An electrochemical model of an amperometric NOx sensor

Sensors & Actuators: B. Chemical 290 (2019) 302–311

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Sensors and Actuators B: Chemical journal homepage: www.elsevier.com/locate/snb

An electrochemical model of an amperometric NOx sensor a,⁎

a

a

b

c

Masoud Aliramezani , Charles Robert Koch , Marc Secanell , Robert E. Hayes , Ron Patrick a b c

T

Mechanical Engineering Department, University of Alberta, Edmonton, Canada T6G 1H9 Chemical Engineering Department, University of Alberta, Edmonton, Canada T6G 1H9 ECM, Los Altos, CA 94023, USA

ARTICLE INFO

ABSTRACT

Keywords: Electrochemical gas sensor Physics-based model Solid electrolyte Pt-Au NOx Polarization loss Diesel engine Spark Ignition Engine

To help design future amperometric NOx sensors, a physics-based sensor model that includes diffusion and electrochemical submodels is developed. It is shown that NO is partly reduced in the O2 sensing chamber which affects NO concentration in the O2 sensing and in the NOx sensing chamber. Therefore, the electrochemical model is developed to simulate partial reduction of NOx on the O2 sensing electrode and reduction of NOx on the NOx sensing electrode. A transport model that simulates diffusion of the gas species through the sensor diffusion barriers and sensor chambers is coupled to the electrochemical submodels. A fully controlled sensor test-rig that provides controlled gas mixtures is employed to carry out experiments to estimate model parameters. Then, the sensor is installed on the exhaust system of a medium duty Diesel engine and then on a port injection spark ignition engine. Experiments at different engine operating conditions with different NOx concentrations from 0 to 2820 ppm have been performed to validate the model accuracy at different operating conditions. Through the validation process, the NOx sensing cell voltage is changed experimentally at different NOx concentrations to evaluate the model accuracy at different cell voltages. The model results closely match the experiments with the maximum 12% error for the NOx sensing pumping current.

1. Introduction Solid-state electrochemical gas sensors have many remarkable properties that make them useful for a variety of applications such as automotive [1–3], biotechnology [4,5], medical [6] and many other industries [7–9]. The reliability, small size, fast response and low price of amperometric solid-state electrochemical sensors make them ideal for the automotive industry [10–12]. Amperometric NOx sensors which simultaneously measure the O2 and NOx concentration [13] are increasingly used in commercial combustion engines to meet the stringent emission regulations [14,15]. They are used in conjunction with the methods that have been developed to improve combustion efficiency and reduce engine emissions [16–19]. Amperometric sensors generate an output that is linearly dependant on the concentration of the measuring gas, which makes them suitable for detecting high gas concentrations [7,20]. Both the electrode reactions inside the sensor chambers and the diffusion of exhaust gas species through the sensor diffusion barriers affect the sensor output [21–24]. Typically, when the sensor reaction dynamics are much faster than the diffusion of species into the sensing chambers, the sensor is also called limiting current type [25]. In this case, the sensor output is proportional to the diffusion of species ⁎

through the sensor barriers into the sensor chambers. The diffusion mechanism of species through an amperometric NOx sensor has been found to be the normal multi-component diffusion mechanism through the sensor diffusion barriers [20]. Although an amperometric sensor is typically operated at a temperature and cell voltage range so that the diffusion of species is the dominant process that affects the sensor output [12], a more comprehensive understanding of the electrochemical cell is beneficial for optimizing sensor performance, durability and sensitivity [26,27]. The effect of electrochemical properties of electrodes on the sensing behavior of solid state electrochemical gas sensors has been studied in detail [28–31]. The sensitivity of amperometric sensors to gas species and the reliability of these sensors are directly affected by electrochemical properties [32]. These properties include electrode material [33–35], electrode potential [36], electrolyte properties [37–39] and properties of diffusion barrier [12,20,40,41]. These previous studies are mostly experimental and do not include quantitative physics-based models of the main relevant processes that simultaneously take place inside the sensor and affect the sensor output. An understanding of solid-state amperometric sensors that includes both the electrochemical properties and species diffusion is essential to provide further insight into the performance of these sensors and is the

Corresponding author. E-mail address: [email protected] (M. Aliramezani).

https://doi.org/10.1016/j.snb.2019.03.135 Received 22 January 2019; Received in revised form 25 March 2019; Accepted 29 March 2019 Available online 01 April 2019 0925-4005/ © 2019 Elsevier B.V. All rights reserved.

