Adaptive, Demand-Driven Thermal Management of Battery Packs via Branch-Level Flow Allocation

Abstract

Second-life lithium-ion batteries offer strong potential for sustainable stationary energy storage, but their practical reuse is limited by cell-to-cell heterogeneity, non-uniform heat-generation, and the resulting thermal safety risks. Conventional battery thermal management systems (BTMSs), which rely on fixed and uniformly distributed coolant flow, are not well-suited to the asymmetric thermal behaviour of aged battery packs. In this study, an adaptive liquid-cooling framework with locally regulated branch-level flow allocation is proposed for second-life prismatic LiFePO₄ battery modules. A three-dimensional transient conjugate heat transfer model was developed in COMSOL Multiphysics. The analysis was conducted on a 3 × 3 battery module under nine thermal heterogeneity scenarios, followed by a larger 5 × 4 module to evaluate scalability. The results show that thermal severity depends not only on heat-generation magnitude but also on the spatial arrangement of degraded cells. Under the most critical 3 × 3 configuration, the adaptive BTMS reduced the maximum temperature from 37.16 °C to 28.77 °C, corresponding to a reduction of about 8.38 °C, while limiting the cell-to-cell temperature difference to approximately 1.16 °C. A comparison with a conventional constant-flow cooling configuration in the larger 5 × 4 module further showed that adaptive branch-level coolant redistribution improves thermal uniformity under heterogeneous thermal loading by selectively directing cooling capacity toward thermally stressed regions. The results demonstrate the potential of demand-driven flow allocation as a distributed thermal-management strategy for heterogeneous second-life battery systems.

1. Introduction

The rapid expansion of renewable energy systems and electrified transport has substantially increased the demand for reliable electrical energy storage [1]. Lithium-ion batteries have emerged as the dominant storage technology due to their high energy density, long cycle life, low self-discharge, and mature production ecosystem [2,3,4]. As a result, large volumes of lithium-ion batteries are now reaching the end of their first life in electric vehicle (EV) applications, driven not only by capacity fade but also by warranty limitations, performance requirements, and safety considerations.

Market analysis projects continued strong growth in lithium-ion battery deployment over the coming decade [5,6]. This growth directly implies a rapidly increasing stock of EV batteries that retain substantial remaining capacity but no longer meet automotive performance requirements. Recycling alone does not adequately resolve this challenge. While recycling technologies recover valuable materials, reported cost differences between recycling batteries at high and low states of health (SOH) remain relatively limited [7], and premature recycling of high-SOH batteries eliminates functional capacity that could otherwise be reused. In parallel, capacity–retention thresholds around 80% are often used in practice as indicative benchmarks for continued serviceability in less demanding applications, although the specific criteria depend on battery chemistry, application, and evaluation framework [8,9,10,11,12,13,14,15,16,17,18,19,20,21].

Based on prior studies [9,10,11,12,13,14,15,16,17,18,22], lithium-ion second-life batteries (SLBs) have strong potential for stationary energy storage applications, including grid support, renewable energy integration, and building-level energy management. In practice, second-life EV batteries may be repurposed either at the cell level or at the module level, depending on the condition of the returned packs, the intended application, and the required level of disassembly and requalification. Although the present study focuses on cell-level thermal heterogeneity within a module, the proposed demand-driven cooling methodology is equally applicable to repurposed module-level configurations that exhibit non-uniform thermal loading.

Although second-life battery systems often operate at moderate average C-rates in stationary storage applications, some emerging use cases may expose the batteries to intermittent high-power demand conditions and localised thermal stress. One important example is grid-support battery energy storage deployed alongside EV charging infrastructure, where repurposed battery systems may be charged during low-demand periods and subsequently deliver high transient power during peak charging events. Such applications are particularly relevant in regions with constrained grid infrastructure, where stationary BESS units may help avoid costly grid upgrades. Similar transient high-power conditions may also arise in future electrified heavy-duty transport and industrial machinery applications. Therefore, thermal-management systems for second-life batteries should also be evaluated under elevated thermal-load conditions rather than only nominal operating scenarios.

However, the literature also consistently identifies unresolved technical and safety challenges that prevent SLB solutions from being deployed at scale. These challenges are summarised in

Table 1. Key challenges in SLB deployment.

Challenge	Description	References
Thermal Stability	Battery degradation can lead to performance loss, thermal imbalance, and an increased risk of thermal runaway—a rapid, uncontrollable temperature increase that can result in fire or explosion.	[23,24,25]
Cell Inhomogeneity and State Estimation	As batteries age, their capacity to store charge decreases, and their internal resistance increases, leading to inefficient power delivery and overheating risks. Variations between individual cells can create imbalances that reduce system efficiency and may cause safety issues.	[23,24]
Compatibility	Different manufacturers, chemistries, and electrical attributes can complicate mixing and matching SLBs.	[23,24]
Lack of Standards and Regulatory Gaps	There is no universal framework for evaluating, classifying, or reusing SLBs, and regulations on transport, storage, and safety remain incomplete.	[12,17,26]

, based on prior studies and industrial experience.

As shown in Table 1, four interrelated challenges dominate SLB deployment: thermal stability, cell-level inhomogeneity and state estimation, compatibility across chemistries and manufacturers, and the absence of standardised evaluation and regulatory frameworks [12,17,23,24,25,26]. Among these, thermal instability and cell-level inhomogeneity are consistently identified as the most safety-critical factors, as they directly influence failure escalation mechanisms and the risk of thermal runaway [23,24,25].

In practical second-life battery applications, cells or modules are typically screened, graded, and grouped according to their electrical and thermal characteristics before reuse. Consequently, repurposed battery systems do not necessarily consist of severely degraded or randomly mismatched cells. However, even in carefully matched second-life systems, residual heterogeneity remains due to manufacturing tolerances, ageing history, thermal exposure, and differences in prior operational conditions. These variations may still produce non-uniform heat generation and localised thermal asymmetry, particularly under transient high-power operating conditions. The present work, therefore, focuses on the thermal-management implications of heterogeneous thermal loading rather than on detailed electrochemical degradation modelling or prediction of resistance evolution.

