Evaluation Procedure for the Exhaust Gas Measurements

Gas Components

A.1 Evaluation Procedure for the Exhaust Gas Measurements

that the concentrations of H2, CO, and HC are zero. The offset of theλsensor is basically unknown. However, it is assumed to be quite accurate far lean from stoichiometry and downstream of the TWC. Its maximum offset is assumed to be 0.5%. Theλmeasured at this point isλ^cal. In order to determine the theoretically correct concentrations of CO2, H2O, N2, and O2at this calibration point, a very simple combustion calculation is performed. It is assumed that only CO₂, H₂O, O₂, N₂, and NO occur in the exhaust downstream of the TWC atλ= 1.1. The gasoline consists of carbon (C), hydrogen (H) and oxygen (O) at ratios obtained from an analysis performed at the Swiss Federal Laboratories for Materials Testing and Research (EMPA). Traces of N and other compounds are neglected. The composition of the humid air is assumed to be

(21−0.21ϕ)·O2+ (79−0.79ϕ)·N2+ϕ·H2O (A.1) where ϕis the water concentration in % in the humid air. Thus, the overall combustion reaction can be expressed as follows:

1

³

α+^β₄ −^γ2

´ λ

CαHβOγ+ O2+79−0.79ϕ

21−0.21ϕN2+ ϕ

21−0.21ϕH2O→ ν1CO2+ν2H2O +ν3O2+ν4N2+ν5NO (A.2) Thereby, the coefficients are

ν1 = α

³

α+^β₄ −^γ2

´ λ^cal ν2 =

β 2

³α+^β₄ −^γ2

´λ^cal

+ ϕ

21−0.21ϕ ν3 = 1− 1

λ^cal −1 2ν5

ν4 = 79−0.79ϕ 21−0.21ϕ−1

2ν5 (A.3)

The coefficient for NO,ν5, is obtained from measurements, which are corrected in this point as follows:

y_NO^cal =y_NO^meas(1 +qH2Oy^cal_H2O+qCO2y_CO^cal₂) (A.4) qH2OandqCO2 denote the correction factors for the compensation of the dis- tortion occurring from the quenching of the chemiluminescence reaction in the NO measurement device by H₂O and CO₂.

As has been shown in Section 2.2, CO2 is measured in the dehumidified exhaust gas. The relation of the dry and the wet CO2mole fractions is

y_CO^dry₂ = y^wet_CO2

1−yH2O+y_H^dry_2O (A.5) y_H^dry_2Ois the mole fraction of the water in the dehumidified exhaust gas. It corre- sponds to the saturation concentration of water at5^◦C, which is the operating temperature of the Horiba device.

The mole fractions of CO2and H2O can be obtained from (A.3) and inserted in (A.5), leading to

y^dry_CO2 = ν1 5

X

i=1

νi

· 1

1− ν2 5

X

i=1

νi

+y^dry_H2O

= α

½µβ 4 +γ

2

¶³

1 +y^dry_H2O´

−β 2 +· · · +100(1 +y^dry_H2O)−ϕ

21−0.21ϕ µ

α+β 4 −γ

2

¶ λ^cal

)−1

(A.6) This allows the calculation of a correction factor for the CO₂ concentration, i. e.,

y^dry_CO2 =fCO2y^meas_CO2 (A.7) Fromλ^cal, the calibration mole fractions of water, nitrogen, and CO₂can be calculated:

y^cal_H2O = ν2(λ^cal)

5

X

i=1

νi(λ^cal) y_N^cal₂ = ν4(λ^cal)

5

X

i=1

νi(λ^cal) y^cal_CO2 = ν1(λ^cal)

5

X

i=1

νi(λ^cal)

(A.8)

The calibration mole fraction of NO has already been calculated in (A.4). The sum of the mole fractions obtained from the mass spectrometer (ξi) is scaled to 1. Notice that there are no hydrogen or hydrocarbons present, hence:

ξ_N^cal₂ = y_N^cal₂ 1−y_CO^cal₂−y_NO^cal ξ^cal_H2O = y^cal_H2O

1−y_CO^cal₂−y_NO^cal (A.9)

The goal is now to derive correction factors fH2O andfN2 for the water and the nitrogen mole fractions. These will be valid in all operating conditions, since the ion currents are proportional to the partial pressures in the focused operating region, see also Section2.2.