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subject of this paper. To do this, a physics-based model of an amperometric NOx sensor is developed that includes diffusion of multiple sample gas species into the sensor chambers through the sensor diffusion barriers. The model also includes an electrochemical submodel that calculates the sensor pumping current as a function of sensing cell voltage. To experimentally validate the model at different NOx concentrations and NOx sensing cell voltages, the sensor is first mounted in the exhaust system of a medium duty Diesel engine, then in a port injection spark ignition (SI) engine exhaust system and finally in a controlled gas mixture test bench. The diffusion model is used to calculate the mole fraction of O2 and NOx in the sensor chambers. The mole fractions are then used in the NOx sensing cell electrochemical model to define the sensor pumping current from the cell voltage. The diffusion and the electrochemical model are coupled and solved simultaneously. The ohmic loss, the activation loss and the concentration loss are also included in the model. The NOx sensor output is then calculated as a function of the sensing cell potential and NOx concentration.

3. Sensor model Typical current-voltage response of an amperometric sensor as the cell voltage (VP) is varied is shown in Fig. 4 [20,43,44]. The response is divided into six regions that depend on the sensing cell voltage. In region I, the cell voltage is lower than the open-circuit potential and not high enough to cause reduction of species on the cathode. Therefore, the reverse reaction (oxidation) takes place causing a negative cell current. When the cell voltage is just above the open-circuit potential, activation polarization becomes the rate-determining step. This phase is labeled as region II. In region III, the cell current varies almost linearly with voltage, based on Ohm's law. In this region, the Ohmic loss dominates the cell voltage-current relation. As the cell voltage increases further, the current-voltage relation becomes non-linear in region IV and then the cell current finally reaches a saturation level determined by the diffusion rate of oxygen into the chamber (region V). In region V the pumping rate of (O2−) ions from the cell has reached its maximum level since all available reducible molecules are being reduced on the cathode as soon as they reach the electrode surface. The transition phase from ohmic behavior to the limiting current region is marked as region IV. Further increases in the cell voltage, results in the partial pressure of the reduced species (in this case, NO) becoming lower according to the Nernst equation [45]. If the partial pressure of the reduced species (NO) decreases to less than a certain value (≈10−33 atm), electrolyte decomposition takes place [46]. This causes a further increase in the O2− pumping rate and therefore the cell current increases (region VI) which can potentially damage the sensor. The sensor behavior in this region is not studied in this work as it is not useful when measuring NOx.

2. Amperometric NOx sensor 2.1. Working principle of amperometric NOx sensor A production amperometric NOx sensor (ECM, P/N: 06-05 [42]) has been studied in this work. The sensor consists of two sensing chambers as shown schematically in Fig. 1. The first chamber is the O2 sensing chamber while the second chamber is the NOx sensing chamber. The first chamber has a Platinum (Pt)-based electrode (typically, PlatinumGold (Pt-Au) alloy) while the second chamber has a Pt electrode [21]. In the first chamber, O2 is reduced and the resulting O2− ions are pumped out with a pumping current proportional to the oxygen concentration in the sampling gas. The Pt-Au electrode has a higher activation energy for NO reduction than the Pt electrode and this enables selective O2 and NO2 reduction inside the first chamber at the typical sensor operating condition [21]. Therefore, NO2 is reduced to NO inside the first chamber. The remaining species diffuse into the second chamber through the second diffusion barrier. In the second chamber, NO is completely reduced and the resulting O2− ions are pumped out through an electrochemical cell into the reference chamber as shown in Fig. 2. Therefore, the pumping current of the second chamber cell is proportional to the total NOx (NO2 and NO) concentration. The working principle of the NOx sensing cell is schematically shown in Fig. 2.

