Analytical modeling and optimization of electrostatic particle sensors for particle number detection: incorporating particle size influence

Developing measurement devices and methods to track increasingly stringent emission standards, including particle number, is crucial. This paper presents a novel analytical model to describe the signal response of an electrostatic particle sensor not only to particle mass concentration, but also to the particle number concentration of in-flowing particles. The uniqueness of this model lies in its ability to calculate the signal as a function of particle diameter, enabling the determination of particle number concentration from the signal. The model considers the effects of aerosol flow, electrode voltage and temperature, and can be used for the optimization of the sensor geometry parameters, length, width, and electrode gap. The model was designed to optimize the sensor’s geometry and signal retrieval as well as the optimization of the electric field between the electrodes. Comparative analysis was conducted between the proposed model and a model from the literature as well as experimental data from literature and experimental data collected in this paper.


Introduction
As the prevalence of environmental pollution from various combustion processes continues to increase, respiratory diseases are becoming more common in the population [1,2].Inhaling particles smaller than 1 µm can lead to a range of deleterious health effects in the human body.Such particles can penetrate deep into the lung, causing inflammation, oxidative stress and other harmful reactions that can impair respiratory function and overall health [3].To lower the health risks and decrease air pollution it is essential to first identify the source of pollution and control the amount of pollution emitted to the environment.One viable way to accomplish this is by deploying well-designed particle detectors to measure the particle number concentration (PN) of airborne particles from anthropogenic sources.Such detectors can serve as the foundation for a comprehensive testing system, to provide information for compliant operation of vehicles with the current legislation [4,5].PN is a mandatory emission standard established within the legislative framework known as the particle measurement program (PMP).The laboratory gold standard for such evaluations is a condensation particle counter (CPC).In addition to laboratory tests, PN emissions must also be measured under real driving conditions.Hence, the development of lightweight, robust, and cost-efficient onboard diagnostic sensors is crucial for conducting Real Driving Emission (RDE) measurements in compliance with the PMP [6,7].On-board sensors can be utilized to detect particle filter failures and defeat devices, enhancing the overall effectiveness of emission control measures.Depending on the sensitivity and the temporal resolution adequacy, such a sensor could also serve for fleet emission monitoring [7].
Widely used methods to measure particles are the before mentioned CPC, optical measurement techniques like light scattering, laser-induces incandescence detectors and electrical measurement techniques like diffusion charging [8].CPCs and diffusion chargers are most commonly utilized for the periodic technical inspection.A CPC operates by introducing sampled exhaust particles into a supersaturated vapor, typically composed of butanol.As particles collide with the vapor, they grow into droplets, and the resulting droplets are counted individually as they pass through an optical detection system.This capability allows CPCs to achieve high sensitivity by counting single particles.In contrast, a Diffusion Charger employs the principle of charging particles by allowing them to naturally collide with ions in a controlled electric field generated by a corona wire or needle.In this process, particles gain a net charge and are subsequently transported to the electrometer via the sample flow in the device.The resulting current is proportional to the lung-deposited surface area.In certain particle size ranges, the current can also be proportional to the number of particles, offering a method for measuring the number of charged particles present in the sampled exhaust.
In contrast to CPCs, diffusion chargers present several noteworthy advantages.They boast a compact design and lower power consumption, rendering them more efficient in terms of size and energy usage.Furthermore, their operation is free from the requirement of working fluids, enhancing simplicity and sustainability.The insensitivity to orientation allows for versatile deployment, accommodating various spatial configurations.Additionally, diffusion chargers exhibit minimal sensitivity to fluctuations in temperature.However, these advantages are counterbalanced by certain drawbacks.Notably, diffusion chargers tend to be less accurate, displaying a larger dependence on the morphology of particles.Charging efficiency may be contingent upon the composition of the carrier gas, introducing a potential variable in the charging process.Moreover, their functionality relies on the initial charge state of the aerosol, and they demonstrate heightened sensitivity to variations in flow rate.The efficiency curve of diffusion chargers is characterized by less smooth contours, impacting the predictability of their performance [7,9,10].
Many sensors available in the market require preconditioning and dilution stages, as seen with condensation particle counters, for instance.While these counters prove suitable for laboratory applications, they face inherent limitations when applied to RDE measurements.Although there are ongoing research efforts exploring the use of CPCs in harsh environments with alternative working fluids, these innovations have yet to reach the market [11].