Even new lithium-ion battery packs exhibit cell-to-cell variability because microstructural differences cause uneven capacity fade and resistance growth; simulations show that a 1 µm difference in active-material particle size can alter ageing rates and current sharing, with lower-resistance cells carrying higher currents and degrading faster. Parallel and series configurations amplify these differences, and temperature or SOC imbalances exacerbate them; for example, imposing a small SOC mismatch can make one cell draw 60% more current than expected [27].

In second-life packs, the spread is far larger: experimental studies on retired or field-aged LiFePO₄ cells have reported that internal resistance can remain low (~0.5–0.6 mΩ) for 10,000 cycles but may rise steeply once the state of health falls below 60%. Resistance differences of 30–60% have also been measured across cells, reducing round-trip efficiency. Retired modules often lose 20–30% of capacity due to cell degradation and another 20–25% due to balance issues, highlighting the need for matching and balancing [28]; sorting cells by internal resistance and placing the highest-resistance cells at module ends has been shown to reduce voltage deviation without extra hardware. Because cells with higher resistance dissipate more heat, mismatched second-life packs develop thermal gradients that accelerate degradation, and state-of-health estimation for repurposed batteries therefore relies heavily on tracking internal-resistance rise and capacity fade through impedance measurements [29].

The operating temperature of a lithium-ion battery module and the temperature variation between individual cells significantly affect the cycle life, usable capacity, and safety of the battery module. During operation, considerable heat is generated due to exothermic reactions and internal resistance [30,31]. Thermal gradients accelerate heterogeneous degradation. A recent review of hybrid cooling technologies emphasised that lithium-ion batteries should operate between 25 °C and 40 °C, and the cell-to-cell temperature difference should be kept below 5 °C [32]. Uneven heat generation and insufficient cooling create local hotspots that degrade the most vulnerable cells first, leading to the “weakest-link” effect. Managing temperature and thermal uniformity is therefore critical to improving battery life and avoiding safety incidents [33].

Battery energy storage systems (BESSs) also face important thermal challenges in cold environments, where low temperatures reduce electrochemical performance, increase internal resistance, and may require active heating to maintain safe and efficient operation [34,35,36,37].

Accordingly, effective battery thermal management systems (BTMSs) are essential for safety and performance [38]. BTMSs can be categorised into passive systems (relying on natural convection or conductive materials), active systems (using forced air or liquid coolant) or hybrid systems that combine both [39,40].

Air-cooled BTMSs offer simple design and low cost but provide limited heat dissipation and poor temperature uniformity [32,41,42,43]. Liquid-cooling, implemented through cold plates or immersion cooling [44], provides higher thermal conductivity and improved temperature uniformity [43]. Phase change material (PCM) systems exploit latent heat to buffer temperature rise [45], and coupled or hybrid BTMSs combine two or more methods (e.g., liquid cooling with PCMs or heat pipes) to balance efficiency, cost, and weight. Recent hybrid designs use composite fins, heat-pipe augmentation or topology-optimised channels to further reduce peak temperature and pumping power. In microchannel cold plates, serpentine or hexagonal channels provide strong cooling but incur high pressure drop, whereas pumpkin-shaped or streamline channels lower the pressure drop. Researchers also demonstrated that increasing the number of cooling plates and optimally placing them reduces peak temperature and temperature variation, and topology optimisation can refine channel geometry to enhance cooling while minimising flow losses [46].

Air-cooled systems are extensively utilised in residential and small-scale commercial applications, such as the Tesla Powerwall, whereas liquid-cooled systems are predominantly employed in large-scale energy storage projects, including Hornsdale Power Reserve (Australia), Tesla Megapack installations, Mira Loma BESS (USA), Noor Power Station (Morocco), and the Microsoft data centre in Ireland [38].

An optimised BTMS expands driving range or energy efficiency by up to 25% and prevents power derating by maintaining cells within an ideal temperature window [39]. However, most existing BTMSs are designed for uniform cell populations and cannot address the heterogeneity and thermal imbalance inherent in SLB systems [47,48]. In second-life BESS, coolant flow in conventional designs cannot be precisely matched to the heterogeneous heat generation of individual cells. Reviews of cylindrical battery packs note that the flow rate of liquid coolant cannot be aligned with varying cooling requirements, and researchers have proposed nonlinear optimisation strategies to tune coolant flow based on heat generation [49]. There is debate over whether uniform cooling suffices or whether cell-level control is necessary. Passive BTMS favour simplicity and low cost but may lack the heat dissipation capacity needed for high-power or aged cells, while active and hybrid systems offer better control at the expense of complexity [39]. The optimal approach for second-life packs remains an active research question.

Conventional BTMSs typically rely on fixed or uniformly distributed coolant flow through the module. However, due to heat exchange along the flow path, coolant temperature increases as it travels, leading to different thermal conditions between upstream and downstream cells. This flow-path-dependent behaviour, combined with cell-to-cell heterogeneity, makes uniform cooling inherently mismatched to the underlying thermal dynamics. In this work, we formulate battery thermal management as a demand-driven flow-allocation problem, in which thermal-management capacity is dynamically distributed across parallel branches according to local cell-level thermal demand.

The novelty of the present work is not the use of liquid cooling itself, since liquid-cooled BTMSs are already well established, but rather the reinterpretation of battery thermal management as a demand-driven thermal-distribution problem. The proposed approach adapts a principle commonly used in hydronic building heating and cooling systems, where coolant is redistributed among parallel branches according to local thermal demand. In the present battery application, independently regulated parallel cooling branches are used to redistribute thermal-management capacity according to local cell-level temperature conditions, rather than relying on fixed or uniformly distributed coolant flow.