3.1. NOx sensing cell electrochemical model The components of an overall NOx sensor model will be described. 3.1.1. NOx sensing cell electrochemical model A simple model of the reactions that take place in the NOx sensing cell is:

Cathode: NO + 2e

2.2. Sensor dimensions using X-ray tomography

1 N2 + O 2 2

(1a)

Anode: O2

1 O 2 + 2e 2

(1b)

Overall: NO

1 1 O2 + N2 2 2

(1c)

The Nernst voltage is then [45]:

The exact dimensions of the internal components of the NOx sensor are measured to parameterize the sensor model, using X-ray tomography with spatial resolution of 5176 × 1450 × 1051 pixels (0.007, 0.007 and 0.005 mm/pixel). The sensor top view using the x-ray tomography image is shown in Fig. 3 detailing the sensor components.

EN = Esco +

Sox red,sc (T 2F

To )

¯ RT ln 2F

0.5 0.5 O2,rc pN2,rc

(p p

NO,sc

)

(2)

where, pNO,sc, pO2,rc and p N2,rc are the partial pressure of NO in the second chamber, and the partial pressures of oxygen and nitrogen in the reference chamber, respectively. Note: Subscript rc is the reference chamber while subscript sc is the second chamber. R¯ is the universal gas

Fig. 1. NOx sensor working principle and diffusion model structure. 303

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Fig. 2. Boundary conditions of the NOx sensing cell (second chamber).

constant, F is the Faraday constant, T is the sensor temperature (in Kelvin), To is the reference temperature, Esco is the open-circuit potential at the standard state (1 atm; 298.15 K) and ΔSox−red is the change in entropy of products − reactants at the operating temperature (the index sc stands for the second chamber). In this case, it is approximated as [45]:

Sox

o Sox

red

red

=

i

Sio

Prod

j

S jo

(3)

Reac

where, νi and νj are the stoichiometric coefficient of species i and j respectively. When kinetic, ohmic and transport losses are included, the second chamber cell voltage (VP2) becomes [45,47]:

VP 2 =

Esco

+

Sox red,sc (T 2F

To) EN

pO0.5,rc pN0.52,rc ¯ RT ln( 2 ) 2F pNO,sc

Fig. 4. Typical electrode current vs potential relation.

VP 2 = Esco +

Sox red,sc (T 2F

To)

¯ x O0.5,rc x N0.52,rc RT ln( 2 ) 2F xNO,sc

EN

ac

ac

(5)

where, x O2,rc , x N2,rc and xNO,sc are the mole fraction of O2 in the reference chamber, the mole fraction of N2 in the reference chamber, and the mole fraction of NO in the second chamber respectively.

(4)

where, ηΩ and ηac, are Ohmic and kinetic-transport polarization losses respectively [45,47]. According to Dalton's law of partial pressures [48], the total mixture pressure is equal to the sum of partial pressures of all species in the mixture. Therefore, dividing by the total pressure of the mixture, Eq. (4) becomes:

3.1.2. Ohmic losses (ηΩ) The ohmic loss is calculated as follows:

= R cell iP 2 Ae2

(6)

where Rcell is the cell ohmic resistance, iP2 is the NOx sensing cell

Fig. 3. X-ray tomography of the NOx sensor ceramic (top view). 304

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M. Aliramezani, et al.

Fig. 5. NO mole fraction gradient inside the NOx sensor at different NO concentrations in the sample gas.

current density and Ae2 is the cross sectional area of the NOx sensing electrode. The ohmic resistance of Yttria Stabilized Zirconia (YSZ) electrolyte significantly decreases with temperature regardless of its thickness [49]. The ohmic loss, ηΩ, is neglected in this model due to the high sensor operating temperature (1023 K) and the very low value of cell current.

Ct =

xNO,sc o xNO,sc

[exp ( RT¯ F ) a2

ac

exp

(

c2 F

¯ RT

ac

)]

IP 2 = iP 2 Ae2 =

(7)

xNO,fc

Cathode: NOfc + 2e

1 N2 + O 2 2

1 N2 2

NOenv + 2e

Anode: O2 +

Overall: NOfc

(8)

xNO,sc LD2

xNO,sc LD2

(11)

3.2.1. O2 sensing cell electrochemical model Although NO is usually assumed not reduced in the first chamber, our experiments show that NO is indeed partly reduced in this chamber. This can affect the sensor behavior, at very low NOx sensing cell pumping current (IP2 ≈ 0). As the NOx sensing cell current reduces to zero, the molar flux of NO through this cell also reduces to zero, according to Eq. (8). This means that at zero NOx sensing cell current, there is no gradient in NO through the second diffusion barrier, according to Eq. (9) (also see Fig. 5). A model is developed to estimate the NO concentration that would reach the second chamber to consider the effect of partial reduction of NO in the first chamber on the NOx sensing cell behavior. In the first chamber, NO is reduced through the following reactions: (12a) (12b) (12c)

