## Appendix C 1D Method with Pressure Loss-Based Junction Model  
In this case, a collocated one-dimensional finite volume method is used for all the calculations inside the pipes. The Euler equations of fluid dynamics simplified for one-dimensional flow in a straight uniform duct can be expressed as:  
\[\frac{\text{d}\overline{\mathbf{W}}_{i}}{\text{d}t}=\frac{\text{d}}{\text{d}t} \left(\begin{array}{c}\rho\\ \rho u\\ \rho e_{0}\end{array}\right)_{i}=\frac{A(\mathbf{F}_{i-1,i}-\mathbf{F}_{i,i+1})}{V_{i}} \tag{102}\]  
Here, \(\overline{\mathbf{W}}_{i}\) is the cell-averaged state vector of cell \(i\), \(\mathbf{F}_{i-1,i}\) and \(\mathbf{F}_{i,i+1}\) are the inter-cell fluxes between cells \(i-1\) and \(i\) and between \(i\) and \(i+1\), respectively, and the other symbols refer to the same magnitudes as in Appendix B, with \(u\) being now the axial velocity of the flow. The inter-cell fluxes are computed by an approximate solution of the Riemann problem as described by [PERSON] et al. [38]. The state vector is extrapolated to the cell boundaries to compute the fluxes by means of a Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) approach as described in [39], while the solution is propagated in time using [PERSON]'s method, leading to a second order in time and space, total variation diminishing scheme.  
While the main one-dimensional flow inside the ducts is simulated, the effects of the geometry of the junction are modelled. The connections of the ducts to the junction are solved using an auxiliary small zero-dimensional element. Each of the one-dimensional branches is connected to that zero-dimensional element making use of the Riemann variables to compute the fluxes at their corresponding boundary. At each connection, it is assumed that a certain amount of stagnation pressure is lost, depending on the angle of the junction and of the ratio between the outflow mass flow \(\dot{m}_{out}\) passing through the branch of interest and the inflow mass flow \(\dot{m}_{in}\). The pressure loss coefficients are computed following the expressions given in [15; 16], and are defined as the ratio of the difference in stagnation pressure between the outflow branch and the inflow branch to the dynamic pressure (\(\rho u^{2}/2\)) of the inflow branch. Finally, the total pressure loss coefficient \(K\) for a three-branch junction with the same section in all the branches can be estimated as:  
\[K=\left(\frac{\dot{m}_{out}}{\dot{m}_{in}}\right)^{2}-\frac{3}{2}\frac{\dot{m }_{out}}{\dot{m}_{in}}+\frac{1}{2} \tag{103}\]  
when the branch is collinear with the inflow branch, and,  
\[K=\left(\frac{\dot{m}_{out}}{\dot{m}_{in}}\right)^{2}+1-2\frac{\dot{m}_{out}}{ \dot{m}_{in}}\cos\left(\frac{3}{4}\theta\right) \tag{104}\]  
for the lateral branch, when the flow is split between a collinear and a lateral branch. In this case, \(\theta\) is the angle between the lateral branch and the axial outflow branch, so that \(0^{\circ}\) means that both outflow branches are parallel. The same expression applies when the inflow branch is not parallel to any of the outflow branches: in that case, the angle \(\theta\) is measured between the inflow branch and the other outflow branch.  
In the auxiliary zero-dimensional element, the gas is again considered as a perfect gas, and the mass and energy equations are solved:  
\[\frac{\mathrm{d}m}{\mathrm{d}t}=\sum\dot{m} \tag{10}\]  
\[\frac{\mathrm{d}(mc_{\flat}T)}{\mathrm{d}t}=\sum\dot{m}h_{0} \tag{11}\]  
where \(m\) is the mass trapped in the zero-dimensional element, \(\dot{m}\) is the mass flow, positive when it enters the element, and \(h_{0}\) is the specific stagnation enthalpy associated with the mass moving inside or outside of the element. These two equations set an additional limitation to the maximum possible time step.