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 k_m for a mixture of three gases 1, 2 and 3 can be expressed as:

k_m = 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 A_12, A_13, 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:

k_m ≈ a₁ x₁ + a₂ x₂ + a₃x₃                                                          (2)

Where a₁, a₂ and a₃ are constants related to the thermal conductivities k_1, 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 U_seebeck can be expressed as:

U_seebeck = S k_m                                                                            (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:

U_seebeck = 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.