Performance Optimization of Step-Like Divergence Plenum Air-Cooled Li-Ion Battery Thermal Management System Using Variable-Step-Height Configuration

Several studies on air-cooled battery thermal management systems (BTMSs) have shown that improvement can be achieved through redesign of the BTMSs. Recent studies have achieved improvements in managing the temperature in the system, but mostly with an increase in pressure drop. It is therefore imperative to carry out an extended study or redesign of the existing designs to overcome these challenges. In this work, a standard Z-type BTMS, which has a flat divergence plenum, was redesigned to have a step-like divergence plenum of variable step height. Computational Fluid Dynamics (CFD) approach was adopted to investigate the thermal and airflow performance of the BTMSs. The CFD methodology was validated by comparing its results with experimental data in the literature. Various step height configurations were considered for 3-step and 4-step models. Findings from the result revealed that the variable step height design enhances the cooling performance of the battery pack. For instance, a 3-step model with step heights of 3, 6, and 6 mm offered the least pressure drop and maximum temperature difference, and when compared with the model with a constant step height of 5, 5, and 5 mm, it yielded reductions of 3.4% and 21.6%, respectively. By increasing the inlet airflow velocity, the 4-step cases generally improved. The best cooling improvement was seen in case 26 at velocities over 3.7 m/s for maximum temperature and velocities over 4.8 m/s for maximum temperature difference.