NOenv

where NOfc is the NO molecule inside the first chamber and NOenv is the NO molecule in the sample gas. Using a similar approach to that in Section 3.1.1, the effect of NO reduction on the first chamber cell voltage (VP1) can be expressed by:

in Amperes where NNO,sc[ mol ] is the molar flow of NO through the s second diffusion barrier to the second chamber. The relative molar flux of NO can be estimated using Fick's law [51]:

Ct D NO AD2

xNO,fc

Due to the observed NO reduction current in the first chamber at low IP2 currents, a model of the first chamber is needed to be coupled to the second chamber.

3.1.4. Diffusion Diffusion plays a pivotal role in the operation of amperometric sensors and must be included as one of the main components of the model. It is assumed that O2 has been completely removed by the first chamber and all NO2 molecules are converted to NO. The only species that is reduced in the second chamber is NO which diffuses through the second diffusion barrier. Each NO molecule transfers two electrons according to Eq. (1a). Therefore, according to Faraday's law, the pumping current of the NOx sensing cell is:

NNO,sc =

2FCt D NO AD2

3.2. Modeling NO reduction in the first chamber

In Eq. (7), αa2 and αc2 are the charge transfer coefficient of anode o are the NOx sensing and cathode respectively, while iP2, iPo2 and xNO,sc cell current density; the reference exchange current density; and the reference mole fraction of NO in the second chamber that corresponds to the reference NO concentration in the sample gas. The overpotential ηac is defined as ηac = VP2 − EOC2, where VP2 and EOC2 are the potential and the open circuit (zero-current) potential of the NOx sensing cell. The concentration ratio is raised to the power of γ which reflects the effect of concentration losses. The reference NO concentration in this model, is constant and corresponds to 2820 ppm of NO in the sample gas.

I p2 = iP 2 Ae2 = 2FNNO,sc

(10)

where P is the absolute pressure in Pa. The limiting current value, is the maximum possible cell current determined from diffusion which according to Eq. (9) corresponds to the lowest molar fraction of NO in the second chamber (xNO,sc = 0). Combining Eq. (8) and (9), the pumping current of the NOx sensing cell becomes:

3.1.3. Activation-transportation polarization (ηac) The activation-transportation polarization, is calculated using the Butler-Volmer equation [50]. According to the Butler-Volmer equation, the relation between the second chamber pumping current density (iP2) and the activation-concentration polarization is:

iP 2 = iPo2

P ¯ RT

(9)

VP1 = Efco +

where NNO,sc, xNO,fc and xNO,sc are the molar flow vector and the mole fraction of NO in the first and the second chamber respectively, LD2 is the length of the second diffusion barrier, AD2 is the cross sectional area of the second diffusion barrier, DNO is the diffusion coefficient of NO calculated in [20], and Ct is the total molar concentration in mole which m3 according to the ideal gas law is:

Sox red,fc (T 2F

To) EN

¯ RT ln 2F

( pp ) NOenv NOfc

a

(13)

where, pNOenv and pNOfc are the partial pressure of NO in the sample gas and in the first chamber respectively. In this case, Efco and ΔSox−red,sc are both equal to zero [45]. Again, using Dalton's law of partial pressures [48], Eq. (13) becomes: 305

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VP1 = Efco +

Sox red,fc (T 2F

To)

¯ RT ln 2F

( xx

NO,env NO,fc

)

The vector β contains three constants used in the model:

a

=

(14)

EN

All the model parameters and constants are listed in Table 3. The objective of this model is to find the variable vector x for the given model input vector u. Since the variable vector x has four components, four equations are needed to find the variables. The four equations are obtained as follows:

where, xNO,env and xNO,fc are the molar fraction of NO in the sample gas and in the first chamber respectively. 3.2.2. Activation overpotential The first chamber cell potential is kept at high values (> 0.42 V). Therefore, the activation is calculated using a Tafel approximation which is valid at high cell voltages [45]: a