The sum of corrected fractions has again to be scaled to 1. Hence, the correc- tion factors have to adjust the fractions after scaling, they cannot be obtained by simply dividing the measured (ξ_i^meas) by the calculated mole fractions (ξ_i^cal).

The new mole fractions after scaling are

ξ_H^cal_2O = fH2Oξ^meas_H2O

ξ^meas_O2 +fN2ξ_N^meas₂ +fH2Oξ^meas_H2O ξ^cal_N2 = fN2ξ_N^meas₂

ξ^meas_O2 +fN2ξ_N^meas₂ +fH2Oξ^meas_H2O (A.10)

Solving for the correction factors yields the desired result:

fH2O = ξ_O^meas₂ ξ_H^meas_2O

ξ_H^cal_2O −ξ^meas_H2O Ã

1 + ξ^cal_N2 ξ_H^cal_2O

!

fN2 = ξ_O^meas₂ ξ_N^meas₂

ξ^cal_N2 −ξ^meas_N2 Ã

1 +ξ_H^cal_2O ξ^cal_N2

! (A.11)

With these correction factors, the scaled mole fractions obtained from the mass

spectrometer can be calculated as follows:

ξH2 = ξ_H^meas₂

ξ^meas_H2 +ξ_O^meas₂ +fH2Oξ_H^meas_2O +fN2ξ_N^meas₂ ξO2 = ξ_O^meas₂

ξ^meas_H2 +ξ_O^meas₂ +fH2Oξ_H^meas_2O +fN2ξ_N^meas₂ ξH2O = fH2Oξ_H^meas_2O

ξ^meas_H2 +ξ_O^meas₂ +fH₂Oξ_H^meas_2O +fN₂ξ_N^meas₂ ξN2 = fN2ξ_N^meas₂

ξ^meas_H2 +ξ_O^meas₂ +fH2Oξ_H^meas_2O +fN2ξ_N^meas₂ (A.12) Wet CO/CO2Concentrations

The CO/CO₂measurement device only provides concentrations related to the dehumidified exhaust gas. Since water is measured, as well, the CO/CO₂con- centrations related to the wet, i. e., total exhaust can be obtained. However, the calculation is somewhat awkward, since the water concentration has to be obtained from the mass spectrometer measurements, which in turn makes the use of the CO/CO2concentrations necessary.

The wet CO/CO2mole fractions are calculated as follows:

y^wet_CO2 = y^dry_CO2(1−yH2O+y_H^dry_2O)

y_CO^wet = y^dry_CO(1−yH2O+y^dry_H2O) (A.13) The water fraction is obtained from the mass spectrometer measurements. These have to be scaled to the total exhaust gas:

yH2O = ξH2O(1−y_CO^wet₂−y^wet_CO −y^corr_NO −yHC)

= ξH2O(1−y_CO^wet₂−y^wet_CO −...

−y^meas_NO (1 +qH₂OyH₂O+qCO₂y^wet_CO2)−yHC) (A.14) The simplest way to solve for the H₂O, CO, CO₂, and NO concentrations, is to start with the water. Inserting (A.13) in (A.14) and solving for the desired water fractionyH2Oyields after some algebra

yH2O=ζ₁^NOξH2O(1−(ζ₂^NOy^dry_CO2+y^dry_CO)(1 +y_H^dry_2O)−y_NO^meas−yHC) 1−ζ₁^NOξH2O(ζ₂^NOy^dry_CO2+y^dry_CO)

(A.15)

where

ζ₁^NO = 1

1 +qH2Oy_NO^meas (A.16) ζ₂^NO = 1 +qCO2y_NO^meas. (A.17) With the known water concentration, yCO2 andyCO can be obtained easily from (A.13).

Corrected NO Concentration

Now that the H₂O and the wet CO₂concentrations are known, the corrected NO concentration can be calculated as follows:

y_NO^corr=y^meas_NO (1 +qH2OyH2O+qCO2y^wet_CO2) (A.18) Rescaling of the Mass Spectrometer Measurements

The sum of mole fractions provided by the mass spectrometer is scaled to 1.

Since four species are not measured by this device, the fractions have to be rescaled such that the sum of all species becomes 1. This is done in a straight- forward manner:

yi|i=H2,H2O,N2,O2=

ξi|^i=H2,H₂O,N₂,O₂·(1−y_CO^wet₂−y_CO^wet−y_NO^corr−yHC) (A.19)