1-Introduction
In recent years, electric vehicles (EVs) have gained increased popularity as a substitute for fossil fuel-driven vehicles in the transportation sector due to the massive release of harmful pollutants such as carbon dioxide by fossil fuel-driven vehicles, which has adverse effects on the environment.One important component of the EV is the battery pack, whose performance has a significant impact on the vehicle's power output, mileage, and market pricing [1].While the battery pack is in operation, a significant amount of heat is produced, which could result in an extremely high battery temperature and a wide variation in temperature among the batteries in the pack.These problems affect the battery's performance and can reduce its lifespan.Lithium-ion batteries are one of the most frequently utilized batteries for EVs because of their great capacity and efficiency.The battery operates best at a temperature between 25 and 40 °C [2], and the battery pack's temperature gradient should not be more than 5 °C [3].Hence, a battery thermal management system (BTMS) that will provide an optimal battery temperature and keep a steady temperature distribution inside the battery pack is required.In that regard, the research and development of BTMS made use of different cooling technologies, such as heat pipe cooling [4,5], liquid cooling [6][7][8], phase change material (PCM) cooling [9,10], and air cooling [11][12][13][14][15].When considering the cost of production, operation, and maintenance, aircooling technologies are the best and more preferred technology globally.However, due to the low specific heat of the air, it becomes vulnerable to large temperature variations and high temperatures inside the battery packs.Hence, researchers made several attempts to improve the air-cooled BTMS's cooling performance, either by adding more components, redesigning its structure, or a combination of the two.Some researchers looked into the effects of the BTMS's inlet and outlet positions [13][14][15][16][17][18], while others investigated number of inlet and outlets [18][19][20][21].For instance, Chen et al. [13] studied BTMSs with various outlet and inlet positions.It was discovered that the symmetrical BTMS with an outlet and inlet in the middle of the plenums yielded improved thermal performance.When the optimized BTMS was compared to the standard Z-type BTMS, the maximum temperature (  ) and maximum temperature differential (∆  ) decreased by 4.5 K and 7.7 K, respectively.In another study, Jiaqiang et al. [16] investigated several air-cooling techniques by using a baffle and varying the distance between the BTMS inlet and outlet.The study reported that by positioning the air flow inlet and outlet locations on different sides of the module, and using baffles, optimum performance of the BTMS was attained.The size and position of the secondary vent's effect on BTMS performance were investigated by Hong et al. [18].Findings from the study indicated that secondary vent's position has a significant impact on the battery pack's   and ∆  .Shahid & Agelin-Chaab [19] used an input plenum as a supplementary inlet with axial airflow in a passive battery pack to enhance temperature uniformity.By evaluating the battery pack's cooling efficiency against a baseline scenario, the average   of batteries reduced by 4% and temperature homogeneity increased by 39%.Zhang et al. [20] further examined the effect of number and size of baffles and secondary outlets on BTMS cooling capacity.Based on the findings from the study, the   and ∆  were minimized after optimization and when compared to the conventional Z-type BTMS, reduced by 1.84°C and 3.66°C, respectively.The cooling channel was also equipped with a baffle according to the initial optimization model, and it was found that when compared with standard Z-type BTMS,   and ∆  were reduced by 4.49 °C and 2.17 °C, respectively.
Other researchers focused on battery spacing distribution [22][23][24] and battery cell architecture [25,26].In a study by Fan et al. [25], a battery system with 32 cylindrical lithium-ion batteries arranged across, laterally, and vertically was investigated.The layout with the best thermal homogeneity and performance was shown to be aligned, then staggered, and finally cross-shaped.The alignment configuration consumed around 23% less energy than the cross arrangement.Similarly, Peng et al. [26] examined the effects of various battery configurations, air intake and outlet locations, as well as the number of outlets and inlets, on a BTMS.They observed that a moderate length-to-width ratio is preferable for improving cooling system performance.Chen et al. [22] modified battery spacing, which was reported to have improved the thermal performance air-cooled BTMS.The study further adopted an optimization approach, which was reported to have significantly improved the performance of the system, resulting in a 42% decrease in the   and a slight decrease in ∆  without increase in the total pressure difference in the system.In another study, battery module with 36 lithiumion cells, was analyzed by Li et al. [23].From the analyses, it was discovered that increasing the overall mass flow rate may cause the profile of the channel mass flow to be more uneven and that a large channel spacing size may worsen temperature stability on the battery walls.Also, when the system's cooling channel size was optimized, the   was reduced from 23.9 K to 2.1 K and the ∆  from 25.7 K to 6.4 K.
Several studies also investigated effect of the inclination of the plenum on the performance BTMS [27][28][29][30].In the study by Chen et al. [25], inlet air temperature and flow rate, and battery pack C-discharge rate were investigated.After optimizing the system, it was found that the cooling performance of the BTMS was significantly improved, resulting in a 42% decrease in the   and a slight decrease in ∆  without increase in the total pressure difference of the system.Similarly, Oyewola et al. [28] investigated the effects of inlet and outlet plenum angles of standard Z and U-type flow BTMS to optimize the flow and thermal distribution in the systems.The results demonstrated that by adjusting the inlet plenum angle to 150° and the outlet plenum angle to 120°, the performance of standard Z-type flow cooling improved.  reduced by 1.6 °C, the ∆  reduced by 3.7 °C, and the power consumption reduced by 29%.The standard Utype flow BTMS's cooling efficiency also improved by keeping the outlet plenum angle at 135° and leaving the inlet plenum angle perpendicular.The power consumption reduced by 11%, the   reduced by 0.4 °C, and the ∆  reduced by 1 °C.
In addition to changing the air-cooling BTMS structure, researchers also attempted to install additional materials such as plates [31,32], heat sinks [33,34], spoilers [35,36], and baffles [16,20] to alter the airflow distribution.Zhang et al. [36] added spoilers to the airflow path of the inlet plenum to enhance its cooling efficiency.The effects of spoiler quantity and placement on the thermal behavior of a BTMS were examined.The   and ∆  were decreased by 1.86 K and 2.51 K, respectively, when compared with the original model.The effects of parallel plate height and length on the thermal efficiency of the BTMS were examined by Wang et al. [17].The plates were placed parallel along the system's cooling channels.Results show that the cooling module with a pair of parallel plates exhibited excellent thermal performance within the allowable power consumption loss range.Also, the impacts of the parallel plate's height and length were examined, with the optimal values being 1.5 mm and 30 mm, respectively.
In order to further improve the cooling performance of existing BTMSs, researchers now adopt the combination of two existing cooling strategies, known as hybrid cooling strategies (HCS), to optimize the performance of the designs with a single cooling strategy [37][38][39][40].For instance, Mousavi et al. [37] adopted a hybrid design that combines phase change material (PCM) and mini-channel cold plates (MCPs).The study reported that the maximum temperature in the battery pack was reduced by 30 K in the optimum system.Additionally, under multiple pulsed heat generation, the difference in averages of maximum temperature was less than 1 K between the two cooling systems.In another study, Yang et al. [38] proposed a hybrid battery thermal management system that provides a compact, lightweight, and energyefficient solution.The results show that the optimum hybrid cold plate design, which only weighs half of the baseline cold plate, can provide more than 50% reduction in the total pumping power while achieving the same cooling performance.More so, Zare et al. [39] developed a new hybrid thermal management system that combines the use of fins and PCM to enhance the performance of the system.The performance of the hybrid BTMS was better when compared to the system with natural air cooling and the PCM cooling system without fins.After the PCM complete melting process, the BTMS with 4 internal-external fins reduced the temperature of the batteries by 9.90 and 17.45 K compared to the PCM cooling system without fins at discharge rates of 3C and 5C, respectively.In order to further enhance the use of hybrid cooling strategies, Khan et al. [40] experimentally studied a hybrid li-ion battery thermal management system with eutectic PCM-embedded heat transfer fluid.The eutectic PCM comprises stearic acid and lauric acid, with a melting temperature of 33.29 °C, a thermal conductivity of 0.356 W/mK, and a latent heat of 151.76 J/g.Experimental results show that with natural air cooling, the maximum temperatures of the battery packs reach 66.9 °C, 57.9 °C, and 45.6 °C when charging and discharging at 2°C, 1.5°C, and 1°C rates, respectively.Furthermore, when compared to natural air cooling at the 2C rate, heat transfer fluid cooling reduced the maximum temperature by 22.42%, eutectic PCM cooling by 40.90%, and hybrid cooling by 46.18%.
