Measuring Carbon Dioxide Concentration in Humidity Air

Tu Xiang Zheng

POSIFA’s thermopile thermal conductivity sensors are manufactured by MEMS and CMOS technologies which offer real "best-in-class" performance for drift accuracy, linearity and repeatable performance, as well as lower cost-of-ownership. It can be widely used for determining the gas concentrations of humid binary mixtures of gases. It can also be calibrated to measure a single component of a multi-component gas mixture.

As a MEMS device, a POSIFA’s thermopile thermal conductivity sensor comprises a hot plate for minimizing its power consumption and an integrated thermopile for measuring the temperature difference between the hot plate and the body of the sensor. The hot plate is crated in a silicon substrate and suspended over a cavity recessed in the substrate so as to have its three edges supported by the body of the substrate and the rest edge free. In order to make the hot plate being hot a resistive resistor is positioned near its free edge, which is heated by applying a voltage. After heating a temperature difference is established between the heater and the body of the substrate since the heat transfer from the heater to the body is resistance by the thin film structure of the hot plate. The thermopile is configured to have its hot junctions along one side of the heater and its cold junctions on the edges of the body and so the temperature difference can be measured by the thermopile.

In operation, a measurement sensor and a reference sensor are typically connected in a differential amplifier circuit and generally operated at a constant voltage. When the sensors are all running in dry air, the sensors loose heat at a similar rate resulting in a zero differential signal between the two sensors providing to the amplifier. When the measurement sensor is exposed to the gas mixture of dry air and carbon dioxide, the atmosphere around the sensor changes, resulting in a lesser amount of heat being lost from the sensor, leading to an increase in the temperature difference of the thermopiles. The increase is dependent on the mixture thermal conductivity being less than the thermal conductivity of the dry gas. The reference sensor being sealed does not show this effect.

The output voltage of each thermopile is proportional to the total thermal conductance. Neglecting radiation effects, the total thermal conductance is equal to the sum of the thermal conductance of the thin film structure, the thermal conductance through the gas and the heat loss by convection. Since the thin film structure conductivity and the heat loss by convection are identical in both sensors and keeping both sensors at the same temperature, the output voltage difference of the two thermopiles depends only on the thermal conductance difference between the measurement gas mixture and the reference dry air.

The output voltage difference ΔV of the two thermopiles can be expressed as

ΔV = NS (T_h -T_d) (λ_m –λ_a)                                                                 (1)

Where N is the number of the thermocouples, S is the Seebeck coefficient of the thermocouple, T_h is the temperature of the heater of the sensor, T_b is the temperature of the body of the sensor, λ_m is the thermal conductivity of the gas mixture, λ_a is the thermal conductivity of the dry air.

The thermal conductivity of the gas mixture can be calculated by

λ_m  = λ_a n_a + λ_CO2 n_CO2 = λ_a (1- n_CO2 )+ λ_CO2 n_CO2                               (2)

Where n_a is the volume fraction of the dry air, n_CO2 is the volume fraction of carbon dioxide gas, and λ_CO2 is the thermal conductivity of carbon dioxide gas.

The thermal conductivities of carbon dioxide gas and dry air can be calculated by

λ_CO2 = -2.400 x 10^-5 + 2.16 x 10^-7 T – 3.244 x 10^-11 T²                      (3)

λ_a = kA₀ + kA₁ T+kA₂T² + kA₃T³ +kA₄T⁴ +kA₅T⁵                                             (4)

Where KA0 = 2.276501 x 10³, = 1.2598485 x 10⁴, KA2 = 1.4815235 x 10⁷,

KA3 = 1.73550646 x 10¹⁰, KA4 = 1.066657 x 10¹³ and KA5 = 2.47663035 x 10¹⁷.

With equation (1) and expressions (2), (3) and (4) the concentration of the carbon dioxide gas in dry air can be calculated based on the measured differential voltages of the thermopiles of the two POSIFA’s thermopile thermal conductivity sensors. 

Carbon dioxide gas is usually mixed with humid air instead of dry air. Tsilingiris suggests the following expression, which was original proposed by Wassiljewa, as the basis of the calculation of the thermal conductivity of a humid air.

λ_mh = { [1-RH (p_sv/p₀)]λ_a } / { [1-RH (p_sv/p₀)]+[RH (p_sv/p₀)Φ_av] }+

{ [RH (p_sv/p₀)λ_v] } / { RH (p_sv/p₀)+[1-RH (p_sv/p₀)]Φ_va }                   (5)

Where

RH is the relative humidity of the humid air,

P_sv is the saturated vapor pressure of water,

p₀ is the total atmospheric pressure,

λ_v is the thermal conductivity of water vapor,

Φ_av is the interaction parameter between air and water vapor and

Φ_va is the molecular interaction parameter between water vapor and air.

The saturated vapor pressure of water was expressed as

Psv = E₀ + E₁ t + E₂ t ² + E₃ t ³ + E₄ t ⁴                                          (6)

Where E₀ = 0.7073034146, E₁ = 2.703615165 x 10², E₂ = 4.36088211 x 10³,

E₃ = 4.662575642 x 10⁵ and E₄ = 1.034693708 x 10⁶.

The molecular interaction parameters were expressed as

Φ_av = 2^1/2 (1 + M_a/M_v)^-1/2[ 1 +(μ_a/μ_v)^1/2(M_v/M_a)^1/4 ]²                       (7)

Φ_va = 2^1/2 (1 + M_v/M_a)^-1/2[ 1 +(μ_v/μ_a)^1/2(M_a/M_v)^1/4 ]²                       (8)

Where M_a and M_v are the molecular weights of air and water vapor, μ_a and μ_v are the viscosity of the air and water vapor.

The viscosity of dry air and water vapor are offered by the following correlations

μ_a = MA₀ + MA₁ T+ MA₂  T²+ MA₃ T³+ MA₄ T⁴                            (9)

Where MA₀ =-9.8601 x 10^-1, MA₁= 9.080125 x 10^-2, MA₂ = -1.17635575 x 10^-4,

MA₃ =1.2349703 x 10^-7and MA₄ = -5.7971299 x 10^-11.

μ_v = MV₀ + MV₁T                                                                          (10)

Where MV₀ = 8.058131868 x 10¹ and MV₁ = 4.000549451 x 10^-1.

The thermal conductivity of water vapor was expressed as

λ_v = KV₀ + KV₁ T + KV₂ T²                                                             (11)

Where KV₀ = 1.761758242 x 10¹, KV₁ = 5.558941059 x 10^-2 and KV₂ = 1.663336663 x 10^-4.

The unknown parameters in the equation (5) are the relative humidity RH and the temperature T which are commonly measured by using a relative humidity sensor and a temperature sensor. Then the concentration of the carbon dioxide gas in the humid air can also be determined based on the date provided by the POSIFA’s thermopile thermal conductivity sensors.