# A Review of Hydrogen Sensors for ECLSS: Fundamentals, Recent Advances, and Challenges
## 3 Heat Conduction Hydrogen Sensors
### Principle of Concentration Measurement
Ideally, all the heat of the sensor is generated via the internal resistance in the rated state, i.e., \(Q_{i}=I^{2}r\). It is known from physics that there are three cases of heat transfer: heat conduction, heat radiation, and heat convection. The length of the selected detection element resistive wire is often much larger than its diameter, meaning the heat loss in the axial direction can be ignored. Considering also that heat convection does not exist in practical applications, the heat loss is, therefore, composed of only two parts: heat conduction loss of heat \(Q_{1}\) and heat radiation loss of heat \(Q_{2}\). \[Q_{0}=Q_{1}+Q_{2} \tag{4}\]  
where the expression for \(Q_{1}\) is  
\[Q_{1}=\lambda S(T-T_{0}) \tag{5}\]  
the thermal conductivity of the gas is \(\lambda\), the sensor temperature is \(T\), the ambient temperature is \(T_{0}\), both \(T\)s are Kelvin temperatures, and the surface area of the sensor is \(S\). According to [PERSON]'s law, the expression for \(Q_{2}\) is  
\[Q_{2}=A\sigma S\left(T^{4}-T_{0}^{4}\right) \tag{6}\]  
the radiation coefficient is \(A\), while \(\sigma\) is the [PERSON] constant. According to the law of conservation of energy, the heat produced is dynamically balanced with the heat lost; therefore,  
\[Q_{i}=Q_{0}=Q_{1}+Q_{2} \tag{7}\]  
\[I^{2}r=\lambda S(T-T_{0})+A\sigma S(T-T_{0}) \tag{8}\]  
When the target gas concentration is 0, the Wheatstone bridge is in equilibrium and the initial operating current of the sensor is \(I_{0}\). The thermal equilibrium equation at this point is  
\[I_{0}^{2}\ r=\lambda_{0}S(T-T_{0})+A\sigma S(T-T_{0}) \tag{9}\]  
When the target gas measurement is carried out, the thermal conductivity of the target gas mixed with air is \(\lambda_{1}\); the ambient temperature \(T_{0}\) is kept constant; the sensor is detected at a constant temperature, meaning that the temperature \(T\) is also kept constant; and the initial current \(I_{0}\) is transformed into the operating current \(I\). The above equation can be combined to obtain the constant temperature detection equation of the sensor:  
\[I^{2}=I_{0}^{2}+\frac{(\lambda_{1}-\lambda_{0})S(T-T_{0})}{r} \tag{10}\]  
From the above equation, it can be seen that, in practice, \(r\), \(T_{0}\), \(T\), \(S\), \(\lambda_{0}\), and \(I_{0}\) are definite values; thus, \(I_{2}\propto\lambda_{1}\). The difference between the set operating current and the initial test circuit is \(\Delta I\). \[I=I_{0}+\Delta I \tag{11}\]  
Substituting Equation (10) into Equation (9), while ignoring the small change \(\Delta I\), it is true that  
Figure 3:0  
From this equation, it can be seen that the current \(I\) is linearly related to the thermal conductivity \(\lambda_{1}\) of the target gas when \(\Delta I\) does not change much. Next, using [PERSON]'s law, this operating current flows through the resistor and becomes a potential signal output, which represents the change in the thermal conductivity of the gas mixture. According to the thermodynamic knowledge, when the target gas is in the background gas of air, the \(\lambda_{1}\) of the gas mixture is linearly related to the concentration \(C\) of the target gas, and the expression is  
Figure 3:1  
Therefore, it is only necessary to know the thermal conductivity \(\lambda_{1}\) of the gas mixture to obtain the concentration \(C\) of the target gas and complete the measurement of the gas concentration.