Aside from the benefits derived from the HCSs, some researchers are also looking into further optimizing existing single cooling strategy designs, which could later be combined with other designs to further enhance performance of already existing HCSs [41][42][43][44][45]. Shen et al. [44] developed a new BTMS by modifying the Z-shaped conventional design, which was investigated to analyze its thermal performance.By comparing the modified system with the conventional Zshape the maximum temperature of the battery pack was reduced from 38.15 °C to 34.14 °C, which implies a decrease of 10.5%, while the temperature difference was reduced from 2.59 °C to 1.97 °C, which implies a decrease of 23.9%,In the same vein, Oyewola and Idowu [45] installed steps to the divergence plenum of four flow pattern BTMSs.The results revealed that only one of the designs with step installed; yielded improvement, with maximum temperature and maximum temperature difference reduced by 3.2 K and 7.6 K, respectively when compared to the design without step installed.Additionally, Alzwayi & Paul [41] designed a spiral and vertical fin which was used to enhance the performance of thermal performance of the system by reducing the maximum temperature of batteries.Effects of the fin's number, thickness, rotation, length, and position were investigated at various current rates.The orientation of the fin also has a significantly impact on the heat transfer between the cell and air cooling, with the cell temperature rising by 1.5 °C when compared to the half-length of a longitudinal fin.Furthermore, compared to the longitudinal fins, the spiral fins reduce the cell temperature by 3.2%, resulting in a 65.6% reduction in material usage.Chen et al. [42] developed a control strategy for efficient battery thermal management of an air-cooled system.The control strategy of the BTMS is based on the difference in temperature among the batteries, proposed as the J-type flow.The results with high current discharge rate and varying random operating conditions revealed that the developed system ensured the temperature difference was maintained below 0.5 K after several switches of flow type.The average temperature difference among the batteries in the developed system was reduced by more than 67% when compared to J-type flow alone.In another study, Fini et al. [43] experimentally investigated the effect of pressure on the performance of PCM in li-ion battery thermal management system.The results revealed that at discharge of 7 C, the cell was able to discharge for almost twice as long when subjected to the pressure of 500 kPa when compared to under atmospheric pressure.It was further reported that by increasing the pressure from 100 to 500 kPa increased the depth of discharge by almost 10 %, while doubling the extracted energy.
Aside from the benefits derived from HCSs, some researchers are also looking into further optimizing existing single cooling strategy designs, which could later be combined with other designs to further enhance the performance of already existing HCSs [41][42][43][44][45]. Shen et al. [44] developed a new BTMS by modifying the Z-shaped conventional design, which was investigated to analyze its thermal performance.By comparing the modified system with the conventional Z-shape, the maximum temperature of the battery pack was reduced from 38.15 °C to 34.14 °C, which implies a decrease of 10.5%, while the temperature difference was reduced from 2.59 °C to 1.97 °C, which implies a decrease of 23.9%.In the same vein, Oyewola & Idowu [45] installed steps to the divergence plenum of four flow pattern BTMSs.The results revealed that only one of the designs with a step installed yielded an improvement, with the maximum temperature and maximum temperature difference reduced by 3.2 K and 7.6 K, respectively, when compared to the design without a step installed.Additionally, Alzwayi & Paul [41] designed a spiral and vertical fin that was used to enhance the performance of thermal performance of the system by reducing the maximum temperature of the batteries.The effects of the fin's number, thickness, rotation, length, and position were investigated at various current rates.The orientation of the fin also has a significant impact on the heat transfer between the cell and air cooling, with the cell temperature rising by 1.5 °C when compared to the half-length of a longitudinal fin.Furthermore, compared to the longitudinal fins, the spiral fins reduce the cell temperature by 3.2%, resulting in a 65.6% reduction in material usage.Chen et al. [42] developed a control strategy for efficient battery thermal management in an air-cooled system.The control strategy of the BTMS is based on the difference in temperature among the batteries, proposed as the J-type flow.The results with a high current discharge rate and varying random operating conditions revealed that the developed system ensured the temperature difference was maintained below 0.5 K after several switches of flow type.The average temperature difference among the batteries in the developed system was reduced by more than 67% when compared to J-type flow alone.In another study, Fini et al. [43] experimentally investigated the effect of pressure on the performance of PCM in a li-ion battery thermal management system.The results revealed that at discharge of 7 °C, the cell was able to discharge for almost twice as long when subjected to a pressure of 500 kPa when compared to under atmospheric pressure.It was further reported that increasing the pressure from 100 to 500 kPa increased the depth of discharge by almost 10% while doubling the extracted energy.
Different cooling strategies are being developed by researchers in recent years to widen the adaptation and application of BTMS.Tian et al. [46] carried out the design and experimental study of wave-type micro-channel cooling plates for large-capacity marine battery thermal management.The study reported that the designed system with non-linear wavetype microchannel could relief the contradiction between cooling capacity and the temperature uniformity.For all the working conditions, the system could maintain the maximum temperature and temperature standard deviation of batteries below 41.5 °C and 0.96 °C, respectively.Also, Zhang et al. [20] optimized the cooling performance of air-cooling lithium-ion battery thermal management system by adopting multiple secondary outlets and baffle.Findings from the study showed that, when compared with the conventional Z-type BTMS, the maximum temperature (  ) and maximum temperature difference (∆  ) were reduced by 1.84 °C (4.20%) and 3.66 °C (75%) after optimization, respectively.By further introducing a baffle in the cooling channel, the performance of the system was enhanced, and when compared with the conventional Z-type BTMS, the optimized   and ∆  were reduced by 2.17 °C (4.95%) and 4.49 °C (91.89%), respectively.Furthermore, Weragoda et al. [47] conceptualized a new BTMS based on capillarydriven evaporative cooling.In the design, a structure was directly integrated onto the battery's surface to enable direct cooling.Experimental study was carried out by affixing a copper foam to an emulated battery block, and using ethanol and Novec 7000 as the cooling media.Findings from the study revealed that copper foam with higher pore density performed better than the others due to its greater wetting height.More so, the maximum battery surface temperature was maintained around 40 °C for a continuous 50W heat input.
According to the aforementioned literatures, the main factor influencing   and ∆  of a battery pack is uneven airflow distribution.Hence, by modifying the plenum design of the BTMS, the airflow distribution in the system can be altered, thereby enhancing the cooling efficiency of the BTMS.Consequently, several researches redesigned the divergence plenum of Z-type air-cooled BTMS.One of the new designs is the step-like divergence plenum, developed by Oyewola et al. [48].The step-like BTMS was reported to significantly improve cooling efficiency, when compared with the conventional Z-type BTMS.However, the study reported that improvement in the cooling efficiency is often accompanied with increase in pressure drop (i.e. increase in power consumption).In this study, the scope of research on existing step-like divergence plenum design of Z-type BTMS [48], with equal height of steps was extended, by varying the height of steps, with the aim of minimizing both the ∆  and ∆ .Variable step heights optimization was employed and additional studies were carried out to minimize the pressure drop while still achieving reasonable cooling efficiency in terms of maximum temperature in the battery pack.Furthermore, under different inlet flow velocities, the behaviors of each selected case's performance were examined.It is expected that findings from this study will advance system sustainability by providing different concepts of constructing efficient air-cooled BTMS in the future.In summary, the structure of the article comprises; abstract, introduction, methodology, result and discussion, conclusion, and references.