=

¯ RT ln 1F

(i i ) P1,NO o P1

Eq. (5)&(7)combined Eq. (11) Eq. (18)

f1 (x, u , , ) =

f2 (x, u , , ) = x 4 f3 (x, u, , ) =

(16)

NNO,fc =

2FCt DNO AD1

(18)

The vector ζ contains 23 model parameters: red,fc

Sox

red,sc x N2,rc x O2,rc A e1 A e 2 AD1

o AD2 R cell xNO,sc iPo1 iPo2

( )

+

2

2 1

5

2 1

(

sinh

23 12

1

(

x1

(

0.5 0.5 7 8

x2

x4

x2 ) 17 14

)

)=0

(19)

=0 21

3

ln

2 16 (

x2 22

)

1

2 1

2 1

ln

(20)

( ) u3 x1

21 18 1

ln

( ) x3

15

9

+ x4

10

+2

1

2 21

23 11

u3

x1 21

=0

(22)

To make sure that the sensor behavior is not affected by the other exhaust gas species, specially O2 and NO2, a sensor test rig was built based on a design from ECM [42]. The test rig consists of six externallycontrolled 2-way valves, three externally-controlled 3-way valves, four humidifying tanks and six CCR MKS-GE50A mass flow controllers, connected to gas cylinders filled to known concentrations. All of the setup actuators are externally controlled by a PC as schematically shown in Fig. 7. To have a better control of NOx concentration, the water tanks are bypassed for all the tests carried out on the sensor test rig. The sensor tests are replicated with the sensor test rig using NO/N2 mixture at different NO concentrations.

xNO,fc xNO,sc ip1,NO ip2

Sox

21 19 + 20 1 2

1

21

3

4.2. Gas mixture test rig

The vector x contains four model variables:

o o fc Esc

3

)

1

To study the sensor behavior over a wide range of NOx and other exhaust emission concentrations, the sensor was first mounted in the exhaust pipe of a four cylinder medium duty Tier III diesel engine (Cummins QSB 4.5 160) and then in the exhaust pipe of a four cylinder port injection spark ignition (SI) engine fueled with natural gas (GM Vortec 3000) as shown in Fig. 6. The tests were carried out at different engine operating conditions with different NOx concentrations in the exhaust system as listed in Table 1. The sensor was installed upstream of the engine aftertreatment systems to measure the engine raw emissions. The NOx sensor used in the experiments was a production ECM NOx sensor (P/N: 06-05). The sensor parameters and operating condition were changed using the corresponding control module (ECMNOxCANt P/N: 02-07) connected to a computer via Kvaser Light HS CAN interface. All measurements were carried out once the engine and the sensor reached steady state operating conditions.

u = [Vp1 Vp2 xNO,env ]

[T PE

+2

(

4.1. Internal combustion engines

To develop a comprehensive NOx sensor model, diffusion and electrochemical equations for both chambers must be solved simultaneously. A set of coupled nonlinear equations is used to present the model in a standard form of f(x,u, ζ, β)=0. The model inputs, variables and parameters are vectors which are listed next. The vector u contains three model inputs:

=

2 1

4. Experimental setup

3.3. NOx sensor model by coupling the diffusion and the electrochemical models

x=

10

u1 +

f4 (x , u , , ) = x3

(17)

xNO,env xNO,fc LD1

6

+

(21)

where, NNO,fc is the molar flux vector through the first diffusion barrier, xNO,env is the mole fraction NO in the sample gas, AD1 is the cross sectional area of the first diffusion barrier, and LD1 is the length of the first diffusion barrier. Combining Eq. (16) and (17):

i p1,NO Ae1 + i p2 Ae2 =

4

=0

where, Ae1 and Ae2 are cross sectional area of the first and the second chamber electrodes respectively. The current I p1,NO only considers the effect of NO on the first cell pumping current. In the presence of O2, the value I p1,NO is at least three orders of magnitude lower than the overall Ip1 value [20] and does not significantly effect the sensor sensitivity to O2. The model parameter estimation tests are carried out in the absence of O2 in the sample gas. The relative molar flux of NO can be estimated using Fick's law [51]:

xNO,env xNO,fc Ct D NO AD1 LD1

u2 +

13 x 4 10

i p1,NO Ae1 + i p2 Ae2 = 2FNNO,fc 2

f3 (x, u, , ) = 0

f4 (x, u, , ) = 0

with,

3.2.3. Diffusion All the NO that comes through the first diffusion barrier is reduced either in the first or the second chamber. Therefore, according to the Faraday's law, the molar flux of NO through the first diffusion barrier, NNO,fc has the following relation between the first and the second cell pumping currents: Ip

f1 (x, u, , ) = 0

f2 (x, u , , ) = 0

Eq. (14)&(15)combined

(15)

where, iP1,NO and iPo1 are the current density and the exchange current density of the first cell respectively and α1 is the charge transfer coefficient of the first cell.

I p1,NO

F R¯ To

]

1 a2 c 2 LD1 LD2 DNO

306

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Fig. 6. Experiment setup - Internal Combustion Engines.

5. Results and discussion

the absence of O2 at different NO concentrations balanced with N2. The IP1 measurement system is designed by the sensor manufacturer for measuring O2 concentration and therefore does not have an appropriate resolution for measuring low currents. Therefore, the tests are carried out only at points that have a recognizable effect on IP1. IP1 increases with the environmental NO concentration as shown in Table 2. The test results confirm that NO is partly being reduced in the first chamber affecting NO concentration gradient inside the sensor compartments including the NOx sensing cell, when IP2 is equal to zero. Using the data in Table 2 and the process above, the resulting activation polarization parameters for the first chamber were determined and are listed in Table 3. The use of Eq. (2) is needed because the first sensing cell voltage is not fixed and changes slightly to keep the reference cell voltage constant through a closed loop controller. Activation polarization parameters for NO reduction in the first chamber are found based on the experimental open-circuit potential of species in the second chamber for three NO concentrations along with the corresponding first cell voltage, VP1, as listed in Table 2. The value of VP1 is linearly interpolated for the concentrations between the measured points listed in Table 2.

The sensor was tested over a wide range of operating conditions to study the behavior of the NOx sensing cell and to observe the effect of all important factors on the sensor output. To do this, the sensor temperature was kept at the stock sensor temperature in normal operation (T= 1023 K). Then, the NOx sensing cell voltage, VP2 was changed to cover regions (I to V) in Fig. 4 (not region VI). These tests were performed at different engine operating points with different NOx concentrations up to approximately 2800 ppm (xNO,env = 2800 × 10−6). 5.1. Activation polarization parameters for the first cell (O2 sensing cell) To investigate partial reduction of NO in the first chamber (O2 sensing chamber), several tests were carried out with the sensor test rig in the absence of O2. First, the open-circuit (zero-current) potential of the second cell (NOx sensing cell) was found through the experiments at different NOx concentrations in the sample gas. Theoretically, it is equal to the Nernst potential, EN, defined in Eq. (5). Since the second cell pumping current (iP2) is equal to zero, the mole fraction of NO in the first and the second chamber are equal. Then, using the experimental open-circuit potential of the second cell at different NOx concentrations, the mole fraction of NO in the second chamber, xNO,sc, can be estimated using Eq. (5) with ηac and ηΩ set to zero. Then, the kinetic parameters for the first chamber, α1 and iPo1, are obtained by fitting iP1 using Eq. (15) where Eq.(2) is used to estimate the overpotential, ηa as illustrated in Fig. 8. The effect of NO concentration on the first sensing cell current, IP1,NO, was studied to evaluate the role of the first sensing cell in NO reduction at zero NOx sensing cell current (IP2=0) and the results are shown in Table 2. All the tests were carried out on the sensor test-rig in

5.2. Activation polarization parameters for the second chamber Generally, the activation polarization becomes the rate-determining step of the species reduction/oxidation when the current density is very low and the concentration of reduced/oxidized species is high. Therefore, the NOx sensing cell behavior should be studied at very low current densities to find the activation polarization parameters for the model. As discussed in Section 5.1, the first sensing cell plays a pivotal role in determining the open-circuit potential of the second sensing cell by changing the mole fraction of NO through the sensor compartments as schematically shown in Fig. 5. The first cell potential, VP1 increases with the NOx concentration in the environment as shown in Table 2. This reduces mole fraction of NO inside the first and the second chamber according to Eq. (2) which makes the open-circuit potentials to converge at high concentrations as shown in Fig. 9 and schematically illustrated in Fig. 5. The cell behavior at high concentration (2820 ppm) and very low current densities (< 0.2 [A/m2]) was used to find the parameters of activation loss, i.e. iPo2 , αa2 and αc2 as shown in Fig. 10, considering the corresponding mole fraction of NO at this operating condition as the o reference mole fraction ( xNO,sc ). The cell overpotential is defined as η = Ecell − EN,exp, where EN,exp is the experimental open-circuit cell