In the realm of particle detection, light-scattering detectors encounter restrictions determined by the wavelength of the lasers they utilize.Additionally, laser-induced incandescence presents a notable drawback due to its high energy consumption.As the search for more effective and environmentally robust sensor technologies continues, it remains crucial to address these limitations and seek alternative solutions for accurate and practical emission measurements in diverse settings [4].
In this paper we investigate the sensor effect of an simple electrostatic particle sensor as discussed by Diller et al [12] and Bilby et al [13].Electrostatic particle sensors have been used extensively in the detection of exhaust particles.The parallel plate capacitor arrangement is the simplest form of electrostatic particle sensor and will be used to explain our model approach.In literature it is suggested that the measured current in electrostatic sensors is caused either by the charging of particles due to the recent high voltage applied, generating a measurable leakage current [14], or the dendrite formation and a resulting equilibrium state of incoming and outgoing fractions of the breaking dendrites [13].Regarding the first explanation, the particles are ionized due to the presence of the high-voltage electrode.The particles move towards the ground electrode and the resulting leakage current can be detected.However, no exact mechanism is described in [14].Regarding the second explanation, to our knowledge the first model implementing dendrite formation in an electrostatic sensor model was published by Bilby et al [13].
A high electric field is generated between two parallel copper electrodes and as aerosols are conveyed into this high voltage region, naturally charged particles from combustion deposit on these electrodes.By this particle deposition, dendrites are formed on the surface of the electrodes, aligned with the electric field lines.Due to the contact with the electrodes, charges accumulate on the surface.After the dendrites grow to a certain size, fragments break off, carrying a high number of charges and redeposit on the electrode with the opposite polarity, thereby transferring their charge causing a measurable current [15,16].
The resulting current and voltage drop are dependent on the frequency of this charge transfer and therefore the sensor signal is proportional to the mass concentration of particles [4,13].The signal can be detected by a bench multimeter, making the electrostatic particle sensor a cost-effective and straightforward approach to detect exhaust particles.
When considering the advantages of an electrostatic sensor over a diffusion charger, several notable aspects come to light.Firstly, electrostatic sensors are often smaller and the electronic circuitry is less complex, benefiting from signal amplification caused by dendrite fragmentation.Moreover, these sensors are cost-effective and easy to manufacture, enhancing their practicality.Unlike diffusion chargers, electrostatic sensors eliminate the need for precharging.Additionally, they do not require preconditioning and dilution, making them highly suitable for deployment in harsh environments where maintaining precise conditions may be challenging.However, it is imperative to acknowledge certain drawbacks in comparison to diffusion charging.Electrostatic sensors tend to be less accurate, and need preconditioning prior to initial use.Furthermore, they display heightened sensitivity to fluctuations in flow rate and electrode voltage, introducing factors that may impact their precision under variable conditions [4,14,17].
In this work we present the underlying physical principle as well as performance characteristics of such an electrostatic particle sensor with the help of a simple analytic model.
In their work Bilby et al [13] suggest a coupled system of differential equations to model an average dendrite population per sensor area.Where n represents the number of possible height levels of the dendrites.This dendrite population further produces break-off fragments and the concentration of breakoff fragments carries a specific number of charges with the particle surfaces.The sum of these charges is considered as the measured current that can be directly correlated to particle mass [13].The model successfully describes the influence of flow and voltage but has no spatial resolution.It is simulated whether a particle enters the sensor and deposits on the electrode or passes through the sensor.However the information about the position of the particle are lost which would be valuable for sensor optimization and signal retrieval.Also n represents the number of possible height levels of dendrites, hence the number of equations and therefore the size of the system is set by the discretization dendrite.As such, it can cause a high computational effort.Factors for dendrite formation, such as drag force, electrostatic forces, and intramolecular binding forces, as well as adhesion, contact charging, and electrostatic precipitation must be known, which makes the model complex.The model from Bilby et al [13]. is based on a particle size distribution, but the particle positions are distributed over a uniformly distributed area (free dendrite spaces).Lastly, parameters such as temperature, fragment size, and geometric parameters such as length and width are included in the model, but no optimization has been performed.
While there are other studies focusing on the modeling of dendrite-based sensors, these primarily address a different sensor type, the so-called resistive sensor.In this sensor type, soot particles form conductive paths between the electrodes of the sensor.By measuring the resistivity of the soot bridges, the soot concentration can be approximated.Compared to electrostatic sensors, the dendrite-bridge formation in resistive sensors involves a longer time frame [17,18].