2-1-Physical Geometry
A Z-type air-cooled BTMS system was considered for investigation in this study and depicted in Figure 1.The air and cell properties were obtained from Chen et al. [13] and presented in Table 1.The BTMS consists of nine cooling channels and eight prismatic cells.Each cooling channel has a thickness, d, set to 3 mm, while the dimensions of each battery (length × width × height) are 27 × 70 × 90 mm.For this study, an inlet air temperature and velocity of 299.15K and 3.5 m/s, respectively, were used.According to the findings from the literature [13,30,33], air flow velocities in each cooling channel of the standard Z-type air-cooled BTMS are not uniform.The studies observed that the cooling channels closest to the outlet manifold absorb a significant amount of the airflow compared to those closest to the inlet manifold, which is not suitable for cooling homogeneity and could result in a significant difference in temperature between the cells.To influence the airflow distribution, Oyewola et al. [48] redesigned the divergence plenum into a step-like structure, covering the path that connects all of the cooling channels, thus altering the airflow path and creating an avenue for adequate flow of air to be sucked into the respective cooling channels, which in turn produces better cooling performance.The study focused on constant step heights of 2.5, 4, 5, and 10 mm derived from the height of the divergence plenum and the number of steps: 7 steps, 4 steps, 3 steps, and 1 step, respectively.The height of the step must be an integer to achieve steps of equal height.A detailed description of the design of the steps in the step-like BTMS can be obtained in the work of Oyewola et al. [48].However, the study did not consider variable step height in all the cases.In the current study, variable step heights were considered for BTMS with 4 steps and 3 steps.Figure 2 displays the schematics of a 3-step BTMS model.

