Transient Nonlinear Thermal Model Development Using Solver

The attached spreadsheet is an example of using the Excel Solver function to estimate parameters for the nonlinear transient thermal model of a battery used on a hybrid vehicle.

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:

Mc_pdT/dt = Q - hA(T-T_a) where Mc_p is the thermal mass, Q the heat load, hA the heat transfer coefficient times the surface area, and T_a the ambient temperature.

The heat load, Q, was defined by Q_fI² = Q_charge I² for the charge cycle and Q_dischargeI².  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 C_h(T-T_a)^x(T-T_a).  This is based on natural convection heat transfer coefficients being estimated by CR_aⁿ where R_a is the Rayleigh number and the exponent n ranging from .25 to .35 depending on the surface type.

The overall equation looks like

Mc_pdT/dt = Q_fI² - C_h(T-T_a)^1+x.

This can be setup in a finite difference form in time as

T^p+1 = T^p + (Q_fI²/Mc_p)/dt - (Ch(T^p - T_a)^1+x/Mc_p)/dt.

The spreadsheet is setup with this form of the model.

Solver can now be used to find the best fit values for Q_f, C_h, c_p, C_h 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 Q_f term set to 0.0 to find fits for C_h, x and cp.  The heat-up section can then be used to find Q_f charge and discharge using the previously determined C_h and x values.  The c_p 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.