Given the intricate and multifaceted nature of the physical mechanisms of the sensor effect (especially the dendrite formation), it is critical to gain a understanding of these processes to develop highly effective sensors.Therefore, the possibility of measuring the PN concentration with an electrostatic sensor is investigated and a model was developed and experimentally validated.Our analytical model simulates the sensor response per particle diameter, more precisely, it tracks the path of a particle with a specific particle size until its deposition on the electrode.After deposition, the individual particle becomes part of a dendrite fragment that traverses the sensor and bounces between the electrodes.The flight paths of these fragments are calculated.Each impact with the electrode walls is counted as a bounce.These bounces are further weighted with a decay constant and the surface charge probability is approximated by a Boltzmann charge distribution, resulting in a signal per particle diameter and particle charge.The model considers the effects of aerosol flow, voltage and temperature and can be used for the optimization of the sensor geometry parameters: length, width and electrode gap.Two specific cases are examined: first an electric field gradient corresponding to electrode segmentation, and second an arbitrary electrode shape.The model was designed to optimize the sensor's geometry and enables the calculation of optimal electrode voltage.
This study provides a comprehensive understanding of the physical processes within the simple electrostatic sensor.Physical variables such as voltage, flow velocity, and temperature are considered, and additional influencing factors like fragment size, decay constant, and various sensor geometries are comprehensively and methodically addressed for the first time.Furthermore, insights from the model for optimizing sensor geometry can be utilized, and disturbances such as flow, temperature, and voltage fluctuations can be subtracted from the signal, thereby enhancing the performance of such an electrostatic sensor.Additionally, the development of a simple analytical model has the advantage of keeping computational effort minimal and allowing for simplifications that enable a different perspective on the physical processes in the sensor.However, these advantages come with disadvantages, as the model makes some very coarse assumptions and introduces uncertainties, which will be discussed in detail later.Nevertheless, the primary goal of the paper is not to detail every physical phenomenon but to accurately capture trends of an electrostatic sensor with respect to particle number measurement.

Materials and methods
In this section, an analytical model to simulate the output signal of an electrostatic particle sensor, based on a plate capacitor arrangement, is presented.In addition to that an experimental evaluation procedure is described.The simplest form of an electrostatic sensor is represented by section A in figure 2(the parallel case).A diagram of the sensor and the signal measurement is illustrated in figure 1.When aerosol is directed into the high voltage region between the two copper electrodes, naturally charged particles adhere to electrodes of opposite polarity.Dendrites develop on the electrode surface and gain a charge through contact.Over time, fragments of these dendrites detach from the electrodes, relocating to the opposite polarity electrode, transferring their charge.These fragments carry numerous charges.Since the electrode is grounded, the deposited charge induces a current, resulting in a voltage drop across the bench multimeter's internal resistance.The more frequent this process occurs over a given period, the greater the current and, consequently, the higher voltage drop [4].

Description of the sensor model
Figure 2 shows three types of electrode arrangements.In each arrangement the electrodes of the sensor, which also form the aerosol flow path, are depicted in brown.The aerosol enters the sensor's flow path from the left.The inflowing particles (cyan line) deposit onto the electrodes.The deposition position, from which the particle is considered part of a dendrite fragment, is further referred to as the 'conversion point'.At this point, not only the diameter of the particle under consideration changes, but also its charge and the nature of the dominant charging process.From the conversion point on, particles travel as part of a dendrite fragment (purple lines) through the sensor.
We consider three sensor cases: (A) a parallel plate capacitor with a constant electrode gap and a constant voltage (parallel case), (B) a parallel plate capacitor with a constant electrode gap and a field gradient along the x-axis of the sensor (segmented case), and (C) an arbitrary form of electrodes (exemplified by a linear opening of the electrodes at a slight angle) and a constant voltage (arbitrary case).As the electrodes serve as the boundary of the flow channel, altering the electrode shape in the arbitrary case concurrently modifies the flow profile and the electric field within the channel.Any physically optimized shape is possible here, but depending on the complexity of the shape, the model complexity also increases.
In the segmented case the electrodes are separated into several elements, each lying on a different potential.In the case of an infinite number of electrode segments the electrostatic potential of the whole sensor can be described by a continuous function, whereby any function meeting the physical requirements of an experiment is possible.Throughout this paper, we will assume a simple linear dependence on the x-position according to equation ( 1) where a is the slope of the function and U 0 is the initial voltage at the sensor inlet between the electrodes and x are the xcoordinates for the particle or fragment.In this example, the parameters a and U 0 can be used to optimize the segmented sensor designs.