2-2-Numerical Solution 2-2-1-Boundary Conditions and Assumptions
The velocity and temperature boundary conditions for the BTMS at the inlet were set to 3.5 m/s and 299.15 K, respectively, while the pressure-outlet boundary condition was assigned with atmospheric pressure as the surrounding pressure.An adiabatic and non-slip condition was assigned to the surrounding wall of the system, while a non-slip condition was assigned to the battery walls.The numerical solution also considered the following assumptions: (1) the physical properties of air and batteries are constant; (2) pressure and temperature remain constant in the surrounding environment; and (3) the batteries were considered to be a constant heat source, having a constant heat generation rate of 11.8 W.

2-2-2-Mesh Generation and Grid Sensitivity Study
A structural mesh with five inflation layers of first layer height (y + ) of 0.1mm at both the system and battery walls, to account for boundary layers, was generated.Based on the settings, a grid sensitivity analysis was carried out to assure a solution-independent grid.Figure 3 shows the selected grid of the standard BTMS.In order assess the sensitivity of the grid, the ∆  and   of the standard BTMS were employed.As the grid number increases above 118,016, as shown in Figure 4, ∆  and   become more stable with deviations less than 0.01 K and 0.02 K, respectively.Hence, a model with a 118,016 mesh size was chosen and used in this study.