While distributed thermal-management principles are well established in other engineering domains, their application to heterogeneous battery systems remains largely unexplored. Battery modules introduce additional challenges due to strong thermal coupling, rapid transient behaviour, and safety-critical operating conditions. The present work, therefore, investigates the thermal feasibility and behaviour of adaptive branch-level coolant redistribution using a simulation-based proof-of-concept framework. The focus is on evaluating the resulting thermal behaviour under heterogeneous thermal loading rather than on developing a fully optimised real-time control architecture. Experimental validation, hydraulic optimisation, sensing architecture, and advanced controller development are left for future work.

The main research question addressed in this work is: how can branch-level coolant redistribution improve thermal uniformity in battery modules subjected to spatially heterogeneous heat generation? The specific contributions of this study are: (i) formulation of battery thermal management as a demand-driven branch-level flow-allocation problem; (ii) definition and comparison of multiple cell-level thermal heterogeneity scenarios to identify the influence of spatial arrangement on hotspot severity; (iii) demonstration of adaptive coolant redistribution in both 3 × 3 and 5 × 4 battery-module configurations; and (iv) clarification of the limitations and future requirements for hydraulic implementation, sensing, controller development, and experimental validation.

The paper is structured as follows. Section 2 presents the mathematical formulation of the thermal model and the governing equations for heat transfer and coolant flow, together with the numerical implementation in COMSOL Multiphysics 6.4. The modelling assumptions, boundary conditions, and the branch-level adaptive flow-allocation strategy are also described. Section 3 discusses the thermal performance of the proposed BTMS under different heat-generation scenarios representing second-life heterogeneity, including temperature distribution, peak temperature reduction, and cell-to-cell temperature variation. Finally, Section 4 summarises the main findings and highlights the implications of branch-level adaptive cooling for safe and reliable deployment of SLB energy storage systems.

2. Methodology

2.1. Physical Description

A liquid-cooled prismatic LiFePO₄ battery module is considered in this study (Figure 1). In practical SLB applications, cells exhibit variations in internal resistance and degradation state, which lead to non-uniform heat generation at the module level and asymmetric thermal loading. Similar non-uniform thermal behaviour may also arise when second-life EV batteries are repurposed and reused at the module level rather than fully disassembled to the cell level. The same adaptive cooling concept, therefore, remains applicable in such configurations. The main investigation, therefore, considers a nine-cell module subjected to nine thermal heterogeneity scenarios to examine the effect of cell-to-cell non-uniformity on the thermal response of the system. Additionally, a 20-cell module arranged in a 5 × 4 configuration is considered in the final stage of the study to provide a clearer demonstration of coolant flow redistribution in a larger module. Each cell has dimensions of 63 mm × 118 mm × 13 mm and a rated capacity of 7 Ah.

In the present model, each prismatic LiFePO₄ cell is represented as a three-dimensional module-scale thermal domain. The external geometry of the cells, cooling plates, and coolant channels is modelled in three dimensions, and heat conduction in the solid domains together with conjugate heat transfer to the coolant is solved in the full 3D geometry. Therefore, the model is a three-dimensional module-scale thermal model rather than a pseudo-2D representation. The internal electrochemical structure of the cell, including the electrode stack or jelly roll winding, separator layers, tabs, and local current-density distribution, is not explicitly resolved. Instead, each cell is treated as a homogenised solid domain with effective thermophysical properties and prescribed volumetric heat generation [50,51,52,53,54,55,56]. The temperature signal used for each cell was calculated as the volume-averaged temperature of the corresponding solid cell domain [57,58,59,60], rather than as a point measurement.

An aluminium cold plate, sandwiched between two consecutive cells and containing coolant flow passages, is modelled in COMSOL Multiphysics 6.4. The channel type is rectangular, with dimensions 63 mm × 118 mm × 2 mm, flow depth 1 mm, channel width 4 mm, and number of channels 7.

The proposed cooling structure is designed to enable branch-level flow regulation, allowing the coolant to be redistributed according to local thermal demand. Each cooling branch operates as a locally regulated thermal unit, avoiding the sequential flow-path dependence present in conventional shared-flow configurations.

Unlike configurations where coolant passes sequentially through multiple cells, the proposed system employs locally regulated parallel branches, rather than relying on a shared sequential path. A conceptual schematic illustrating the parallel, independently regulated structure of the proposed BTMS is shown in Figure 2.

The cooling system is modelled as a closed-loop circuit driven by a circulator pump and coupled to a heat-rejection unit (e.g., heat exchanger or reservoir). The coolant leaving the pump and entering the parallel branches is referred to as the supply flow, while the fluid exiting the branches constitutes the return flow, providing a convenient location for temperature monitoring and flow regulation. Although physically part of a continuous loop, distinguishing between supply and return improves clarity in describing the flow distribution and thermal behaviour. Each cooling branch operates as an independently regulated thermal unit arranged in parallel, enabling localised regulation of heat transfer. Unlike conventional configurations where coolant flows sequentially through multiple cells, the proposed architecture eliminates flow-path-dependent thermal imbalance by decoupling the thermal behaviour of individual cells.

The proposed concept is inspired by hydronic thermal distribution systems commonly used in building heating and cooling applications, where a central circulating loop supplies multiple parallel branches and local thermal demand is regulated through branch-level flow control. In the present battery application, the cooling branches should therefore be interpreted as locally regulated parallel flow paths within a common coolant circulation system rather than as independently pumped subsystems. The branch-level flow-allocation BTMS redistributes coolant flow according to local thermal demand using controllable branch-level flow regulation, analogous to temperature-controlled radiator circuits in hydronic systems.

The model assumes the following:

• The volumetric heat generation is spatially uniform within each individual cell; however, different heat-source magnitudes are assigned to different cells to represent cell-to-cell heterogeneity.
• In the adaptive-flow configuration, the coolant mass flow rate in each parallel cooling branch is independently regulated and dynamically varied, rather than applying a single fixed flow rate to the entire module.