In the arbitrary case, the geometry of the electrode gap is altered.The wall function (g(x)) is chosen to be linear for simplicity reasons, but any function describing the shape of the flow path could be used.Our considered example of a linear wall function is given in equation ( 2) (2) In our example, g 0 represents the distance between the electrodes at the sensor inlet and k the slope of the electrode.
Because the distance between the electrodes changes in the xdirection, the field strength changes and so does the flow velocity in the x-direction.In the case of a linear function as in equation ( 2), flux velocity and field strength also change linearly.Due to the opening angle of the electrodes, the electric field lines close to the electrodes are no longer parallel to the y-axis.However, the field lines are presumed to be parallel to the y-axis, since the electrode opening angles are kept small.In this model we use a 2D representation where charged particles are considered to pass through a gap between two electrodes with a potential difference between them, see figure 2. Each particle size has a specific electric mobility according to its surface charge, which is defined by the Boltzmann charge distribution [19].For simplification, an unipolar charge distribution is assumed in all cases,  so that contributions from negative charges are neglected.The electrical mobility and the drag force acting on the particle are considered to calculate the conversion point, after which a particle travels as part of a dendrite fragment (see figure 2(A): cyan line).This point is a critical point in the model where not only the size of a particle but also the dominant charging mechanism changes.It is assumed that after the conversion point the particle travels within the dendrite breakoff fragment.The particle is considered as a part of the dendrite fragment moving further through the sensor in a way that it will come in contact with both electrodes multiple times (see figure 2(A): purple line), depositing its surface charge on each contact.Fragments deposit and separate several times before leaving the sensor.Depending on the aerodynamics of the particle and the surface charge, the conversion point varies between particle sizes.The longer the distance to the conversion point, the shorter the distance available for bounces.This leads to a weakening of signal with larger particle diameters.However, larger particles carry multiple charges which, in return, increases the signal.The resulting current is proportional to the number of bounces (electrode contacts) a dendrite fragment goes through until it leaves the sensor.However, the contribution of the initial particle within the fragment decays with each bounce due to particle losses and therefore a decay constant is introduced into the signal calculation [13].Moreover, a small particle with a smaller surface area contributes less to a dendrite fragment than a large particle with a larger surface area.To account for this, the ratio of the initial particle diameter and fragment diameter is included.This term describes that larger particles have a greater influence on the total signal than smaller particles.The sensor signal is obtained by multiplying the number of bounces by the corresponding charge per bounce and the probability from the Boltzmann charge distribution for a wide range of possible surface charges.The signal is averaged over the different starting positions a particle of a specific size can occupy and the result is a calculated current that can be compared to experimental values.

Model details
We now consider the simplest case of a parallel electrode arrangement, as shown in figure 2(A).To calculate the earlier introduced conversion point, the velocity in the x-and the y-direction have to be determined.The geometry and the flow through the sensor are known and the electrical mobility (see equation (19) in the appendix and list of symbols) can be calculated using the mean free path of the gas (see equation (17)) and the formula for the Cunningham slip correction factor (see equation ( 18)).
The velocity in the x-direction (v x (t)) is calculated according to equation ( 3) where Q is the flow rate of the sensor and A is the sensor crosssection and w is the width of the sensor.In parallel and segmented case, g(x) is considered a constant.
The velocity in y-direction (v y (t)) can be calculated according to equation ( 4) The electric field E is expressed as U(x) divided by g(x) [20].
Where U(x) can be a function for the change in the electric field in the x-direction (segmented case) or a constant electrode potential in the other cases.g(x) is the wall function of the electrodes (arbitrary case) or a constant electrode gap in the other cases [20].The initial acceleration acting on the particle is neglected for both velocity in x-and y-direction.
The y-component of the particle flight path (y(t)) can be calculated by integrating the velocity in the y-direction.y is the y-coordinate for the particle or fragment according to equation ( 5) dt. ( The integration yields y = ZU g t in the parallel case and y(t) = Z g (av x t 2   2 + U 0 t + y 0 ) in the segmented case.For the parallel and segmented case the time (t) where y = ( g 2 − p) is satisfied represents the time where the particle reaches the conversion point.p is the start position of the particles and has to be subtracted from the fixed electrode position.The x-coordinate of the conversion point can be calculated by rearranging the equation to yield the time when the particle hits the electrode and then insert it in equation (3).For symmetric reasons, only one-half of the gap is calculated so p is varied between zero and the half of g(x).
The x-component of the particle flight path (x-coordinate of the conversion point) (x(t)) can be calculated by integrating the velocity in x-direction according to equation ( 6) where t is the time it takes for the particle to reach the electrode.
