The heat flux was kept at 785 W/m² and the coolant velocity was kept at 0.2 m/s. The coolant inlet temperature was set to 25 °C, 30 °C and 35 °C. The temperature along the flow direction increases continuously from the inlet to the outlet in all three cases because the coolant absorbs heat as it travels through the plate. The inlet-to-outlet temperature difference is almost identical for the three cases and is about 4.17 °C. The maximum plate temperature, however, changes substantially. When the coolant temperature is 25 °C, the maximum plate temperature is about 30.46 °C. When the coolant temperature is increased to 35 °C, the maximum temperature rises to about 40.41 °C. Therefore, the absolute temperature of the marine traction battery module can be shifted by adjusting the coolant inlet temperature, but the temperature rise across the plate is dominated by the heat flux and the coolant mass flow rather than by the coolant inlet temperature.
The transient temperature evolution also depends on the coolant temperature. With a 25 °C coolant, the temperature field reaches its steady state after about 150 s. With a 30 °C coolant, the steady state is reached after about 200 s. With a 35 °C coolant, about 250 s is needed. This means that a lower coolant temperature not only reduces the final temperature of the liquid-cooled plate, but also shortens the thermal response time of the traction battery system. The main findings are summarized in the following table.
| Coolant temperature | Tmax (°C) | Steady-state time (s) | Temperature-control trend |
|---|---|---|---|
| 25 °C | 30.46 | 150 | Strongest cooling |
| 30 °C | 35.51 | 200 | Balanced cooling |
| 35 °C | 40.41 | 250 | Weakest cooling |
Influence of coolant velocity
Coolant velocity is one of the most convenient control variables in a liquid-cooled battery thermal management system because it can be changed by adjusting the pump speed. I performed simulations at coolant velocities of 0.1 m/s, 0.2 m/s and 0.3 m/s while keeping the heat flux at 785 W/m² and the coolant temperature at 30 °C. The results show that the maximum temperature decreases substantially when the velocity increases from 0.1 m/s to 0.2 m/s. A further increase from 0.2 m/s to 0.3 m/s produces only a small temperature reduction. The maximum temperature is about 39.25 °C at 0.1 m/s, 35.51 °C at 0.2 m/s and 34.12 °C at 0.3 m/s. The corresponding inlet-to-outlet temperature differences are 8 °C, 4.17 °C and 2.84 °C, respectively.
The transient steady-state time also changes with velocity. At 0.3 m/s, the temperature field reaches equilibrium after only about 100 s. At 0.2 m/s, equilibrium is reached after about 150 s. At 0.1 m/s, the coolant heats up faster and requires about 250 s to reach equilibrium. A higher velocity therefore makes the thermal management system more responsive to sudden changes in traction battery load. The cost is an increase in pressure drop and pumping power. Since the improvement between 0.2 m/s and 0.3 m/s is relatively small, 0.2 m/s is a reasonable operating point for this marine traction battery application. The table below summarizes the effect of velocity.
| Coolant velocity | Tmax (°C) | Inlet-to-outlet temperature difference | Steady-state time (s) |
|---|---|---|---|
| 0.1 m/s | 39.25 | 8.0 °C | 250 |
| 0.2 m/s | 35.51 | 4.17 °C | 150 |
| 0.3 m/s | 34.12 | 2.84 °C | 100 |
Influence of traction battery heat flux
The heat generation of a marine traction battery is not constant during operation. It depends on the charge and discharge rate, the state of charge and the internal resistance. In order to quantify the robustness of the bionic liquid-cooled plate, I changed the surface heat flux from 400 W/m² to 1200 W/m² while keeping the coolant velocity at 0.2 m/s and the coolant temperature at 30 °C. The maximum temperature increases monotonically with heat flux. At 400 W/m², the maximum plate temperature is about 32.77 °C. At 1200 W/m², the maximum temperature increases to about 38.3 °C. The inlet-to-outlet temperature difference also increases with the heat flux because a larger heat load raises the coolant temperature along its path. The temperature difference is 2.12 °C at 400 W/m², 4.17 °C at 785 W/m² and about 6.37 °C at 1200 W/m². The steady-state time is almost unchanged for heat fluxes below about 785 W/m², but it increases slightly for higher heat fluxes. Over the whole heat-flux range, the optimized bionic liquid-cooled plate is able to maintain the maximum temperature below 40 °C, which satisfies the safety requirement for a traction battery. The results are summarized in the table below.
| Heat flux (W/m²) | Tmax (°C) | Inlet-to-outlet temperature difference |
|---|---|---|
| 400 | 32.77 | 2.12 °C |
| 785 | 35.51 | 4.17 °C |
| 1200 | 38.30 | 6.37 °C |
Discussion
The results show that the internal geometry of a liquid-cooled plate plays a central role in the thermal management of a marine traction battery. The particular contribution of the present bionic design is not that it replaces a straight channel with a single complex channel, but that it combines the shape efficiency of hexagonal obstacles with the topological distribution of tree roots. This combination yields a beneficial division of the flow domain into two regions. The first region near the inlet is relatively open and serves as a distributor, while the second region contains most of the heat-transfer surface and creates repeated boundary-layer renewal. Because the enhanced heat-transfer zone is placed downstream, it receives fully developed flow and disperses the incoming stream over a wide area before the coolant reaches the outlet corners.