2.2. Mathematical Model of BTMS

2.2.1. Validation

A baseline 3D thermal model was first developed and validated against the reference study of Monika et al. [61] for a liquid-cooled prismatic LiFePO₄ battery module under 5C discharge. In the reference study, a five-cell module was analysed using mini-channel cold plates placed between consecutive cells, with five cooling channels of 4 mm width, an inlet coolant temperature of 25 °C, and an ambient temperature of 25 °C under fixed-flow cooling conditions. Their results showed that the proposed cooling configuration could effectively regulate the module temperature and provide a relatively uniform longitudinal thermal distribution.

Figure 3 compares the temporal evolution of the average battery temperature obtained from the present baseline model and the reference study. As observed, the predicted temperature profile closely follows the reported trend throughout the entire discharge period.

The deviation in average temperature over the 0–720 s interval remains below 0.2% relative to the overall temperature rise. This close agreement supports the correct implementation of the governing equations, boundary conditions, and numerical settings. Therefore, the validated baseline model was used as the basis for the extended simulations in the present study, including spatially non-uniform heat-generation conditions and demand-driven branch-level flow allocation.

2.2.2. Boundary Conditions

The cooling system is modelled as a multi-branch configuration in which each cooling plate is assigned an independent coolant inlet mass flow rate, ${\dot{m}}_i$. Although the present proof-of-concept model does not impose a fixed total-flow or pumping-power constraint, the intended physical interpretation is redistribution within a common circulating coolant loop rather than unrestricted independent flow generation in each branch. The adaptive branches should therefore be interpreted as locally regulated flow paths, for example, using controllable valves or variable hydraulic resistance elements, which redistribute available coolant flow according to local thermal demand. The purpose of the present model is to investigate the resulting thermal behaviour of demand-driven flow redistribution rather than to optimise the hydraulic system itself. Consequently, the present study does not include detailed manifold design, valve-loss modelling, pump operating curves, or parasitic pumping-power analysis. These aspects should be addressed in future prototype-level investigations.

Each branch is allowed to vary independently between ${\dot{m}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\dot{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ according to the corresponding battery temperature signal. Therefore, the cooling flow rate is not constant in the adaptive configuration, and the coolant velocity in each branch varies according to the imposed branch mass flow rate and the channel cross-sectional area. To verify the laminar-flow assumption, the Reynolds number was calculated at the prescribed maximum branch mass flow rate. This conservative check gave $\mathrm{R}\mathrm{e}\approx 2040$, which is below the conventional laminar-flow threshold of 2300.

This formulation allows non-uniform flow distribution across the module, in contrast to conventional single-inlet configurations. Because the module consists of alternating battery cells and cooling plates, the intermediate cells are thermally coupled to two adjacent cooling plates, whereas the outer cells are cooled from one side only.

At the channel outlets, a static pressure boundary condition of $p_0=0$ Pa gauge pressure is imposed, corresponding to atmospheric pressure at the outlets. The coolant is assumed incompressible, and the flow remains in the laminar regime throughout the operating range. The no-slip condition is applied at all solid–fluid interfaces.

The inlet coolant temperature is fixed at 25 °C. The initial temperature of the entire computational domain is also set to 25 °C.

Table 2. Boundary conditions and operating parameters for the demand-driven BTMS simulations.

Parameter	Value
Coolant	Water
Coolant inlet temperature (°C)	25
Ambient temperature (°C)	25
Discharge rate (C)	5
Rated cell capacity (Ah)	7
Equivalent discharge current (A)	35
Simulation time (s)	720
Maximum branch mass flow rate, ${\dot{m}}_{max}$ (kg s−1)	0.005
Minimum branch mass flow rate, ${\dot{m}}_{min}$ (kg s−1)	0.0001
Outlet pressure (gauge) (Pa)	0
Flow regime	Laminar (Re < 2300)

summarises the selected boundary conditions and operating ranges adopted in this work.

For the 7 Ah cell considered in this study, the 5C condition corresponds to an equivalent discharge current of 35 A and a full-discharge duration of 720 s. In the present module-scale thermal model, this condition was represented by prescribed constant volumetric heat generation inside the battery-cell domains.

Thermally, the external surfaces of the module are modelled using a convective boundary condition with ambient temperature $T_{amb}$. Radiative heat transfer is neglected.

In the no-BTMS reference cases, no cooling channels or active coolant flow were included, while external convective heat exchange with the ambient environment was retained.

Within the solid domains, volumetric heat generation is imposed inside each battery cell. The heat source is spatially uniform within an individual cell; however, its magnitude varies from cell to cell in order to represent asymmetric thermal loading.

The volumetric heat-generation rates used in the adaptive heterogeneity simulations were prescribed as effective thermal-load levels. They are not intended to represent a unique electrochemical ageing pathway or directly measured cell-specific heat-generation values. Instead, they were used to create controlled cell-to-cell thermal heterogeneity and to evaluate the response of the adaptive flow-allocation strategy under spatially non-uniform thermal loading. The prescribed heat-generation levels and material properties used in the simulations are summarised in

Table 3. Prescribed heat-generation levels and thermophysical properties used in the simulations.

Parameter	Value
Normal-cell heat generation (W m−3)	$5\times {10}^4$
Intermediate-cell heat generation (W m−3)	$7\times {10}^4$
High-heat-cell heat generation (W m−3)	$1\times {10}^5$
Coolant density at 25 °C (kg m−3)	997
Coolant specific heat capacity at 25 °C (J kg−1 K−1)	4181
Coolant thermal conductivity at 25 °C (W m−1 K−1)	0.606
Coolant dynamic viscosity at 25 °C (Pa s)	8.9 × 10−4
Aluminium cold-plate density (kg m−3)	2719
Aluminium cold-plate specific heat capacity (J kg−1 K−1)	871
Aluminium cold-plate thermal conductivity (W m−1 K−1)	202
Battery-cell density (kg m−3)	2500
Battery-cell specific heat capacity (J kg−1 K−1)	1000
Battery-cell thermal conductivity (W m−1 K−1)	3

.

