The temperature data was collected on a rack of batteries that were repeatedly charged and discharged. The batteries were cooled by natural convection. The data used in the example analysis was collected over roughly a 24 hour period. The ambient temperature at the rack ranged from ~ 32.5 to 37.5 C.
The temperature trace used was from a thermocouple located on the case of one of the batteries.
The first pass model was a single lumped mass of an individual battery with natural convection cooling and joule heating. This was represented by:
McpdT/dt = Q - hA(T-Ta) where Mcp is the thermal mass, Q the heat load, hA the heat transfer coefficient times the surface area, and Ta the ambient temperature.
The heat load, Q, was defined by QfI2 = Qcharge I2 for the charge cycle and QdischargeI2. The electrical current is I and ranges from 240 amps for the charge and 300 amps for the discharge. Note: For this example it is assumed that any heating from chemical kinetics is secondary and captured by the Q terms.
The natural convection, hA, is defined by Ch(T-Ta)x(T-Ta). This is based on natural convection heat transfer coefficients being estimated by CRan where Ra is the Rayleigh number and the exponent n ranging from .25 to .35 depending on the surface type.
The overall equation looks like
McpdT/dt = QfI2 - Ch(T-Ta)1+x.
This can be setup in a finite difference form in time as
Tp+1 = Tp + (QfI2/Mcp)/dt - (Ch(Tp - Ta)1+x/Mcp)/dt.
The spreadsheet is setup with this form of the model.
Solver can now be used to find the best fit values for Qf, Ch, cp, Ch and x. Solver will find local minima so several starting values should tried. The constraints can also be used if reasonable values are known. For example, the convective heat transfer coefficient, x, should be somewhere between . 2 and .35 based on textbook values for flat surfaces with natural convection. Reference: "Fundamentals of Heat and Mass Transfer" 4th Edition by Incropera and DeWitt pgs 492 - 501.
The values could be determined by using the full time period, however, more accuracy can be gained by breaking the regression into sections. For example, the cool-down section can be used with the Qf term set to 0.0 to find fits for Ch, x and cp. The heat-up section can then be used to find Qf charge and discharge using the previously determined Ch and x values. The cp value can either be used from the cool-down or see what value fits the heat-up.
The full model can then be checked with the best fit values from each of the sections. Some refinement is necessary to better fit the overall time period and this was done manually.
Each of the worksheets used for this approach is given in the attached workbook.
Battery Transient Model Regression Analysis Using Excel Solver
The video shows more detail of the actual analyses and results.