212 at the output 513 of the oxidant conveyor 205. The oxidant 212 may be conveyed to the cathode space 520 by means of an oxidant cooler 509. At the output of the oxidant cooler 509 may also be disposed a branch of oxidant 212 to the bearing pathway 514. Of the mass flow of oxidant 212 at the output of the oxidant cooler 509, it is thus possible to conduct a (relatively large) portion into the cathode space 520 and a (complementary) portion to the one or more bearings of the motor 510 of the oxidant conveyor 205.

At the output of the oxidant cooler 509, there may be disposed a temperature sensor 507 configured to detect temperature measurements T_AirBearⁱⁿin relation to the temperature of the oxidant 212 at the output of the oxidant cooler 509 and/or at the input to the one or more bearings of the motor 510.

The reaction product outlet 403 from the cathode space 520 may be directed to the outlet 519 of the cathode subsystem 500. In addition, the cathode subsystem 500 may have a bypass pathway 515 for oxidant 212 that runs from the output 513 of the oxidant conveyor 205 to the outlet 519 of the cathode subsystem 500, in order to be able to route excess oxidant 212 (for example in the case of startup of the fuel cell stack 102) directly out of the cathode subsystem 500. On the bypass pathway 515, there may be disposed a physical mass flow sensor 506 (for measurement of the mass flow rate of oxidant 212 on the bypass pathway 515) and/or a bypass valve 508 (for control of the mass flow rate of oxidant 212 on the bypass pathway 515). When the bypass valve 508 is open, a portion of the oxidant 212 at the output 513 from the oxidant conveyor 205 flows into the inlet 516 to the oxidant cooler 509, and a complementary portion into the bypass pathway 515.

The mass flow rate of oxidant 212 into the cathode space 520 can be controlled by open-loop or closed-loop control using the mass flow rate measurements from the physical mass flow rate sensor 502 in order to ascertain the actual mass flow rate value. However, mass flow rate measurements are typically subject to a relatively high level of noise, especially in the case of relatively high mass flow rate values, which leads to impairment of the accuracy and/or dynamics of the open-loop and or closed-loop control of the mass flow rate, and hence to impairment of the power dynamics of the fuel cell stack.

The (control) apparatus 103 of the fuel cell system 100 may be configured, using a base estimation model of the oxidant conveyor 205, to estimate the value {dot over (m)}_Cmprof the oxidant mass flow rate through the oxidant conveyor 205. An illustrative basis estimation model is:

${\dot{m}}_{Cmpr} = {\dot{\tilde{m}}}_{Cmpr} \frac{δ_{Cmpr}^{i n}}{\sqrt{θ_{Cmpr}^{i n}}}$

$δ_{Cmpr}^{i n} = \frac{p_{Cmpr}^{i n}}{p^{ref}}$

$θ_{Cmpr}^{i n} = \frac{T_{Cmpr}^{i n}}{R^{ref}}$

where p^refis a reference pressure (e.g. p^ref=1 bar) and T^refis a reference temperature (e.g. T^ref=298.15 K) (defined by the reference conditions). The normalized mass flow rate {tilde over ({dot over (m)})}_Cmprcan be found as follows:

${\dot{\tilde{m}}}_{Cmpr} = Φ_{Cmpr} ρ_{Air}^{ref} \frac{π}{4} d_{Cmpr}^{2} U_{c}$

$Φ_{Cmpr} = \frac{k_{3} Ψ_{Cmpr} - k_{1}}{k_{2} + Ψ_{Cmpr}}$

$U_{c} = \frac{π}{60} d_{Cmpr} {\tilde{n}}_{Cmpr}$

where d_Cmpris the diameter of the oxidant conveyor 205, where ρ_Air^refis the density of the oxidant 205 under the reference conditions (e.g. ρ_Air^ref=1.1638 kg/m³), where Φ_Cmpris the normalized mass flow rate, where ψ_Cmpris a dimensionless parameter for description of the oxidant conveyor 205, where k_i, with i=1, . . . , 3, are fitting parameters, where U_cis the outside speed of the rotor of the oxidant conveyor 205, and where ñ_Cmpris the normalized speed of rotation of the oxidant conveyor 205. The diameter of the oxidant conveyor 205 may be, for example, d_Cmpr=0.06 m.