The baseline value represents the normal thermal-load condition and was selected consistently with the validated reference thermal response [61]. The intermediate heat-generation level was defined by increasing the baseline value by 40%, based on an experimental ageing study on LiFePO₄ cells that reported an approximately 40% increase in internal resistance during ageing [62]. Since the ohmic contribution to heat generation is directly related to internal resistance under a given current, this increase was used as a representative intermediate thermal-load level. The high-heat level was then prescribed to represent a more severe local thermal-loading condition and to test the response of the adaptive flow-allocation strategy under stronger thermal stress.

2.2.3. Governing Equations

The thermal behaviour of the battery module is modelled using a transient three-dimensional conjugate heat transfer framework. The solid domains, including the battery cells and the aluminium cooling plate, are assumed to be homogeneous in density and specific heat capacity. The heat generation within each battery cell is considered volumetrically distributed and spatially uniform inside an individual cell, while it may differ from cell to cell to represent second-life heterogeneity.

The transient energy conservation equation governing the solid domains, including both the battery cells and the cooling plate, is expressed as:

$${\displaystyle \frac{\partial \left({\rho }_s{c}_{p,s}{T}_s\right)}{\partial t}}=\nabla ·(k_s{\nabla T}_s)+Q$$

(1)

where $Q$ represents the volumetric heat-generation term. For the battery cells, $Q=Q_i$, denoting the heat-generation rate of cell $i$, while for the cooling plate, $Q=0$.

The coolant is assumed Newtonian, incompressible, and laminar, with constant thermophysical properties. Radiation heat transfer and thermal contact resistance between the battery and the cooling plate are neglected.

The governing conservation equations for mass, momentum, and energy describing the fluid flow are given as follows:

$$\nabla ·\overrightarrow{v}=0$$

(2)

$${\rho }_f\left[{\displaystyle \frac{\partial \overrightarrow{v}}{\partial \mathrm{t}}}+(\overrightarrow{v}·\nabla )\overrightarrow{v}\right]=-\nabla P+{\mu \nabla }^2\overrightarrow{v}$$

(3)

$${\rho }_f{c}_{p,f}{\displaystyle \frac{\partial T_f}{\partial \mathrm{t}}}+\nabla ·\left({\rho }_f{c}_{p,f}\overrightarrow{v}{T}_f\right)=\nabla ·(K_f\nabla T_f)$$

(4)

To capture temperature-dependent redistribution of coolant among parallel cooling branches, the mass flow rate in branch $i$, ${\dot{m}}_i\left(t\right)$, is updated dynamically toward a temperature-dependent target value ${\dot{m}}_{i,\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\left(t\right)$ using a first-order response law. Here, the adaptive flow-allocation rule is introduced as a physically motivated response mechanism to demonstrate the concept, rather than as a fully developed real-time control algorithm:

$${\displaystyle \frac{d{\dot{m}}_i}{dt}}={\displaystyle \frac{{\dot{m}}_{i,\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}-{\dot{m}}_i}{\tau }}$$

(5)

where $\tau$ is the characteristic response time of the flow-allocation mechanism.

The target mass flow rate is bounded between predefined limits ${\dot{m}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\dot{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and is defined as a smooth increasing function of a local temperature signal $T_i$ which represents the monitored temperature associated with branch $i$ (taken here as the average temperature of the corresponding cell region):

$${\dot{m}}_{i,\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}={\dot{m}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+\left({\dot{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\dot{m}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right){\displaystyle \frac{1+\tanh \left(\frac{T_i-T_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}{\propto }\right)}{2}}$$

(6)

where $T_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}$ is the activation threshold, and α is the temperature smoothing scale. In this study, α was set to 2, which controls the steepness of the hyperbolic tangent transition and prevents an abrupt change in the target mass flow rate near $T_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}$. The hyperbolic tangent function varies when its argument changes approximately from −3 to +3. Therefore, α = 2 spreads this main transition over a temperature-difference range of approximately −6 to +6. As shown in Figure 4, this produces a smoother change in the target mass flow rate near $T_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}$, instead of a sharp change around the activation threshold. In addition to providing a smooth and bounded mapping between temperature and mass flow rate, this formulation avoids discontinuous flow adjustment while progressively increasing coolant allocation as the local temperature exceeds the threshold.

The hyperbolic–tangent response law used in Equation (6) is introduced here as a simple bounded demand–response function intended to demonstrate the physical behaviour of branch-level coolant redistribution. The objective of the present study is not controller synthesis or optimisation, and the proposed response law should therefore not be interpreted as a fully developed real-time control algorithm. In a practical implementation, the same demand-driven principle could be realised using temperature sensors together with controllable valves or other flow-regulating elements within a common circulating coolant loop, analogous to hydronic heating and cooling systems used in buildings.

Figure 4 demonstrates the behaviour of Equation (6). It shows how ${\dot{m}}_{target}$ changes as a function of the temperature difference between the volume-averaged temperature of the cell ($T_i$) and $T_{high}$. In other words, the figure illustrates how ${\dot{m}}_{target}$ varies between ${\dot{m}}_{min}$ and ${\dot{m}}_{max}$ according to the relative temperature signal used for branch-level flow regulation. The hyperbolic tangent function, therefore, provides a bounded and continuous transition between the minimum and maximum allowable flow rates, avoiding an abrupt switching response.