In the arbitrary case, the conversion point cannot be calculated as simple as in the other two cases.An interception point between the particle path and the wall g(x) has to be calculated according to equation ( 7) s(x) defines the particle position within the sensor.The particle path is assumed to follow a straight line, with a slope defined by the velocity of the particle in y-direction and in xdirection.(They-velocity is calculated as Z * U 0 /g 0 and the xvelocity is calculated as Q/w * g) Assuming that both electric field and velocity field are linear s(x) can be calculated according to equation ( 8) We chose g(x) according to equation ( 2) and inserted it together with equation ( 8) into equation (7).By solving the resulting equation for x(t) we get equation ( 9) Table 1 summarizes the main equations describing all three cases.The x-and y-components of velocity and particle positions are calculated by using equation (1) for the segmented case and equation (2) for the arbitrary case.
The conversion point is the starting point for the resulting dendrite fragment further through the sensor.At the conversion point, a pivotal juncture is reached, where various physical properties undergo significant changes, with a particular emphasis on the alteration of the dominant charging mechanism.In the following section, we will go through the dendrite fragment flight path calculation.We call the point where the dendrite fragment hits the opposite electrode 'bounce point' and note the distance between two bounce points in x-direction as x f (see figure 2(C)).
The charge of the dendrite fragment (c f ) is calculated according to equation (10), adapting the field charging theory from [20] (equation (15.25)) to our system where ε is the relative permittivity, U(x) is the electrode voltage, N i is the ion concentration, Z i is the mobility of the ions, t c is the charging time, d f is the fragment diameter and K E is a constant of proportionality.The fragment diameter is assumed to be constant for all particle size and the resulting charge number in the field charging regime is contingent on the square of the parameter d f .For the parallel case, electrode gap and voltage are constant throughout the whole sensor, whereas in the segmented case, the voltage U(x) and in the arbitrary case the wall function g(x) is a function dependent on the position in the sensor.
The calculated charge from equation ( 10) is used for the calculation of the electrical mobility of the fragment.In the parallel and segmented case, x f can be calculated according to equation (6).For the arbitrary case, x f can be calculated by equation ( 9).An overview of the equations can be found in table 1.It has to be noted that for the calculation of x f , the diameter of the fragment has to be used instead of the inflowing particle diameter.
When the sensor gap, length, fragment diameter, and fragment charge as well as voltage and flow are kept constant (parallel case), the distance from the conversion point to the bounce point (x f ) is constant throughout the whole sensor.In that case the number of bounces (B), can be calculated with equation ( 11) where L is the sensor length in x-direction.Larger particles have a longer flight path and therefore hit the electrodes less often, this causes the signal to decrease with larger particles.
In the segmented and arbitrary case, the flight path of the fragment for each bounce are not equally spaced due to a change in the electric field and velocity field, so the number of bounces has to be calculated iteratively.Each iterative step n thus describes one bounce and the sum gives the total number of bounces as seen in equation ( 12) This equation is valid for the interception of the flight path with the upper electrode.By inverting the sign of equation ( 9), the interception of the flight path with the lower electrode could be calculated.The iterative solving of equation ( 12) leads to the discretization of the electric field and flow gradient with each step.

Signal calculation
The number of bounces is further weighted by a decay constant α.This constant determines how long the initial particle continues to contribute to the signal generated by the dendrite fragment.The weighted bounces (B w ) are calculated according to equation ( 13) α is the decay constant and B is the number of bounces.The decay constant weights the bounces so that the first bounces have more weight than the later ones.This generally decreases the total signal for all particle sizes, but it also contributes to a sharp drop in signal for larger particles.The sensor signal per size distribution (S sd ) is calculated as the sum of the initial particle signal over all particle charge probabilities.The contribution according to the proportion of a particle within the dendrite fragment particle is calculated by dividing the particle diameter through the fragment diameter and multiplying it with the number of fragment bounces, multiplied by the Boltzmann charge probability function for this particle diameter.This signal is further multiplied by the fragment charges and the elementary charge as well as the throughput time, as shown in equation (14).
where N c is the number of charges, N p is the number of diameter, c is the initial particle charge, d is the corresponding particle diameter, c f is the fragment charge calculated by equation ( 10), t flow is the exhaust flow time.The charge distribution was approximated with a Boltzmann distribution f(d, c) [20].Equation ( 14) combines several physical phenomena in one compact formula.The first part B w (c i , d j ) calculates the bounces within the sensor from equation (13).These bounces include the first contact with the electrode from the initial particle and its physical properties before the conversion point and the properties of the carrier fragment after the conversion point.The main charging mechanism before the conversion point is considered diffusion charging whereas at the conversion point field charging is assumed to be dominant.Multiplying the bounces with the charge probability distribution f(c i , d j ) on the initial particles reduces the overall signal contribution of particles sized that are less likely to occur, like very small and very large particles.The term d j /d f initiates a change in physical particle properties and accounts for the contribution of the particle within the fragment.The term d j /d f follows a more linear trend.The last term t flow ec f purely considers field charging and uses equation (10) to calculate the fragment particle charge.It is used to convert bounces in a charge signal.