Table 1 Engine operating conditions and NOx concentration in the exhaust gas Engine

Engine speed [rpm]

Engine output torque [ft-lb]

NOx mole fraction [×10−6]

Diesel Diesel Diesel SI

2500 2000 1500 2000

30 100 200 118

165 280 590 750, 1310, 1800, 2200, 2820*

* The stoichimetric air/fuel ratio is changed to change the level of NOx emission in the SI engine. 307

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Fig. 7. Sensor test rig schematic.

Fig. 8. Defining the activation polarization parameters for the first cell.

potential. A Butler-Volmer fit has been carried out using a trust-region algorithm [52] with squared correlation coefficient (R2) of 0.9736 and Root Mean Square Error (RMSE) of 0.01512. The cell current density, iP2, is obtained from the experimental measurements. All the ButlerVolmer fit parameters are listed in Table 3.

5.3. Model Validation To validate the model capability to predict sensor behavior at different cell voltages and different NOx concentrations, the sensor is mounted on the exhaust system of a Diesel and SI engine as explained in 308

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Table 2 The effect of NO concentration on the pumping current of the first chamber in the absence of O2 and @ IP2=0 xNO,env

IP1,NO [μA]

VP1 [V]

0 1 2 5

0.375 0.38 0.423 0.465

0 100 ×10−6 1140 ×10−6 2800 ×10−6

Section 4.1 at different engine operating conditions with different NOx concentrations listed in Table 1. All the engine tests are carried out at steady state engine operating condition, where the engine is fully warmed up. To maintain steady state, the engine output torque was controlled by a dynamometer at the set points listed in Table 1, while the engine speed was controlled by the engine controller. After the engine reached steady state operating condition, the second cell potential was changed using the NOx sensor control module via the Kvaser Light HS CAN interface. The NOx sensing cell potential was changed to cover the full operating range at each NOx concentration set point. To do this, the potential was varied from the open circuit potential to the diffusion rate-determining operating condition. The operating conditions and boundary conditions of each test were applied to the sensor model as VP1, VP2 and xNO,env where, VP1 and VP2 are the sensor operating condition inputs and xNO,env is the boundary condition applied to the model. The boundary condition values are listed in Table 1. The actual sensor temperature and cell voltage were measured and used as the inputs for the diffusion model and the electrochemical model. The parameters of the diffusion model are defined in our previous work [20] and are implemented in the physics-based model developed in this study. The model was then used to predict the sensor current at different diesel and SI engine operating conditions with different NOx concentrations listed in Table 1 and at different NOx sensing cell potentials to evaluate the accuracy of the model in predicting the effect of model inputs (VP1, VP2, xNO,env) on the measureable output (iP2). The experimental and the model output results for the NOx sensing cell are shown in Fig. 11 with symbols representing the data and the

Fig. 9. Experimental open-circuit potential of the NOx sensing cell.

lines representing the model. The effect of water vapor and the other exhaust gas species (such as CO2, unburned hydrocarbons, particulate matters, etc) on the model boundary condition is considered since all these species contribute to NOx dilution inside the exhaust gas. However, the cross sensitivity of the NOx sensor to water vapor is neglected in the model which could be one of the factors affecting the model accuracy at some operating conditions shown in Fig. 11. The model results match the experimental data with maximum error of 12% at VP2 = 0.29 V and [NOx]= 1800 ppm as shown in Fig. 11. As the cell voltage increases, more NO is reduced on the cathode electrode inside the sensing chamber which reduces the mole fraction of NO inside the chamber based on Eq. (5). Reduction of NO directly changes the NO gradient inside the sensor and makes is easier to estimate the mole fraction of NO inside the NOx sensing chamber. Finally, the sensor pumping current reaches its maximum value limited by the diffusion of species through the sensor porous diffusion barriers, which is accurately captured by the model as shown in Fig. 11.

