#### 2.1. Injector in PEMFC working principle

$${I}_{{\mathrm{H}}_{2}}=\frac{{\mathrm{rice}}_{In}}{{\mathrm{rice}}_{C}}=\frac{{\mathrm{rice}}_{r}+{\mathrm{rice}}_{C}}{{\mathrm{rice}}_{C}}=\frac{{\mathrm{rice}}_{r}}{{\mathrm{rice}}_{C}}+1$$

#### 2.2. Structural design of PNFN injector

The power rating, mass flow rate, cell voltage, and other geometric parameters of the PEMFC are important for injector design.

$${\mathrm{rice}}_{p}=\frac{{\mathrm{phosphorus}}_{stACk}{\mathrm{medium\; size}}_{{\mathrm{H}}_{2}}}{2{V}_{C}F}$$

Where ${\mathrm{phosphorus}}_{stACk}$ is the rated power of the fuel cell, ${\mathrm{medium\; size}}_{{\mathrm{H}}_{2}}$ is the molar mass of hydrogen, ${V}_{C}$ and $F$ are the single cell voltage and Faraday’s constant respectively.

$${A}_{nt}=\frac{{\mathrm{rice}}_{p}{n}_{Cr}}{k{p}_{Cr}{p}_{p}}$$

$${p}_{Cr}=\frac{{p}_{*}}{{p}_{p}}={\left(\frac{2}{k+1}\right)}^{\frac{k}{k\u20131}}$$

$${n}_{Cr}=\sqrt{2\frac{k}{k+1}}\sqrt{\mathrm{right}{\mathrm{time}}_{{p}^{*}}}$$

Where ${n}_{Cr}$ is the critical speed, $k$ is the ratio of gas specific heat, ${p}_{Cr}$ is relative pressure, ${p}_{p}$ is PF pressure, ${p}_{*}$ is the gas critical pressure, $\mathrm{right}$ is the gas constant, and ${\mathrm{time}}_{p*}$ is the gaseous critical temperature.

For 170 kW (Pe) high-power fuel cells, this study proposes a new four-nozzle injector. Traditional single-nozzle injectors are designed at certain operating points of the fuel cell. When the fuel cell power changes significantly, the performance of traditional single-nozzle injectors will drop sharply and cannot meet the flow requirements. Therefore, we designed multi-nozzle injectors to improve injector performance over the full operating range. We chose four types of nozzles (10%, 20%, 20% and 50% of the fuel cell rated power) because this type can meet the overall power requirements by combining nozzles with different operating modes. If the number of nozzles is less than four, the injector cannot operate within the entire power range of the fuel cell, and when the number of nozzles is greater than four, the structure of the injector will be too complex.

$${A}_{nt2}={A}_{nt3}=20\%{A}_{nt}$$

$${A}_{ntX}=\frac{1}{4}\mathrm{PI}{D}_{X}{}^{2}$$

Where ${D}_{X}$ is the nozzle (X) diameter, X = 1, 2, 3, 4.

$$\mathrm{oh}=\frac{{\mathrm{rice}}_{s}}{{\mathrm{rice}}_{p}}=\frac{{\mathrm{rice}}_{s}}{{\mathrm{rice}}_{F1}+{\mathrm{rice}}_{F2}+{\mathrm{rice}}_{F3}+{\mathrm{rice}}_{F4}}$$

Where, ${\mathrm{rice}}_{F1}$, ${\mathrm{rice}}_{F2}$, ${\mathrm{rice}}_{F3}$ and ${\mathrm{rice}}_{F4}$ are the mass flow rates of N1, N2, N3 and N4 respectively.