One important observation is that the reduction in maximum temperature is not achieved by increasing the velocity of the coolant around the obstacles. In fact, the velocity in the enhanced heat-transfer zone is lower than the velocity in the straight channels of the conventional plate. The better cooling performance is achieved by improving the alignment between the velocity field and the temperature-gradient field, as confirmed by the synergy-angle contours. The concept of field synergy therefore gives designers a useful guide: the geometry should be shaped not to maximize the local velocity, but to produce a flow field in which the coolant moves along paths that actively sweep the heated surface and remove the thermal boundary layer. The hexagonal obstacle shape is particularly favorable because it has no sharp rear corners, which would otherwise create large separated flow zones. The small wake behind each hexagon is enough to regenerate the boundary layer but does not produce excessive pressure loss.
The pressure-drop reduction of 14.2 percent compared with the conventional plate is also important for energy efficiency. In a marine battery cooling system, the coolant pump is a parasitic consumer of battery energy. Every watt consumed by the pump reduces the amount of energy available for propulsion. Therefore, a lower pressure drop is directly beneficial for the overall efficiency of the ship. Since the cooling performance is simultaneously improved, the proposed bionic plate offers a win-win solution rather than a trade-off at the selected operating condition. The multi-objective optimization was necessary because some geometric parameters affect temperature and pressure drop in opposite directions. For example, increasing the transverse spacing \(b\) reduces pressure drop but increases maximum temperature. The optimization found a geometry in which the transverse spacing is small enough to keep the heat-transfer surface dense, while the larger longitudinal spacing and moderate obstacle size prevent excessive flow blockage.
The parametric analysis of coolant temperature, coolant velocity and heat flux provides useful guidance for the design of the control system. In actual operation, the marine traction battery heat-generation rate is determined by the propulsion command, and the coolant inlet temperature is governed by the heat sink, such as seawater or a chiller. The coolant velocity is the main controllable variable. The simulations show that increasing the coolant velocity can suppress a temperature rise caused by an increasing heat load, but the marginal benefit becomes smaller at high velocity. Therefore, a variable-speed pump should be controlled in a stepwise fashion to adapt to the battery load without wasting energy. The data also show that the liquid-cooled plate can keep the traction battery below the recommended upper limit of 40 °C even at a heat flux of 1200 W/m², which gives a safety margin for demanding sailing conditions.
Conclusion
In this study, I designed a novel bionic liquid-cooled plate for a marine traction battery. The plate was inspired by the hexagonal veins of a leaf and by tree-root branching patterns. The thermal and hydraulic characteristics were investigated by numerical simulation, orthogonal experiments, multi-objective optimization and field-synergy analysis. The main conclusions can be summarized as follows.
First, a reasonable division between the ordinary heat-transfer region and the enhanced heat-transfer region significantly reduces both the maximum temperature and the temperature non-uniformity of the cooling plate. The optimized bionic liquid-cooled plate reaches a maximum temperature of 35.51 °C at a coolant temperature of 30 °C, a coolant velocity of 0.2 m/s and a heat flux of 785 W/m². This value is 0.69 °C lower than that of a conventional liquid-cooled plate. The high-temperature zones are confined to the two downstream corners, and the central zone is more uniform.
Second, the hexagonal root-like arrangement in the enhanced heat-transfer region can reduce pressure drop. The optimized bionic liquid-cooled plate has a pressure drop of only 163.79 Pa, which is 14.2 percent lower than that of the conventional liquid-cooled plate. The coolant mass flow is identical because the inlet geometry and velocity are unchanged, so the lower pressure drop directly reduces the pumping energy required by the traction battery cooling system.
Third, the velocity field in the bionic plate is markedly different from that in the conventional plate. The coolant velocity in the enhanced heat-transfer region is lower, but the repeated destruction and regeneration of the boundary layer enhances heat transfer. The synergy-angle contours show that the bionic plate produces a larger region with a favorable synergy angle, which explains why the heat transfer coefficient is higher despite the lower local velocity.
Fourth, the four structural parameters affect the maximum temperature and pressure drop in different ways. The transverse spacing \(b\) has the strongest influence on both objectives. For temperature, the significance order is \(b > a > c > d\). For pressure drop, the significance order is \(b > c > a > d\). The trade-off between temperature and pressure drop can be resolved successfully by the NSGA-II multi-objective optimization algorithm. The selected optimum geometry is \(a = 30\) mm, \(b = 8\) mm, \(c = 50\) mm and \(d = 90\) mm. The CFD result agrees closely with the surrogate-model prediction.
Fifth, the performance of the bionic liquid-cooled plate under different operating conditions is satisfactory. Decreasing the coolant inlet temperature from 35 °C to 25 °C improves the maximum-temperature control and shortens the transient time, while the inlet-to-outlet temperature difference remains almost unchanged. Increasing the coolant velocity from 0.1 m/s to 0.3 m/s reduces the maximum temperature from 39.25 °C to 34.12 °C, but the benefit becomes smaller above 0.2 m/s. Increasing the heat flux from 400 W/m² to 1200 W/m² raises the maximum temperature from 32.77 °C to 38.30 °C. Even at the highest heat-flux level, the proposed liquid-cooled plate can keep the temperature below 40 °C, which satisfies the safety requirement of the marine traction battery.
The present study focuses on a constant property laminar flow and a uniform heat flux. In realistic ship operation, the traction battery heat load is time dependent and may be spatially non-uniform. Future work should therefore include a coupled electrochemical-thermal cell model and transient ship-load profiles. The influence of different coolants, such as water-glycol mixtures, and the effect of manufacturing tolerances on the optimized geometry should also be considered. In addition, the bionic structure may be combined with phase-change material or with a Tesla-valve-like secondary channel to further improve the temperature uniformity of large marine traction battery packs. The field-synergy and multi-objective optimization methodology presented here is general and can be readily adapted to other liquid-cooled heat exchangers.