2-2-3-Numerical Method
In order to simulate the model's flow and temperature profile, ANSYS Fluent, a CFD solver, was used.The   and ∆  of the BTMS serve as the performance indexes for this investigation.The time-dependent flow problem was solved using the governing equations [49], such as continuity (Equation 1) and momentum Equations 2, while the thermal problem of the airflow was resolved using the energy conservation Equation 3: where p, T, and  ⃗ are the static pressure, temperature, and velocity of the cooling air; , , k, and c are the air properties, which are density, dynamic viscosity, thermal conductivity, and specific heat, respectively.The Reynolds number (Equation 4) was estimated to be 6795 based on the inlet flow velocity of 3.5 m/s and the inlet manifold height.Hence the flow is considered to be turbulent, and an enhanced wall treatment [49] with the standard k- model (Equations 5 and 6) was chosen as the turbulence model.
where       are the turbulent kinetic energy generation caused by mean velocity and the turbulent kinetic energy generation as a result of buoyancy effects, respectively;   is the jth component of the velocity vector;  and   are the molecular and turbulent dynamic viscosities; and   and  are the turbulent kinetic energy and turbulence dissipation rate, respectively;    and   are the source terms of   and , respectively;    and   are the inverse effective Prandtl numbers for   and , respectively; and  1 ,  2 ,   3 are empirical constants.YM represents the influence of the fluctuating dilation incompressible turbulent to the sum of dissipation rates.
For the battery cells, the energy conservation equation [27] (Equation 7) is: where Q,   ,   , and   , represents heat generated, the battery's specific heat capacity, its thermal conductivity, and its density, respectively.
In addition, when solving the governing equations, the solver took into account the SIMPLE algorithm.The centraldifferencing and second-order terms, were employed to discretize the convective and diffusive terms.More so, the flow and energy terms convergence requirements were set at 10 -5 and 10 -8 , respectively.Figure 5 shows the flowchart of the solution methodology.

3-1-Validation of the Numerical Model
The experiment by Chen et al. [13] was carried out for the Z-type BTMS and the ∆  and ∆  were measured, from the second and eighth batteries of the battery pack, respectively.The experiments were further done for three inlet velocities of 3, 3.5, and 4 m/s.In the experiment, the batteries were represented by blocks, while K-type thermocouples were placed on the center of each block to measure temperatures.Simulations were conducted with parameters similar to those of the experiment, to test the validity of the numerical model.A comparison of the simulation results and experimental results are shown in Figure 6.The ∆  and ∆  were found to have average errors of 0.9 K and 0.6 K, respectively, which falls within an acceptable range, thus confirming the validity and acceptability of the numerical solution procedure employed.

Fail
Grid Sensitivity study The standard model's temperature and velocity contours are displayed in Figures 7-a and 7-b, respectively.The   , ∆  , and ∆ values are328.86K, 6.96 K, and 22.15 Pa, respectively.It can be seen from Figure 7-a that the first four batteries in the battery pack have a very high temperature compared to the remaining batteries as a result of an uneven distribution of airflow into the cooling channels leading poor thermal homogeneity and decreasing the BTMS's ability to effectively cool the system.

3-2-Influence of Variable Step Height on The Temperature of the Batteries
Generally, the variable height configuration was adopted based on the sum of the step heights for the conventional BTMS models for the 3-step model, the total step height in the divergence plenum is 15 mm, while that of the 4-step model is 16 mm.The basic principle taken into consideration is the reduction in cross-sectional area of air passage due to the step design.To evaluate the impact of variable step height on the system, the pressure drop is being target for reduction, due to the resistance caused by the steps.