The parameters ψ_Cmprfor description of the oxidant conveyor 205 can be ascertained via

$Ψ_{Cmpr} = \frac{2_{C_{p, Air}} T^{ref} [{(\frac{p_{Cmpr}^{out}}{p_{Cmpr}^{i n}})}^{\frac{γ_{Air} - 1}{γ_{Air}}} - 1]}{U_{C}^{2}}$

$k_{i} = k_{i, 1} + k_{i, 2} \times {Ma}_{Cmpr}$

where c_p,Air=1015 J/(kg·K) and where γ_Air=1.4. The parameter Mα_Cmprcan be calculated via

${Ma}_{Cmpr} = \frac{U_{c}}{\sqrt{γ_{Air} R_{Air} T^{ref}}}$

where the gas constant of the oxidant 212 is, for example, R_Air=288.19 J/(kg·K). The fitting parameters can be ascertained experimentally by measurements on the oxidant conveyor 205, for example as

$k_{i, j} [\begin{matrix} 0.1557 & 0.0176 \\ - 1.5296 & 0.0843 \\ 0.1242 & 0.0112 \end{matrix}]$

The normalized speed of rotation ñ_Cmprcan be ascertained on the basis of the speed of rotation n_Cmprof the oxidant conveyor 205 (which can be detected by a sensor) as

${\tilde{n}}_{Cmpr} = n_{Cmpr} \frac{1}{\sqrt{θ_{Cmpr}^{i n}}}$

As set out further up, the oxidant conveyor 205 may comprise a motor 510 with one or more motor bearings that are operated with oxidant 212 that has been conveyed by the oxidant conveyor 205. It is thus possible to branch off a proportion of the oxidant mass flow at the output 513 of the oxidant conveyor 205 for the bearing pathway 514 to the one or more bearings. The mass flow rate {dot over (m)}_AirBearon the bearing pathway 514 can be ascertained using a supplementary estimation model for the one or more bearings, for example as

${\dot{m}}_{AirBear} = C_{AirBear} \sqrt{\frac{p_{AirBear}^{i n}}{R_{Air} T_{AirBear}^{i n}} (p_{AirBear}^{i n} - p_{AirBear}^{out})}$

where p_AirBearⁱⁿis the pressure of the oxidant 212 at the input of the one or more bearings, where p_AirBear^outis the pressure of the oxidant 212 at the output of the one or more bearings, where T_AirBearⁱⁿis the temperature of the oxidant 212 at the input of the one or more bearings, and where the experimentally ascertained fitting parameter C_AirBearis, for example, C_AirBear=8.76·10⁻⁶m². It can be assumed here that p_AirBearⁱⁿcorresponds to the value p_Cmpr^outmeasured by the pressure sensor 505 and/or that P_AirBear^outcorresponds to the value p_Cmprⁱⁿmeasured by the pressure sensor 504.

As set out further up, for the model of the oxidant conveyor 205, the value of the temperature T_Cmprⁱⁿof the oxidant 212 at the input 512 of the oxidant conveyor 205 is required. Because of the fact that the mass flow rate at the input 512 of the oxidant conveyor 205 is a mixture of the oxidant mass flow rate through the filter 501 and the oxidant mass flow rate from the one or more bearings, the temperature T_Cmprⁱⁿof the oxidant 212 at the input 512 of the oxidant conveyor 205 may vary from the temperature measurement T_AirFil^measby the temperature sensor 503. The temperature T_Cmprⁱⁿof the oxidant 212 at the input 512 of the oxidant conveyor 205 can be ascertained as

T
Cmpr

i
⁢
n

=

m
.

AirFil
meas