Table 3 Model parameters and constants Parameter T P Efco

o Esc ΔSox−red,fc ΔSox−red,sc x O2,rc x N2,rc Ae1 Ae2 AD1 AD2 Rcell o xNO,sc

iPo1 iPo2 γ α1 αa2 αc2 LD1 LD2 DNO Constant F R¯ To

Description

value

Sensor temperature Sample gas pressure open-circuit potential at the standard state for the first chamber

Source 1023 [K] 1 [atm] 0 [V]

Measured Measured Calculated [45]

The change in entropy of productsreactants for the first chamber The change in entropy of productsreactants for the second chamber O2 mole fraction in the reference chamber N2 mole fraction in the reference chamber First electrode cross sectional area Second electrode cross sectional area 1st Diffusion barrier cross sectional area 2nd Diffusion barrier cross sectional area Cell resistance Reference mole fraction of NO in the second chamber

0 [J. mole−1. K−1] −12.5325 [J. mole−1. K−1] 0.10 [-] 0.90 [-] 10.15 ×10−6[m2] 2.59 ×10−6[m2] 1.81 ×10−7[m2] 3.2 ×10−8[m2] 0 [Ω] 6.62 × 10−7 [-]

Calculated [45] Calculated [45] Assumption Assumption Measured Measured Measured Measured Estimated Estimated

Second cell exchange current density

0.0471 [A/m2]

open-circuit potential at the standard state for the second chamber

First cell exchange current density

Second cell power of concentration ratio First cell charge transfer coefficient Second cell charge transfer coefficient for anode Second cell charge transfer coefficient for cathode Length of the first diffusion barrier Length of the second diffusion barrier Diffusion coefficient of NO through the sensor barriers Description Faraday constant Universal gas constant Reference temperature

309

0.4487 [V]

0.0109 [A/m2]

0.1 [-] 4.817 [-] 4.15 [-] 7.5 1.66 [mm] 0.71 [mm] 7.1 × 10−6 [m2/s] value 96485.33 [C/mol] 8.314 [J. mole−1. K−1] 298.15 [K]

Calculated [45]

Estimated Estimated

Estimated Estimated Estimated Estimated Measured Measured Estimated Source [53] [53] [53]

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M. Aliramezani, et al.

Fig. 10. Butler-Volmer fit for 2820 ppm NO concentration in the environment (xNO,env = 2820 × 10−6).

from the open-circuit cell voltage, which corresponds to zero current density, to 0.45 V that corresponds to the limiting current of the NOx sensing cell. It is shown that NO partially reduces in the first chamber which affects the NOx sensing cell open-circuit potentials. The experimental open-circuit potential at different concentrations were defined and then implemented in the model to simulate NO reduction in the first chamber at IP2 = 0. The model results for the limiting current and the pumping current of the NOx sensing cell closely match the experiments for a wide range of cell voltages and different engine operating points with different NOx concentrations up to more than 2800 ppm. Acknowledgements This work has been partially supported by the Natural Sciences and Engineering Research Council of Canada (grant number 2016-04646) and Canada First Research Excellence Fund. We would like to express our gratitude to Dr. André Phillion for taking the X-ray tomography images.

Fig. 11. Experiments and simulation results of the sensor pumping current, Ip2 vs cell voltage at different NOx concentrations in the environment.

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The full range of the sensing cell current-voltage curve of an amperometric NOx sensor was experimentally studied and then simulated. First, diffusion of the sample gas species through the sensor porous diffusion barriers and sensor chambers were modeled. Then, the diffusion model was coupled to an electrochemical model to predict the current-voltage curve of the NOx sensing cell. The ohmic loss, activation loss and concentration loss have been included in the electrochemical submodel. To validate the sensor model at different operating conditions, the sensor was installed on the exhaust pipe of a medium duty 4.5 L Diesel engine, a 3 L port injection spark ignition engine and a sensor test-rig that provides fully controlled gas mixtures. Experiments were carried out at different engine operating conditions to provide a wide range of NOx concentrations for NOx sensor model validation. The sensor model consists of 4 non-linear equations with 3 inputs, 23 parameters and 3 constants. The NOx sensing cell voltage was changed 310

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