3-2-1-The 3-step BTMS Model
As stated earlier, three different step heights were selected and assigned to each height of the 3-steps model, while ensuring that the sum of the heights is 15 mm.The first, second and third heights are denoted as H1, H2 and H3, respectively as shown in Table 2. Case 0 is the initial configuration of constant step height for the 3-step model having   , ∆  , and ∆ values of 325.28 K, 1.15 K, and 26.63 Pa, respectively.The temperature contour of case 0 is presented in Figure 8.By observing the temperature contour for standard BTMS (Figure 7-a) and the case 0 BTMS (Figure 8), it can be clearly seen that the latter represent uniform temperature distribution on the battery cells, which indicates thermal homogeneity.Case 1 to Case 13 represent the various step height configurations for this model.As shown in Table 3, cases 1, 2, 10, and 13 show better cooling performance and lower power consumption in terms of thermal homogeneity and pressure drop, respectively, when compared to case 0. These four cases show an improvement from the performance of case 0, with case 1 having ∆  and ∆ values of 0.9 K and 25.72 Pa, respectively.When considering   , the reduction, it was insignificant.Also, when compared to the standard Z-type model, Case 1 performed far better with a reduction of 3.58 K and 87.1% in terms of   and ∆  , respectively.As shown in Figure 9, airflow distribution in each cooling channel is influenced by the height of the steps, especially that of the first and second steps, which cover cooling channels 3 to 7. The first two cooling channels, denoted as index 1 and 2, respectively in Figure 9-a, and the last two cooling channels, denoted as index 8 and 9, respectively, of the four cases behaved in a similar pattern.The second step is higher than the first step in all four configurations, and varying the height of the steps with the consideration of H2 > H1 and with H1 not set relatively high can boost the airflow in the cooling channel at the mid-section of the BTMS.The general performance of the 3-step model as presented in Table 3, revealed series of variation in the predicted values of   , ∆  and ∆.Basically, redesign of the divergence plenum is done to minimize the   while the battery is in operation.This has been achieved by many researchers, and often accompanied by reduction of ∆  , signify thermal homogeneity among the batteries in the battery pack.In the same vein, reduction in the ∆  also comes with increase in ∆.However, a design with the minimum   may not produce the best thermal homogeneity, when compared with another design.For instance, in table 3, BTMS with equal height of step, i.e. case 0, produced the minimum value of   = 325.28, while ∆  = 1.15  and ∆ = 26.63.Considering pumping power consumption, case 0 is not considered economical when compared with case 1 with ∆ = 25.72 .For case 1, the   is 325.44 K, which represent a difference of 0.12 K when compared to case 0. The temperature difference between   of case 0 and case 1 is insignificant, however the difference in ∆ cannot be overlooked as cost of pumping will certainly vary.Cases such as cases 2, 3 and 10 produced lower values of ∆  than the initial case 0 with equal height of steps.Similarly, cases 2, 7, 10, 11 and 13 produced lower values of ∆ when compared to case 0. These findings will be of great significance on systems were improvement in thermal homogeneity and reduction in pumping cost are of primary concern.