In the present study, the local temperature signal is used as an indicator of thermal demand rather than as a diagnostic variable for identifying the physical origin of heat-generation. The imposed cell-level heat-generation rates are prescribed model inputs that represent the combined thermal effect of cell-to-cell heterogeneity, including possible differences in internal resistance and degradation state. Therefore, the model does not attempt to separate operational electrochemical heat from degradation-induced heat. In a practical implementation, this separation would require an additional state-estimation layer, for example, based on impedance monitoring, online internal-resistance estimation, or battery-management-system data. The adaptive rule used here should therefore be interpreted as a proof-of-concept flow-allocation mechanism driven by local thermal demand, not as a complete degradation diagnosis framework.

2.2.4. Definition of Thermal Heterogeneity Scenarios

To investigate the effect of spatial thermal heterogeneity in the module, nine distinct scenarios were defined (Figure 5). In each case, three cells were assigned an elevated-heat-generation level, corresponding to the intermediate heat-generation value in Table 3, while the remaining cells were assigned the normal heat-generation level. The location of these elevated-heat-generation cells was varied to represent different possible second-life configurations.

The scenarios are defined as follows:

S1—Reverse-diagonal distribution

In Scenario S1, three elevated-heat-generation cells are placed along the reverse diagonal of the 3 × 3 module, representing a distributed heterogeneity pattern.

S2—Right-column (edge-aligned vertical)

In Scenario S2, the three elevated-heat-generation cells are aligned vertically along the rightmost column, representing an edge-localised vertical pattern.

S3—Bottom-row (edge-aligned horizontal)

In Scenario S3, the three elevated-heat-generation cells are aligned horizontally along the bottom row, representing an edge-localised horizontal pattern.

S4—Middle-column (centreline vertical)

In Scenario S4, the three elevated-heat-generation cells are positioned along the central column, representing a centreline vertical pattern.

S5—Middle-row (centreline horizontal)

In Scenario S5, the three elevated-heat-generation cells are positioned along the central row, representing a centreline horizontal pattern.

S6–S9—Random heterogeneity distributions

In Scenarios S6–S9, the three elevated-heat cells are randomly positioned within the module to represent irregular second-life thermal heterogeneity.

All scenarios keep the number of elevated-heat-generation cells and the heat-generation magnitudes constant; only the spatial locations of the elevated-heat-generation cells are varied to isolate the impact of heterogeneity placement on the thermal field.

3. Results

The results demonstrate that the thermal severity of the 3 × 3 module is governed not only by the presence of elevated-heat cells, but also by their spatial arrangement when no adaptive cooling is applied. To evaluate the intrinsic thermal impact of cell arrangement, all nine scenarios were first examined under no-BTMS conditions. As shown in Figure 6, although all nine scenarios contain the same number of thermally stressed cells, their peak temperatures differ, confirming that the arrangement of these cells has a measurable effect on module thermal behaviour. Among all cases, Scenario 2 yields the highest maximum cell temperature (37.16 °C), while the milder scenarios remain approximately 0.6–0.7 °C lower. This indicates that the vertical clustering of elevated-heat cells along one side of the module, as observed in Scenario 2, represents the most critical and unfavourable configuration among the investigated arrangements.

These results demonstrate that thermal risk in heterogeneous battery packs is not solely determined by heat-generation magnitude, but also by spatial arrangement, highlighting the importance of geometry-aware thermal-management strategies. For this reason, Scenario 2 was selected for detailed transient analysis under adaptive BTMS operation.

Figure 7 shows the transient temperature response of all nine cells in Scenario 2 under adaptive BTMS operation. All cells exhibit a rapid temperature rise at the beginning, followed by a gradual reduction in heating rate and an approach to quasi-steady behaviour at later times. Despite the imposed thermal heterogeneity, the final cell temperatures remain within a relatively narrow range, from 27.62 to 28.77 °C, indicating that the proposed per-branch cooling strategy effectively limits thermal non-uniformity even under the most unfavourable 3 × 3 arrangement. Cell B3 reaches the highest final temperature, while B1 and B6 also remain among the warmest cells, whereas B8 stays the coolest cell. The final cell-to-cell temperature difference is therefore approximately 1.16 °C. The transient responses further show that the thermal penalty is not limited to the cells with higher heat generation, but also extends to neighbouring cells through conductive thermal coupling and local competition for cooling capacity. This means that the effect of a thermally stressed cell is not purely local, but influences the temperature field of the surrounding region.

In addition, the slight post-peak decline observed in several temperature curves suggests that adaptive flow redistribution contributes to partial removal of the previously accumulated thermal excess before the system reaches quasi-steady conditions.

Figure 8 directly compares the maximum cell temperature evolution in Scenario 2 under no-BTMS and demand-driven BTMS conditions. The no-BTMS case reaches 37.16 °C, whereas the adaptive BTMS limits the maximum temperature to 28.77 °C, corresponding to a reduction of about 8.38 °C.

Figure 9 shows the temporal variation in coolant mass flow rates in the six cooling branches under the adaptive BTMS for Scenario 2. At the beginning of the simulation, the flow rates remain nearly identical because the thermal states of the cells are still similar. As the temperature field becomes more non-uniform, the branch flow rates diverge according to local thermal demand. Branches associated with thermally stressed regions receive higher mass flow rates, whereas branches linked to cooler regions stabilise at lower flow levels. In this way, the adaptive flow-allocation approach creates a clear hierarchy among the six branches, transforming a nominally uniform cooling network into a demand-ranked hydraulic system. The most demanding branch approaches the upper flow limit, while the least demanding branch remains at a substantially lower value, resulting in a final spread of nearly 50% between the hottest and coolest hydraulic paths. The transient flow responses remain smooth, with only mild overshoot and no sustained oscillation. These results explain the thermal behaviour observed in Figure 7: temperature uniformity is improved not by uniformly increasing coolant supply, but by selectively concentrating cooling capacity where it is most needed. At the same time, branches with lower thermal demand operate at reduced flow levels, avoiding unnecessary coolant allocation to low-demand regions within the present simulation framework.

