Experiment design for
measuring component of gas mixtures
using thermal
conductivity sensors
Xiang Zheng Tu
Gas mixture is the combination of two or more gases. One example of
a mixture is air which is made up of nitrogen, oxygen, and smaller
amounts of other gases. The other gases include water vapor (humidity), carbon
dioxide, and methane. These gases may pollute our air at any given time or
place and affect human health and safety or environment protection. To avoid
these things happen, many gas sensors are used to monitor the levels of the
gases in the atmosphere so as to maintain each gas concentrations below a safe
levels. They are recommended for carbon dioxide: 5000ppm, methane: 1000ppm and
humidity: 30% to 50% (relative humidity).
All gases conduct heat to differing degrees, and the amount
of heat transferred by a gas is determined by its thermal conductivity. This
property can be exploited in sensing because each gas has a different thermal
conductivity. POSIFA’s Thermal Conductivity Sensors use this property to
accurately measure one of the three gases present in a pseudo-ternary gas
mixture such as air, water vapor (humidity) and carbon dioxide or air, water
vapor (humidity) and methane. The “pseudo-ternary” means the dry air here is treated
as a simple gas.
The detection principle of thermal conductivity sensors is
as follows. Temperature differences are produced between the hot junctions and cold
junctions of one or tow thermopiles. The hot junctions and cold junctions are
positioned on a hot plate and a frame both are created in a silicon substrate,
respectively. The hot plate is heated by applying a required electrical power
to a resister positioned a long the hot junctions. Heat is transferred from the
hot plate to the substrate via thermal conduction through the gas mixture
filled the cavity under the hot plate. A temperature gradient is established
due to the thermal flow energy in the gas mixture. The temperature difference
or thermopile(s) output for the thermal conductivity sensor, therefore, is a
direct measure of the thermal conductivity of the gas mixture. Heat loss due to
radiation, convection and heat conduction through the terminals of the hot
plate has been minimized by the sensor MEMS structure.
According to Wassiljewa’s equation the thermal conductivity
km for a mixture of three gases 1, 2 and 3 can be expressed as:
km = k1
/ (1 + A12 x2 / x1 + A13 x3
/ x1) + k2 /
(1 + A21 x1 / x2 + A23 x3 / x2)
+ k3 / (1 + A31 x1 / x3 + A32 x2 / x3) (1)
Where k1, k2 and k3 are the
thermal conductivity of gases 1 (air) , 2 (water vapor) and 3 (carbon dioxide
or methane), x1, x2 and x3 are the mole
fraction of gases 1, 2 and 3, and A12, A13, A21,
A23, A31 and A32 are Wassiljewa’s coefficients
which are the functions of the molar masses and viscosities of the two related
gases.
The thermal conductivity of the pseudo-ternary gas mixture does
not vary linearly with the composition of the mixture. As seen from the
equation (1) if the changes of the mole fraction of the gases are small enough
the equation will be reduced as:
km ≈ a1
x1 + a2 x2 + a3x3 (2)
Where a1, a2 and a3 are constants
related to the thermal conductivities k1, k2 and k3 respectively.
For the water vapor (humidity) and carbon dioxide and air, water vapor
(humidity) and methane the above mentioned assumption is true.
The output Seebeck voltage of the thermopile(s) of the
thermal conductivity sensor Useebeck can be expressed as:
Useebeck =
S km (3)
Where S is sensitivity of the sensor, which is the
determined by the input power and the parameters of the sensor structure. It
has been shown that it is the functions of the width of the hotplate, length of
the palate supporting beams and path of the cavity filled with the gas mixture.
It should be noticed that the three component mixture has 2
degrees of freedom and the mole fraction of gas 1 can be calculated by
equation:
x1 + x2
+ x3 = 1 (4)
Combine of equation (3) and (4) results an equation as:
Useebeck =
b1 + b2 x2+ b3 x3
(5)
Where b1 is a bias term in the output Seebeck
voltage of the sensor so that the solution of the equation is no zero solution,
and b2, b3 are sensitive to component 2 and 3 mole
fractions, respectively. Since the output is expressed in unit of V, the
coefficients b1, b2 and b3 are expressed as
the same.
A three factorial experimental design may be used for
determining the coefficients b1, b2 and b3 by
the experimental measurements of the thermal conductivity sensors.The design
includes three treatments x1, x2 and x3 of the
experimental variable, nine levels L1, L2, L3
and L9 of the control variable and nine observations and has 9
different cells as shown below. In the design the means for the columns provide
the researcher with an estimate of the main effects for treatments and the
means for rows provide an estimate of the main effects for the levels. The
design also enables the researcher to determine the interaction between
treatments and levels.
The general linear model is a statistical
linear model. The
general linear model system of equations may be expressed elegantly using
matrix notation as:
Representing the indicated vectors and matrix with single
letters, the form of the general linear model system of equations may be changed
as:
U = X B + E (7)
{\displaystyle
\mathbf {Y} =\mathbf {X} \mathbf {B} +\mathbf {U} ,}
Where U is a matrix with series of multivariate
measurements, X is a matrix that might be a
design matrix, B is a matrix containing
parameters that are usually to be estimated and E is a matrix
containing
errors or noise.
Given the data U and the design matrix X, the
general linear model fitting procedure has to find a set of B values explaining
the data as good as possible. The time course values predicted by the model are
obtained by the linear combination of the predictors:
U = X B (8)
A good fit would be achieved with B values leading to predicted
values which are as close as possible to the measured values u. By
rearranging the system of equations, it is evident that a good prediction of
the data implies small error values:
E = U – X B = U – u (9)
An intuitive idea would be to find those beta values
minimizing the sum of error values. Since the error values contain both
positive and negative values (and because of additional statistical
considerations), the general linear model procedure does not estimate B values
minimizing the sum of error values, but finds those B values minimizing the sum
of squared error values:
E’ E = (U – X B)’ (U – X B) > min (10)
The term E’ E is the vector notation for the sum of
squares. The apostrophe symbol denotes transposition of a vector or matrix. The
optimal B weights minimizing the squared error values are obtained
non-iteratively by the following equation:
B = (X’ X)-1 X’ U (11)
The term in brackets contains a matrix-matrix multiplication
of the transposed, X', and non-transposed, X, design matrix. This
term results in a square matrix with a number of rows and columns corresponding
to the number of predictors. This X'X matrix corresponds to the
predictor variance-covariance matrix. The variance-covariance matrix is
inverted as denoted by the "-1" symbol. The resulting matrix (X'X)-1 plays
an essential role not only for the calculation of beta values but also for
testing the significance of contrasts. The remaining term on the right
side, X'U, evaluates to a vector containing as many elements as
predictors. Each element of this vector is the scalar product of a predictor
time course with the observed voxel time course.