This signal can be calculated for several equally spaced inflow positions along the sensor at a starting point x and y equal to zero.We calculate the mean signal of all starting positions (S mp ) by equation ( 15) where N p is the number of particle positions.

Model validation
To validate our model we compare our results with the model results and experimental data from Bilby et al [13].In order to simulate the signal for a particle number or particle mass concentration, the size distribution specific to the considered measurement case must be incorporated.The signal per particle diameter has to be weighted with the particle concentration of each size bin, divided by the total concentration per size distribution.The particle concentrationdependent signal (S c ) is calculated according to equation ( 16) where C i is the particle concentration of a size bin (either particle number concentration or particle mass concentration).
To calculate the particle number concentration we consider the number weighted size distribution from Bilby et al [13] and used a log-normal distribution to fit the data (with Scipy.optimizecurve fit) which uses non-linear least squares.To convert from particle number to particle mass we, use a 4th degree polynomial fit on the effective density, based on experimental values from Park et al [21].In order to compare our model with the model output from [13] we have to calculate the signal for the captured soot rate.Therefore the number weighted size distribution (n ci ) is multiplied with particle mass and the particle velocity (⃗ v).The particle mass bin is derived by calculating the diameter-dependent particle volume, (V) and particle density (ρ) from [21].As V ⃗ v = t A and A = wg we can also write m ci = Nρ( 43 π ( d 2 ) 3 )/twg for the mass concentration.Where A represents the sensor cross section, w the width of the sensor and g 0 the gap between the electrodes of a parallel sensor configuration.

Experimental validation procedure
The model was validated using a custom-built sensor for PM and the EmiSense PMTrac for PN.The custom-built sensor is described in Wallner et al [4].Its design closely resembles that of a sensor utilized in the parallel case.In the parallel design the particle velocity and electric field are kept constant.A scheme of the experimental setup can be seen in figure 3. A soot generator (miniCAST) was used as an aerosol source.The aerosol was led through a dilution bridge and a static mixer, before being split up and led into a photoacoustic soot sensor (MSSplus, AVL) used as reference instrument for PM or a CPC (TSI Model 3775) used as reference instrument for PN and the device under test.Mass flows of the aerosol parameter are 60 ml/min propane, 1.55 l min −1 oxidation air, 7 l/min nitrogen (quench), and 20 l min −1 dilution air resulting in a size distribution with a geometric mean diameter of 97 nm.The flow rates passing through the sensor were established at 1 l min −1 for the PM measurement and 1.5 l min −1 for the PN measurement.Table 2 contains a comprehensive list of all employed model parameters and sensor specifications.A detailed description of the experimental procedure can be found in [4].

Results and discussion
In this section, we present the outcomes of a comprehensive parameter sweep across all model parameters, aimed at determining the optimal signal within each particle size range.To achieve this, we rigorously adjusted a single model parameter while maintaining constant values for all other parameters, as detailed in table 2. The results are discussed and the model is compared to an existing model from [13] and experimental data.

Impact of model parameters
Fragment diameter and decay constant are quantities that are not easily accessible by measurement.Therefore, the sensitivity of the model to these quantities was investigated.The fragment diameter was varied in the range of 150 nm to 2 µm and the result for different particle sizes is shown in figure 4(a).A detailed discussion regarding the applicability of the used equations for this size ranges can be found in the section 2.2 as well as in the discussion about the model limitations in section 4. The signal increases with increasing fragment diameter.The fragment diameter is linked to the signal via the electric mobility, where it occurs in the exponent as well as in the denominator of equation (18).The fragment diameter therefore influences the velocity in y-direction (see equation ( 4)) and thus the time and distance x f between bounce points.The diameter is also needed for calculating the weighted bounces in equation (13).
