pybamm-team
Multi-layer pouch model
Through homogenization you can show that you can change the parameters in the "2+1D" pouch cell model to be effective parameters that account for the number of layers in the pouch. The model still assumes the temperature is independent of the through-cell direction x but varies in y and z.
Assuming that each layer (electrode pair) behaves identically we can just use the parameter "Number of electrodes connected in parallel to make a cell" to modify the thermal response. The electrochemical model will solve for the single layer.
Starting from a 3D thermal model (along with appropriate boundary conditions) $$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (\lambda \nabla T) + Q,$$ we can integrate in $x$ to find $$N\int_{-L_{cn}}^{L_n+L_s+L_p+L_{cp}}\rho c_p \frac{\partial T}{\partial t} \mathrm{d}x=N\int_{-L_{cn}}^{L_n+L_s+L_p+L_{cp}} \nabla_\perp \cdot (\lambda \nabla_\perp T) \mathrm{d}x + N\int_{-L_{cn}}^{L_n+L_s+L_p+L_{cp}}Q\mathrm{d}x - h_{cn}(T_{left}-T_{\infty}) - h_{cp}(T_{right}-T_{\infty}).$$ Note that the integrals are over each layer, and we end up with a prefactor $N$ to account for the number of layers. Also note the $\nabla_\perp$ operator that takes derivatives in $y$ and $z$ only - integration in $x$ removes the $x$ derivative and a new source/sink term appears due to the boundary conditions in $x$.
We can then replace the integrals with $x$-average, denoted by an overbar, remembering that in PyBaMM x_average only averages over the electrode pair (of total thickness $L$)
$$NL\rho_{eff} c_{p,eff} \frac{\partial \bar{T}}{\partial t} \mathrm{d}x=NL \nabla_\perp \cdot (\lambda_{eff} \nabla_\perp \bar{T}) + NL\bar{Q} - (h_{cn}+h_{cp})(\bar{T}-T_{\infty}).$$
In the final term note that $\bar{T}=T_{left}=T_{right}$ since $T$ is independent of $x$ in the limit we are considering. The material properties are now effective properties, averaged over the layers.
Dividing through by $NL$ gives $$\rho_{eff} c_{p,eff} \frac{\partial \bar{T}}{\partial t} \mathrm{d}x=\nabla_\perp \cdot (\lambda_{eff} \nabla_\perp \bar{T}) + \bar{Q} - \frac{(h_{cn}+h_{cp})}{NL}(\bar{T}-T_{\infty}).$$
In the current implementation, the final cooling term is written $$- \frac{(h_{cn}+h_{cp})A}{V}(\bar{T}-T_{\infty}),$$ where $A=L_yL_z$ and $V=LL_yL_z$ - i.e. the volume is just for a single electrode pair.
We should update the volume used in this calculation to account for the number of parallel electrode pairs.
The quick fix for now is to multiply the numerical values of the parameters $h_{cn}$ and $h_{cp}$ by 1/N.
The x-full model is only valid for a single-layer pouch, so we should raise a warning when processing the x-full model with "Number of electrodes connected in parallel to make a cell" > 1 (thanks to @DavidMStraub for the suggestion).