3-2-2-The 4-step BTMS Model
For the 4-step models, the sum of the step heights for each model is 16 mm Table 4 listed the variable height configurations for the models.Cases 15 to 32 show the various step height configurations, while case 14 represents the initial 4-step model with constant step height, with   , ∆  , and ∆ values of 325.19 K, 0.95 K, and 26.48 Pa, respectively.Figure 10 shows the temperature contour of case 14.Similarly, by observing the temperature contour for standard BTMS (Figure 7-i) and case 14 BTMS (Figure 10), it can be clearly seen case 14 produced a better thermal homogeneity.The variables H1, H2, H3 and H4 represents the heights of the first, second, third and fourth steps, respectively.As shown in Table 5, none of the cases shows improved performance when compared to the default height configuration (case 14).Regarding thermal homogeneity, no case outperformed case 14, though, some cases performed slightly better in the area of   and power consumption, however, their improvements are relatively insignificant.It is worth noting that case 26 (3-5-4-4) having   , ∆  , and ∆ values of 325.32 K, 0.99 K, and 26.18 Pa, respectively performed relatively similar to case 14 with little improvement in power consumption only.However, when compared to the standard BTMS model, case 26 outperformed with an improvement of 3.54 K and 85. 8% in terms of   and ∆  , respectively.This performance shows improvement in design when compared to the work of Zhang et al. [20], where   and ∆  were reduced by 1.84 and 3.66 K, respectively.Although as seen from the table, case 21 performed better than all the cases when considering   , however it does not give the best performance in terms of ∆  and ∆.Figures 11-a and 11-b display the similarity observed in the performances of case 14 and case 26.

3-3-Effect of Inlet Velocity on the Performance of Selected Cases
For some selected cases, such as cases 0, 1, 14, and 26,   of the cases decreases as the flow velocity increases.Though Case 0 offers the lowest   for all velocities, Case 26 significantly improved in   , at a higher velocity surpassing case 14 at velocities above 3.7 m/s as seen in Figure 12-a.For all selected cases and across all velocities, a reduction is seen for the maximum temperature difference.Case 0 provided the highest ∆  across all studied velocities, however, when compared to case 1 at higher velocities (> 4.8 m/s), case 0 performs better.It is worth noting that the reduction observed in the ∆  for case 1 occurs at a slow pace across the studied velocities.Considering the behavior of cases 14 and 26, it was observed that case 14 performed better except at velocities above 4.6 m/s, where case 26 outperformed it.As shown in Figure 12, case 26 improved greatly at higher velocities above 4.6 m/s than the other cases.Generally, for cases with variable step height settings, case 26 provides better performance and improvement in terms of   than case 1 across all velocities, and also, for the ∆  , case 1 outperformed case 26 at lower velocities but performed woefully at velocities above 4.4 m/s.From figure 12-b, it can be observed that case 26 will provide the best thermal uniformity at higher velocities than the other cases.

4-Conclusions
In this study, the scope of research on the existing step-like divergence plenum design of Z-type BTMS [48], with equal height of steps, was extended by varying the height of steps.Studies were carried out to minimize the ∆ while still achieving reasonable cooling efficiency in terms of reducing the maximum temperature of the battery in the battery pack.The CFD methodology was adopted, which was validated by comparing estimated ∆  and   with experimental data from the literature.Two types of step-like models; 3-step and 4-step, each with varying step heights, were considered.Furthermore, under higher inlet air velocities, the performance of each case was examined.The following findings were made from this study: • For a three steps BTMS (3-step model), design with step heights 3, 6, and 6 mm (case 1) offered the minimum ∆ and ∆  , and when compared with the model with constant step height of 5 mm (case 0), yielded reduction by 3.4% and 21.6%, respectively.
• The designs with four steps did not show any significant improvement.However, the configuration with step heights 3, 5, 4, and 4 mm (case 26) showed a relatively similar trait to that of the default model with constant step height of 4 mm (case 14).
• By increasing the inlet airflow velocity, the cooling efficiency of the four steps BTMS cases was improved.The   and ∆  values of all the systems showed a downward trend, however, ∆  for case 1 began to perform poorly at greater inlet flow velocities.The best cooling improvement was seen in case 26 at velocities over 3.7 m/s for   and velocities over 4.8 m/s for ∆  .

Table 4 .
Variable height arrangement for the 4

Figure 11 .Figure 12 .
Figure 11.Airflow and temperature variation of cases 14 and 26 of the 4-step BTMS model

Table 5 . Performance evaluation of various step height settings for a 4-step BTMS model
and   Turbulence kinetic energy generation due to the average velocity gradients and buoyancy effects respectively   Contribution of fluctuating dilatation in compressible turbulence to overall dissipation rate   ,   ,   Model constants   and   Source terms