## 2 Transport Model
The total density of Maxwellized ions is determined as the sum  
\[n_{i}=n_{beam}^{D^{+}}+n_{pel}^{D^{+}}+n_{sep}^{D^{+}}+n_{beam}^{T^{+}}+n_{pel} ^{T^{+}}+n_{sep}^{T^{+}}. \tag{5}\]  
Consequently, the electron density can be determined from the quasi neutrality condition taking into account the impurities and the fast ions,  
\[n_{e}=n_{i}+n_{fast}^{D^{+}}+n_{fast}^{T^{+}}+\sum_{Z=\mathrm{He,Be,Ne}}Z\times n _{Z}, \tag{6}\]  
where \(n_{fast}^{D+}\) (\(n_{fast}^{T+}\)) is the density of fast deuterium (tritium) ions, and \(n_{Z}\) is the density of the impurity with charge \(Z\) (the sum is over all charge states of the He, Be, and Ne impurities). The set of Equation (3) allows us to explicitly obtain the partial ion concentrations and, consequently, determine the \"partial confinement times\" for each plasma fueling system,  
\[\tau_{beam}^{D^{+}/T^{+}}=\frac{\int\limits_{v_{pl}}\left(n_{beam}^{D^{+}/T^{+}}+ n_{fast}^{D^{+}/T^{+}}\right)dV_{pl}}{\int S_{beam}^{D^{+}/T^{+}}dV_{pl}}, \tag{7}\]  
\[\tau_{pel}^{D^{+}/T^{+}}=\frac{\int\limits_{v_{pl}}n_{pel}^{D^{+}/T^{+}}dV_{pl}}{ \int S_{pel}^{D^{+}/T^{+}}dV_{pl}}, \tag{8}\]  
\[\tau_{sep}^{D^{+}/T^{+}}=\frac{\int\limits_{V_{pl}}\left(n_{sep}^{D^{+}/T^{+}} -n_{sep}^{D^{+}/T^{+}}(a)\right)dV_{pl}}{\int S_{sep}^{D^{+}/T^{+}}dV_{pl}}. \tag{9}\]  
Let us also determine the particle and energy confinement times as  
\[\tau_{E}=\frac{\frac{3}{2}\int\limits_{V_{pl}}\left(n_{e}T_{e}+n_{i}T_{i} \right)dV_{pl}}{\int P_{tot}dV_{pl}}, \tag{10}\]  
\[\tau_{p}=\frac{\int\limits_{V_{pl}}n_{i}dV_{pl}}{\int S_{i}^{tot}dV_{pl}}, \tag{11}\]  
\[\tau_{tot}=\frac{\int\limits_{V_{pl}}\left(n_{i}-n_{i}(a)\right)dV_{pl}}{ \int S_{i}^{tot}dV_{pl}}, \tag{12}\]  
where \(\tau_{E}\) is energy confinement time, \(\tau_{p}\) is ion confinement time, \(\tau_{tot}\) is the ion confinement time without recycling, \(P_{tot}\) is the total plasma heating power, and \(S_{t}^{tot}\) is the total source of ions from all fueling systems. To determine the set of Equation (3) unequivocally, it is necessary to provide the transport coefficients. In calculations made in this work, \(D\) and \(V\) were determined as  
\[\left\{ \begin{array}{ll}D=C_{n}\times\left(1-C_{n}^{ped}\times H\left(\rho_ {ped}\right)\right),\\ V=0.0,\end{array} \right. \tag{13}\]  
where \(H(\rho)\) is the Heaviside function, \(C_{n}\) is a constant, and \(C_{n}\)_pd_ is the height of the \"step\" that determines the given value of the density pedestal. The solution of the set of Equation (3) is found by the reverse method, i.e., in the calculation process the values of flows required to provide the given regime of the average density and confinement are selected. The solution is found using the following feedbacks:  
1. Feedback over the average density: the particle flow from the pellets (D + T) is chosen to provide the required value <\(n_{e}\)> = (6-8) \(\times\) 10\({}^{19}\) m\({}^{-3}\). 2. Feedback over energy confinement time: the ion and electron heat conductivity coefficients \(\chi_{e}\) = \(\chi_{i}\) are chosen so as to provide the required value of the factor H\({}_{\text{IPB(y,2)}}\) = 1.3. 3. Feedback over particle confinement time: the value of constant \(C_{n}\) is chosen to satisfy the given value \(\tau_{p}/\tau_{E}\) = 0.75-3.0. 4. Feedback over the fraction of tritium in the pellets: the flow from the pellets is divided between D and T to provide the required value \(f^{T}\) = 0.5. During the calculation, <\(n_{e}\)> and \(\tau_{p}/\tau_{E}\) are considered free parameters and they are varied in the above-mentioned range in order to consider all the most probable DEMO-FNSoperation regimes. It is seen that feedback links (i)-(iii) affect each other, meaning that we have to solve a nonlinear inverse problem.