Ternary Gas Mixture Measurements Using Micromachined Thermal Conductivity Sensors

Xiang Zheng Tu

 

According to Chapman–Enskog theory elastic gases deviation from the Maxwell–Boltzmann distribution in the equilibrium is small and it can be treated as a perturbation.

So the thermal conductivity of the ternary gas mixture can be expressed as  

K_mix = k₁ N₁ / (N₁ + N₂ Φ₁₂ + N₁₃ Φ₁₃) + k₂ N₂ / (N₂ + N₃ Φ₂₃ + N₁ Φ₂₁)

+ k₃ N₃ / (N₃ + N₁ Φ₃₁ + N₂ Φ₃₂)                                                                        (1)

N₁ + N₂ + N₃ = 1                                                                                              (2)

where Φ₁₂, Φ₁₃, Φ ₂₃, Φ ₂₁, Φ ₃₁ and Φ ₃₂ are the Wasiljewa constants, k, k₂, k₃ are the conductivities of air, carbon dioxide and water vapor, and N₁, N₂ and N₃ are the molar fractions of air, carbon dioxide and water vapor.

The Wasiljewa constants can be given by

Φ_αβ = (1/8^1/2) ( 1 + M_α/M_β)^-1/2 [ 1 + (μ_α /μ_β )^1/2 (M_β / M_α )^1/4 ]²                                        (3)

Hear M_α is the molecular weight of species α and μ_α is the viscosity of pure species α. Equations (1),(2) and (3) has been shown to reproduce measured values of the thermal conductivity of mixtures within an average deviation of about 2%.

Equation (1) (2) and (3) are used to predict the thermal conductivity of a gas mixture of CO₂, O₂ and N₂. The following data of the pure CO₂, O₂ and N₂ at 1 atm and 293K can be found from a Physical Handbook.

It is assumed that molecular fractions of CO₂ (1), O₂ (2) and N₂ (3) are 0.133, 0.039 and 0.828 respectively. Using equation (3) it can be found the related values as

                                          N₁+N₂Φ₁₂+N₁₃Φ₁₃=0.763              (4)

N₂+N₃Φ₂₃ +N₁Φ₂₁ =1.057              (5)

N₃+N₁Φ₃₁ +N₂Φ₃₂ =1.049              (6)

Substitution in equation (1) gives

K_mix =(0.133)(383)(10^-7) /0.763+(0.039)(612)(10^-7) /1.057+(0.828)(627)(10^-7) /1.049

        =584(10^-7) cal/cm-s-K                                                      (7)

This is the principle of thermal conductivity sensors able to measure the concentrations of any gas mixtures such as a ternary gas mixture consisting of CO₂, O₂ and N₂. The thermal conductivity sensors manufactured by POSIFA Microsystems Company are shown in the above figure. The sensors are created in a silicon substrate and configured to have a hot plate suspending over a cavity recessed into the substrate, a resistive heater and a plural of hot junctions of a thermopile disposed on the hot plate and a plural of cold junction of the thermopile disposed the frame region of the cavity which is formed by the substrate. An interface circuit of the sensors is also shown in the above figure. The circuit comprises a microcontroller, a pre-amplifier, a measurement thermal conductivity sensor and a reference thermal conductivity sensor. The two sensors are heated by applying PWM to the sensor heaters from the microcontroller. The outputs of the sensors are sent to the pre-amplifier and then to the microcontroller for digital processing. The reference sensor is used to compensate the offset, temperature drift and noise of the measurement sensor.

The quality of air inside a building depends on the concentrations of contaminants which are difficult to measure. However, CO₂ levels, which are easy to measure, can be used in place of other measurements to indicate the indoor air quality. CO₂ is produced when people breathe. Each exhaled breath by an average adult contains 35,000 to 50,000 ppm of CO₂ – 100 times higher than 350 to 500 ppm that is typically found in the outside air.

If a thermal conductivity sensing module is installed in a building it will tell you how clean or polluted your air is, and also actuates a ventilation system to supply the building continuously with fresh air. Other applications of the thermal conductivity sensing modules include:

• 0 – 100% Hydrogen in Air
• 0 – 100% Methane in Air
• 0 – 100% Carbon Dioxide in Methane
• 0 – 100% Helium in Air