The decay constant was varied in the range of 0.1-1 and the results for different particle sizes are shown in figure 4(b).The signal increases with increasing decay constant.The decay constant determines the contribution of a inflowing particle over time to the generated signal.The more often a particle moves through the sensor as part of a dendrite fragment, the less likely it is that this particle will be incorporated again into another dendrite break-off fragment.For this circumstance, the decay constant is used to weigh the signal (see equation ( 13)).By increasing the decay constant the percentage of the remaining contribution of the dendrite fragment particle per bounce is increased and therefore the signal increases.As observed in figures 4(b) and (a), the decay constant and diameter have a significant impact on the result.Therefore, the correct choice of values is of great importance.In our case, we have selected the decay constant and the diameter (as found in table 2) for the best possible match with the experimental results.

Impact of sensor parameters and geometry changes
The influence of flow, temperature and electrode voltage as well as the geometry parameters sensor length, width, size and form of the electrode gap are discussed here.
The flow through the sensor was varied in the range of 0.2 l min −1 -2.9 l min −1 and the result for different particle sizes is shown in figure 5(a).The signal decreases with increasing sensor flow.This is expected since increasing the flow also increases the velocity in the x-direction, thus increasing the distances between bounce points.As a result, the number of bounces is reduced, leading to a decrease in sensor signal.This signal behavior shown with our model is in accordance with findings from Maricq and Bilby [16], where the flow varies in the experiment.Such flow variations can be taken into account by our model by adjusting the corresponding parameters.
The temperature was varied in the range of 20 • C-300 • C and the result for different particle sizes is shown in figure 5(b).The signal increases with temperature.A change in temperature contributes to the signal by changing the length of the mean free path, thus influencing the Cunningham factor and further the electrical mobility.The electrical mobility in turn affects the velocity in y-direction and the time and distance between bounce points.An increase in temperature increases the number of bounces and therefore the signal.This signal behavior is in accordance with findings from Tang et al [14], where the temperature varies in the experiment.Similar to the flow the signal can be corrected for temperature variations with our model.
The voltage was varied in the range of 200 V-1000 V and the result for different particle sizes is shown in 5(c).It is shown that the signal increases with increasing electrode voltage.By increasing the voltage the velocity in y-direction is increased, therefore the time for the particle to hit the electrode is decreased and the distance where the particles hits the electrode is decreased as well, thus increasing the number of bounces as well as the signal.This signal behavior agrees with findings from Maricq and Bilby [16].To increase sensitivity the highest possible voltages for a given sensor design should be chosen.
We simulated the signal for different sensor geometries at a given voltage and flow rate to investigate an optimized Sensor response versus captured soot mass for 3 particle size distributions and 2 voltages with a flow of 1 l min −1 .The solid line shows our simulated sensor signal and the dashed line represents the model calculations from [13].The experimental data is also taken from [13].C1-C3 represents CAST 1 soot to CAST 3 soot (see [13]).
sensor geometry.The sensor length was varied in the range of 5 mm-160 mm and the result for different particle sizes is shown in figure 4(c).With increased sensor length the number of bounces increases and therefore the signal increases as well.
The sensor width affects the sensor signal via the sensor cross-section areas in equation (3).It has less significance since it behaves similarly to the sensor length in affecting the signal and could potentially be replaced by an alternative parameter like radius, depending on the geometry and its form.However, the sensor surface should be designed as large as possible, provided that the sensor can still be easily installed at the application site.
The electrode gap was varied in the range of 1 mm-3 mm and the result for different particle sizes is shown in figure 4(d).The signal decreases with increasing electrode gap.When the distance between the sensor electrodes is increased, the field strength and thus the particle velocity in the y-direction is decreased, which in turn leads to a lengthening of the conversion point and bounce point, which reduces the bounces and thus reduces the signal.The electrode gap should be designed as small as possible, provided that the dielectric strength of air between the pairs of electrodes is considered.

Comparison with an existing model and experimental data
To compare our simulated sensor signal with the reference model and the reference experimental data from Bilby et al [13] we calculated the mass concentration signal for CAST Soot at 750 V and 1250 V (see figure 6).In accordance with the reference model predictions from Bilby et al [13].the sensor signal at 1250 V exceeds the sensor signal from 750 V. Since it is not a monodisperse aerosol but a size distribution that overlaps, the trend that one would expect for a particle size is not so straightforward to discern.Given the simple nature of our model, the overall trend shown in figure 6 is in sufficient agreement with the calculations from Bilby et al [13] and the measured data from Bilby et al [13].This allows for fast and simple signal calculations for the design and optimization purpose of electrostatic sensor.In our model, however, the influence of the electrostatic field is dominant, which differs from the model from Bilby et al [13] and the experimental findings.This is not discussed in detail here but will be addressed in later validation processes.
