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## 4 Optimization Targets for Cooling System Design | Diabatix

November 18, 2022
by
Niels Verdijck

Manually designing a cooling system will require a lot of iterations before reaching the absolute best one. But...

• What is the best design?
• How do we know it is the best?

In this sense, a mathematical formula must translate the end goal to objectively quantify all iterations concerning one another.

This blog post tackles some of the most important targets to include while designing the cooling system for a battery pack:

### 1. Maximum Temperature Constraint

The fear of any battery engineer is a thermal runaway. This phenomenon became well-known to the public when the Samsung Note 7 was released in 2016. The recall occurred after identifying a battery manufacturing defect, causing excessive heat and fires. The automotive industry also has some examples of thermal runaways, for example, GM's recall of Chevrolet Bolts.

Due to the incidents mentioned above, it has become well-known that all battery packs must operate below a certain temperature limit. Going beyond this threshold value will risk thermal runaway. In the best-case scenario, thermal runaway only damages several battery cells beyond repair. In the worse case, the cells explode and start a big fire.

This threshold temperature not only makes it easy to compare the different iterations with one another but also puts a clear limit on whether or not a design is acceptable. If the maximum cell temperature exceeds the threshold, the resulting design is unsuitable for the product. Expressing this into a mathematical formula makes the idea clear:

$T_{\mathrm{max}} \leq T_{\mathrm{threshold}}$

For safety reasons, the threshold temperature will not be the temperature at which thermal runaway will commence. A safety factor (SF) will be taken into account:

$T_{\mathrm{max}} \leq T_{\mathrm{thermal \; runaway}} - \mathrm{SF}$

### 2. Temperature Variance Minimization

The battery pack in EVs consists of multiple modules. Each module is subsequently done out of individual battery cells. The lifetime of the complete battery pack is thus dependent on the lifetime of each of the cells. High temperatures will consume the lifetime of the cells. Therefore, the cooling system for a battery pack is well-designed when the temperature gradient over the battery pack is limited. This fact will ensure the cells within all the modules will thermally degrade at the same pace. Thus, the goal should be to prevent the cells in one of the modules (far from the inlet) surpass their lifespan while the cells in another module (right next to the inlet) are still only halfway through their lifetime.

Translating the above requirements into a stable formulation can initially be a bit daunting. But after careful consideration, one will devise the following penalization scheme. All places that stray too far from the average ($$\bar{T}$$) need to be more heavily penalized than places where the temperature is closer to the average.

The variance is a statistical property with the following mathematical equation:

$\sigma²_\mathrm{T} = \frac{1}{V}\int_\mathrm{V} (T-\bar{T})²\space dV$

where $$V$$ represents the total volume of all the battery cells in the battery pack.

The resulting scalar increases whenever the temperature deviates too far from the average temperature ($$\bar{T}$$). The lower this value, the more uniform the temperature distribution over the component, making it an ideal parameter for comparing the different iterations.

The temperature variance minimization is a very good objective function if it minimizes the temperature gradient throughout all the modules. However, the objective function gives the designer no control over the average temperature value ($$\bar{T}$$). As a worst-case scenario, the average temperature skyrockets if no additional temperature objective and/or constraint counteracts this.

The efficiency of battery cells is temperature-dependent. It might be worthwhile to sacrifice a bit of temperature uniformity if the cooling design can keep the battery cell temperature closer to the optimal temperature ($$T_{\mathrm{optim}}$$).

These requirements translate into the following penalization scheme: the further away the temperatures locally are from the optimal temperature, the more heavily these places need to get penalized during the optimization run. This penalization scheme is almost identical to the one in the previous section; only one modification needs to happen:

$\min \space\frac{1}{V}\int_\mathrm{V}(T-T_{\mathrm{optim}})²\space dV$

where $$V$$ represents the total volume of all the battery cells in the battery pack.

The resulting scalar will again increase whenever the temperature strays too far from the optimal operating temperature ($$T_{\mathrm{optim}}$$). And vice versa, the lower this value, the better the cooling system.

The differences between the temperature spread minimization and temperature variance minimization objectives are, at first glance, very minute. However, the latter (on its own, without any other temperature targets) gives the designer more control over the temperature interval of the cells. It should be noted that the target value for $$T_{\mathrm{optim}}$$ should be physically possible; otherwise, it will never be reached.

### 4. Maximum Pressure Loss Constraint

The maximum temperature constraint is not the only important constraint to consider during the design phase. The design can have an incredibly good thermal performance, but it needs to be practical. For instance, if no pump is available to force the fluid flow rate through the channels, this usually wastes engineering time, and no value is added to the product. The same holds if the pumping power required is impractically high. The energy stored in the batteries should go as much as possible to extend the EV range, not circulate the pump's required fluid.

Similarly to the maximum temperature constraint, the pressure drop through the design needs to remain below a maximum value ($$P_{\mathrm{max}}$$) :

$P_{\mathrm{inlet}} - P_{\mathrm{outlet}} \leq P_{\mathrm{maximization}}$

### Conclusion

Designing the cooling system for a battery pack is a complex process in which many parameters play an important role. Many iterations are needed to come up with the best possible cooling design.

The maximum temperature and pressure constraints clearly limit whether a design is acceptable or not. If a critical limit exceeds, the resulting design is useless (no matter how good the other parameters are). The temperature variance or temperature spread objectives allow for easy comparison of the different designs when both mentioned constraints are met. The design with the lowest value is the best-performing one.

The design process is complex and requires a lot of expertise to critically analyze each iteration's performance. Luckily, the generative design tool, Diabatix's ColdStream, fully automates this process in an approachable way. If you want to know more about this tool, don't hesitate to contact us. And if you would like to know more about other targets, feel free to consult our documentation.

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