The adaptive flow-allocation approach effectively transforms the nominally uniform cooling network into a demand-ranked hydraulic system, where coolant mass flow rates are redistributed across parallel branches according to local thermal demand, enabling targeted allocation of thermal-management capacity while reducing the thermal imbalance associated with sequential flow-path effects.

Overall, the proposed adaptive flow-allocation approach limits hotspot formation and improves thermal uniformity through spatially selective coolant redistribution rather than indiscriminate overcooling of the entire module.

To address the relevance of lower operating rates in stationary second-life applications, an additional 1C sensitivity case was examined for Scenario 2. For this sensitivity case, the prescribed 5C volumetric heat-generation rates were reduced using an approximate $I^2$-based Joule-heating scaling, consistent with $q\propto I^{2}R$ [63]. This was used as a first-order sensitivity assumption for the ohmic heat-generation component.

Under this lower thermal-demand condition, the temperature field remained more uniform than in the 5C case, and the spatial temperature gradients were significantly weaker. The maximum cell temperature in the 1C case was 25.56 °C, and the final cell-to-cell temperature difference was 0.25 °C. As shown in Figure 10, the branch mass flow rates remained close to each other and only limited coolant redistribution was required. This behaviour is physically consistent with the demand-driven nature of the proposed BTMS. Under low thermal-demand conditions, the thermal field remains relatively uniform, reducing the need for strong coolant redistribution. Consequently, the adaptive BTMS naturally converges toward near-uniform flow operation rather than imposing unnecessary hydraulic differentiation.

To further evaluate the robustness of the proposed method under more demanding conditions, the adaptive flow-allocation framework was applied to a larger 20-cell module arranged in a 5 × 4 configuration, as shown in Figure 11. Based on the insights obtained from the previous worst-case analysis, the cells were divided into three categories: a first group of severely aged high-heat cells, a second group of intermediate cells with lower heat generation than the first group, and the remaining healthy cells. To create a more critical thermal condition, the four most stressed cells were intentionally placed in one corner of the module, while the second group was arranged around them, forming both a localised hotspot region and a thermal transition zone. This layout is more severe than the 3 × 3 cases because it combines hotspot concentration with geometric confinement and produces a strong spatial thermal gradient, thereby providing a more demanding test condition for the adaptive flow-allocation framework.

To directly compare the cooling effect in the larger module, Figure 12 presents the maximum cell temperature evolution of the 5 × 4 configuration under no-BTMS and adaptive BTMS conditions. In the no-BTMS case, the maximum cell temperature increases continuously and reaches approximately 41 °C at the end of the simulation. In contrast, the branch-level flow-allocation BTMS limits the maximum cell temperature to approximately 29.23 °C. This comparison confirms that the proposed adaptive flow-allocation strategy remains effective when applied to the larger 5 × 4 configuration.

The grouped thermal response of this larger module is presented in Figure 13. As expected, the first group remains the warmest throughout the simulation, followed by the second group, while the healthy cells maintain the lowest average temperature. Although the temperature rise is sharp during the initial stage, particularly for the first group, all three curves gradually approach stable values without divergence. Moreover, the temperature responses remain relatively close to each other, and the gap between the most stressed group and the healthy cells stays in the order of only about 1 °C. It is also important that the second group remains much closer to the healthy cells than to the most stressed cluster, which indicates that the adaptive BTMS limits the outward propagation of thermal non-uniformity from the hotspot region. These results confirm that, even at the larger pack level and under stronger thermal asymmetry, the adaptive flow-allocation framework maintains the thermal hierarchy without allowing uncontrolled hotspot amplification.

To clarify the specific benefit of adaptive flow allocation, an additional constant-flow BTMS case was simulated for the 20-cell module. In this baseline case, the branch flow rate was kept constant at 0.358 g/s per branch, based on the time-averaged total coolant flow rate obtained from the adaptive simulation. This provides a comparison at the same average total coolant flow rate, allowing the effect of spatial redistribution to be isolated. Under constant-flow cooling, the maximum cell temperature reached 31.17 °C, and the cell-to-cell temperature difference was 3.74 °C. However, the adaptive variable-flow strategy reduced the maximum temperature to 29.23 °C and decreased the cell-to-cell temperature difference to 1.62 °C. This corresponds to a 1.94 °C reduction in peak cell temperature and an approximately 56.5% reduction in cell-to-cell temperature difference compared with the constant-flow case. These results show that the primary benefit of the adaptive strategy is not the general reduction in battery temperature through liquid cooling, which is already well established, but rather the improvement of thermal uniformity under spatially heterogeneous thermal loading. The demand-driven flow-allocation approach selectively redistributes coolant toward thermally stressed regions rather than uniformly increasing cooling intensity throughout the module. Consequently, the primary benefit of the adaptive BTMS is improved thermal uniformity through spatially selective allocation of cooling capacity. This advantage is expected to become increasingly important in larger modules and second-life battery packs, where cell-to-cell variations in internal resistance, ageing state, and heat generation can be more pronounced.

Figure 14 shows the branch-wise redistribution of coolant mass flow rates in the 5 × 4 module. As shown in Figure 15, the temperature distribution remains well controlled across the cells, while the cooling plates connected to the first group receive the highest flow rates and the branches farther from the hotspot stabilise at lower values. This indicates that the adaptive flow-allocation framework allocates cooling capacity according to local thermal demand rather than increasing flow uniformly in all branches. The strongest flow adjustment appears in the subset of branches near the thermally critical corner, forming a localised high-flow region near the thermally stressed corner of the module. As the thermal asymmetry becomes stronger in the 5 × 4 case, the adaptive flow-allocation framework increases its selectivity, and the hottest branches carry roughly three times the flow of the coolest ones, which is a much stronger hydraulic differentiation than that observed in the 3 × 3 module. This redistribution should be interpreted as demand-driven flow allocation within a shared circulating coolant loop, where branch-level flow differences arise from local thermal demand and flow regulation rather than from independent pumping subsystems. The transient responses show a short overshoot followed by smooth settling toward branch-specific steady values. The clearest overshoot appears in the branch with the highest flow rate between 240 and 360 s, while the remaining branches settle smoothly without obvious fluctuations or sustained oscillations. These features indicate that the adopted response law produces smooth and bounded hydraulic adjustment even when the number of branches is increased. The results, therefore, suggest that the adaptive flow-allocation framework remains numerically stable and physically consistent under the larger and more thermally asymmetric module configuration.

