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 (Th -Td)
(λm –λa)
(1)
Where N is the number of the thermocouples, S is the Seebeck
coefficient of the thermocouple, Th is the temperature of the heater
of the sensor, Tb is the temperature of the body of the sensor, λm
is the thermal conductivity of the gas mixture, λa is the
thermal conductivity of the dry air.
The thermal conductivity of the gas mixture can be
calculated by
λm = λa na + λCO2
nCO2 = λa (1- nCO2 )+ λCO2
nCO2 (2)
Where na is the volume fraction of the dry air, nCO2
is the volume fraction of carbon dioxide gas, and λCO2 is the
thermal conductivity of carbon dioxide gas.
The thermal conductivities of carbon dioxide gas and dry air
can be calculated by
λCO2 =
-2.400 x 10-5 + 2.16 x 10-7 T – 3.244 x 10-11 T2 (3)
λa = 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.
λmh = { [1-RH
(psv/p0)]λa } / { [1-RH (psv/p0)]+[RH
(psv/p0)Φav] }+
{ [RH (psv/p0)λv]
} / { RH (psv/p0)+[1-RH (psv/p0)]Φva
} (5)
Where
RH is the relative humidity of the humid air,
Psv is the saturated vapor pressure of
water,
p0 is the total atmospheric pressure,
λv is the thermal conductivity of water vapor,
Φav is the interaction parameter between air and
water vapor and
Φva 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
Φav = 21/2
(1 + Ma/Mv)-1/2[ 1 +(μa/μv)1/2(Mv/Ma)1/4
]2 (7)
Φva = 21/2
(1 + Mv/Ma)-1/2[ 1 +(μv/μa)1/2(Ma/Mv)1/4
]2 (8)
Where Ma and Mv are the molecular
weights of air and water vapor, μa and μv are the
viscosity of the air and water vapor.
The viscosity of dry air and water vapor are offered by the
following correlations
μa = 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.
μv = 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
λv = 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.
