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

Micromachined Thermal Conductivity Sensor

with a Thermopile on a Hot-plate

Xiang Zheng Tu

  

The thermal conductivity sensor with a heater and a thermopile is manufactured by POSIFA Microsystems Company. The sensor is created in a silicon substrate and constructed with a thin membrane suspending over a cavity recessed into the substrate. A resistive heater and a plural of hot junctions of a thermopile are disposed on the membrane and a plural of cold junctions of the thermopile are disposed the top of the substrate which is surrounded the membrane. The cavity is configured to have a bottom surface parallel with the top membrane allowing heat generated by the heater transfers perpendicular through the cavity to the bottom. The path length is optimized to have a maximum heat conduction transfer efficient.

The sensors rely on the thermal conductivity of a gas mixture which affects thermal phenomenon by way of heat conduction transfer that, in turn, is converted into a varying electrical signal capturing the sensor response to its component concentration change. As shown in the top figure, the sensors are thermally isolated so only heat transfer due to thermal conductivity through a cavity. Other heat transfer pathways such as through substrate or electrical leads result in thermal losses that degrade sensor performance and have been minimized in the device design.

In sensor operation the heat P_heat generated inside the heater by a DC voltage U_DC applied to the output terminals of the thermopile sensor follows the equation

P_heat = U ² R_sensor . (1)

We can assume that the thermal contact between the periphery and the ambient is so good that the temperatures of the periphery and the ambient environment are identical. They are equal to T_amb. Since the heat P_heat is generated on the membrane, its temperature T_mem depends on the thermal conductivity λ_mem of the hot plate and the thermal conductivity λ_gas of the gas filled in the cavity as

T_mem = P_heat / (λ_mem + λ_gas) + T_amb. (2)

The generated thermopile voltage U is proportional to the temperature difference between the membrane and the periphery as

U_DC ∝ ΔT = T_mem − T_per ≈ Pheat / (λ_mem + λ_gs) . (3)

Thus,

U_DC ∝ U ² / ( λ_mem + λ_gas) . (4)

Each gas has a known thermal conductivity. The thermal conductivities of some gases can be found in the table below.

Gas	Thermal Conductivity
ACETYLENE	4.400
AMMONIA	5.135
ARGON	3.880
CARBON DIOXIDE	3.393
CARBON MONOXIDE	5.425
CHLORINE	1.829
ETHANE	4.303
ETHYLENE	4.020
HELIUM	33.60
HYDROGEN	39.60
HYDROGEN SULPHIDE	3.045
METHANE	7.200
NEON	10.87
NITRIC OXIDE	5.550
NITROGEN	5.680
NITROUS OXIDE	3.515
OXYGEN	5.700
SULPHUR DIOXIDE	1.950

The sensors can be used not only measure all the gases listed in the table but also to analyze a whole range of binary gas mixtures provided that there are only two gases present and that the two gases have significantly different thermal conductivities.  Example is nitrogen and hydrogen or a pseudo-binary mix. Air is an example of a pseudo-binary mix: it has a fixed proportion of oxygen and nitrogen, both having very similar thermal conductivities and so behaves much like a single gas.

Other examples include:

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

Thermal conductivity sensor with a heater and a thermopile utilizes micromachining technology which is amenable to creating micro-heaters and thermal conductivity sensors with no moving parts required, thus simplifying fabrication and operational design requirements. Another reason for the large interest in thermal conductivity sensors is the advantages gained through miniaturization: low power consumption, higher sensitivity to low conductivity, fast response and ease of use with different modes of operation. 

Diaphragm Pump Controlled by Thermal Flow Sensor

Xiang Zheng Tu

  

A diaphragm pump controlled by a thermal flow sensor is shown in the above figure.

The pump assembly has a thermal flow sensor, a microcontroller, a NPN switch and a solenoid driven diaphragm pump. The microcontroller has a 10 bits ADC, due to noise and other accuracy diminishing factors, its true accuracy is less than 10 bits. This application provides a software-based oversampling technique, resulting in 16 bits resolution. When the diaphragm pump is in operation a fluid flow is driven to pass over the thermal flow sensor. The sensor measures the flow rate and output an electronic signal to the microcontroller. After ADC conversion a pulse width modulation (PWM) signal is generated by the microcontroller.  It is send to the NPN switch for applying a current to the solenoid driven diaphragm pump. The electromagnetic core of the diaphragm pump moves against a spring to slide a diaphragm into the discharge position. When current is removed, the diaphragm slides back into the suction position.

There are two ways for controlling the flow rates of the diaphragm pumps. When the used PWM frequency is in the range of 25 to 200 Hz the solenoid responds (full stroke) over the duty cycle range of control. At zero duty cycle the solenoid does not move, the pump is not opened and therefore the flow is zero. At 50% duty cycle the solenoid moves through full stroke and opens the pump to full flow. Since the pump is only allowing full flow for 50% of the time, the time averaged flow in theory will be 50% of maximum flow. This type of control is called “digital” because the pump is fully open or fully closed, “on” or “off”. Other way is the PWM frequency limited in the range of 200 to 1000 Hz which produces the time averaged current and does not allow the solenoid to fully respond as in digital control. In this case linear position control is realized and any flow rate between zero and maximum can be chosen by the user.

Theoretically diaphragm pumps can produce the same flow at a given speed (RPM) no matter what the discharge pressure. However, a slight increase in internal leakage as the pressure increases prevents a truly constant flow rate. The following figure shows the measured flow rates in one-hour intervals for an infusion pump. The X-axis reference lines showed the acceptable flow rate (5 mL/h ± 15%). In all experiments, pumps initially infused at a rate faster than their nominal flow, and then returned closer to their set rates up to the complete deflation. The percentage of the flow rate error (deviation from 5 mL/h ± 15%) was 100% in the first and second hours of infusion, 96% in the third hour, 60% in the 20th hour and zero percent in the rest of the infusion time. Flow rate error in the initial hours of infusion was due to fast pump flows, and in the 20th hour due to slow infusion rates.