The contour plots in Figure 16 and Figure 17 complete the physical interpretation of the thermal and hydraulic behaviour. Figure 16 shows that, although cell-to-cell temperature non-uniformity is effectively restrained, a clear vertical temperature gradient still remains within the cells, with the lower regions warmer than the upper regions. This suggests that, after demand-driven coolant redistribution suppresses large-scale hotspot spreading, the dominant remaining non-uniformity shifts from inter-cell variation to intra-cell stratification.

Figure 17 confirms the corresponding behaviour on the coolant side: the fluid temperature increases progressively along the channel paths, and the warmest regions appear in the downstream sections where the coolant has absorbed the greatest thermal load. Taken together, these contour plots show that the adaptive flow response is not only numerically stable but also physically consistent across both the solid and fluid domains.

Overall, the combined evidence from Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 shows that the proposed demand-driven BTMS performs more than simple temperature reduction. It responds to concentrated thermal loading by redistributing coolant accordingly and limiting hotspot propagation across the module. This remains true after scaling from the 3 × 3 module to the more demanding 5 × 4 configuration, which supports the potential suitability of demand-based flow allocation for SLB packs with spatially non-uniform heat generation.

4. Conclusions

This study proposed an adaptive liquid-cooling BTMS based on branch-level flow allocation for second-life prismatic LiFePO₄ battery modules. The results show that thermal management in second-life packs should not be treated as a uniform cooling problem, because the location of thermally stressed cells has a strong effect on hotspot formation and on the overall thermal response of the module.

For the 3 × 3 module, the no-BTMS analysis showed that different spatial arrangements of the same number of elevated-heat cells do not lead to the same thermal severity. Among the investigated cases, the edge-aligned vertical arrangement produced the highest peak temperature, which confirms that geometry-dependent heterogeneity is a critical factor in SLB packs. Under this worst-case condition, the proposed branch-level flow-allocation BTMS reduced the maximum temperature from 37.16 °C to 28.77 °C, corresponding to a reduction of about 8.38 °C, while limiting the final cell-to-cell temperature difference to about 1.16 °C.

The hydraulic results showed that this improvement is attainable through selective coolant redistribution rather than uniform overcooling. Branches near thermally stressed regions received higher flow rates, while cooler regions operated at lower flow levels, and the transient response remained smooth and stable without sustained oscillation. This indicates that the proposed adaptive flow-allocation approach produces physically consistent coolant redistribution in response to local thermal demand within the present simulation framework.

The larger 5 × 4 module further confirmed the robustness of the proposed strategy. Even under a strongly localised hotspot condition, the flow-allocation framework maintained a clear thermal hierarchy between cell groups, limited outward hotspot spreading, and produced a structured flow-allocation pattern in which the hottest branches carried much higher flow rates than the coolest ones. At the same time, the contour plots showed that once large-scale inter-cell non-uniformity was suppressed, the main remaining non-uniformity shifted to intra-cell temperature stratification and downstream coolant heating. The comparison with the constant-flow baseline further showed that the adaptive strategy mainly improves thermal uniformity, decreasing the cell-to-cell temperature difference from 3.74 °C to 1.62 °C in the 20-cell module.

Overall, this work demonstrates the potential of adaptive, demand-driven flow allocation to improve thermal uniformity in heterogeneous battery modules. While the present results are based on numerical analysis using volume-averaged cell temperature as a flow-regulation signal, they indicate that distributed thermal-management strategies may provide a promising direction for both second-life and new battery systems.

The primary contribution of the present work is therefore not the demonstration that liquid cooling reduces battery temperature, since this is already well established, but rather the demonstration that demand-driven branch-level flow allocation can improve thermal uniformity under heterogeneous thermal loading. By adapting principles commonly used in hydronic thermal-distribution systems, the proposed approach redistributes coolant according to local thermal demand within a shared circulating loop, enabling spatially selective thermal management rather than uniform cooling independent of local conditions. In this sense, the work should be interpreted primarily as a thermal-management architecture concept and proof-of-concept adaptive flow-allocation framework rather than as a fully developed control-system implementation.

A limitation of the present study is that the coolant mass flow rates in each branch are allowed to vary independently without imposing a fixed total-flow or pumping-power constraint. While this enables a clear demonstration of the adaptive flow-allocation concept, future work should evaluate system performance under global hydraulic constraints (e.g., fixed total-flow or pumping-power limits) and assess the associated energy consumption.

Another limitation of the present study is that the heat-generation rate of each cell is prescribed as a known input. This allows the thermal effect of second-life heterogeneity to be isolated, but it does not address how degradation-induced heat can be separated from normal operational heat in a real battery pack. Future work should therefore integrate the proposed adaptive flow-allocation framework with battery state-estimation methods, such as impedance monitoring, online internal-resistance estimation, or BMS-based diagnostic models, to provide a more realistic thermal-demand signal for practical control.

The present work is a simulation-based proof-of-concept thermal-management study. Therefore, the results should be interpreted as evidence of thermal feasibility and physically consistent flow-redistribution behaviour, rather than as validation of a deployable real-time BTMS prototype. Future work will address flow-regulation strategy development, system integration, and experimental validation. In the present model, cell temperature was used as the flow-regulation input signal. Future work should also investigate the use of branch outlet coolant temperature as an alternative control signal and assess its performance with the present approach.