Figure 7 shows the results of the sensor validation setup (see figure 3).We compare the custom-built sensor signal for a specific mass concentration to the model's calculations for particle mass, with a photoacoustic soot sensor (MSSPlus) as reference.The input parameters specific to this experiment are outlined in table 2 in the 'experiment mass' category.These parameters are integrated into the sensor model through the equations outlined in table 1 under 'Case 1: parallel.'Subsequently, they are converted into particle mass using equation (16).
Figure 8 shows the results of the sensor validation setup.We compare the electrostatic particle sensor signal (PMTrac, Emisense) for a specific mass concentration to the model's calculations for particle mass with MSSplus as reference.The input parameters specific to this experiment are outlined in table 2 in the 'experiment mass' category.These parameters  are integrated into the sensor model through the equations outlined in table 1 under 'Case 1: parallel.'Subsequently, they are converted into particle mass using equation (16).In figure 8 one can see the signal from the custom-built sensor plotted against the particle mass concentration, compared to the simulated signal as a function of particle mass concentration.Figure 9 shows the results of the sensor validation setup.Here we compare the PMTrac sensor signal for a specific number concentration to the model's calculations for particle number, with a CPC (TSI Model 3775) as reference.The input parameters specific to this experiment are outlined in table 2 under the 'experiment number' category.These parameters are again integrated into the sensor model as described above.The conversion into particle number was done with equation ( 16).In figure 9 the signal from the PMTrac sensor is plotted against the particle number concentration compared to the simulated signal as a function of particle number concentration.As the concentration range in question presents a challenge for PMTrac, it is important to acknowledge that the lower measurement range has higher measurement uncertainties.

Conclusion
In this work, we present a new analytical model that can simuthe output for a specific particle diameter enabling PN measurement with an electrostatic sensor.Furthermore, we discuss the influence of difficult to measure quantities like fragment particle diameter and decay constant.With the model, the signal can be corrected for voltage, flow and temperature variations and a favorable geometry can be predicted.Furthermore, two special cases are presented with which one can simulate an electric field gradient and a variable electrode shape.The generated model is compared with the existing model of Bilby et al [13] and validated with experimental data for particle mass and number measurement from Bilby et al [13] and our own experiments .Optimal parameters include a spacious sensor surface, a narrow gap, and elevated electrode voltage.Additionally, the model allows for adapting the signal to variations in flow, voltage and temperature, ensuring accurate readings.
A limitation of the current model is the lack of the complexity and spatial resolution compared to a full 3D Finite Element Method model.Particle interactions manly occur before the conversion point.After the conversion point, the initial particles are considered part of the dendrite.Therefore, we neglect interactions between particles of different sizes, as incorporating such elements would significantly complicate the model, introduce a level of complexity beyond the scope of this paper.However, the signal trend from our model demonstrated good agreement with experimental data and the model from Bilby et al [13].With the simplifications required to maintain the model's simplicity and conciseness, as discussed in section 2.2 (involving significantly simplified particle paths, simplified charging mechanisms and starting positions, no particle interactions between different particle sizes, et cetera ), it becomes unfeasible to make specific assertions about individual particles.However, we can give estimates regarding averaged properties for a given particle size.This estimates are crucial for calculating the signal per particle size and, subsequently, extracting information about the particle count.As a result, we can calculate the signal per particle diameter/size, which is the unique feature of our model.The model's primary benefits stem from its incorporation of spatial dimensions along the flow direction, enabling the calculation of particle size's influence on the signal.This enables us to calculate particle number from a measurement.To our knowledge, this is the first model published to show the signal dependence to a specific particle size as well as to calculate particle number signal for an electrostatic sensor.Our model allows for robust approximations, reliably representing the trend of the system's behavior with minimized computational demands.One limitation that introduces uncertainties into the model is the combination of different charging mechanisms in equation (14).This forces the use of equation (10) outside its intended applicability range.At the conversion point, where a particle of a specific size is taken up into the dendrite, it exists as neither a single particle nor a dendrite fragment.Before that point we consider the major charging mechanism for the initial particles to be diffusion charging.However, it is precisely at this juncture where the dendrites become charged, requiring the use of equation (10) beyond its typical applicability range (unipolar charge carriers, neglecting diffusion charging, particle sizes exceeding 5 µm).Since dendrites are much larger than fragments and exhaust particles, we assume that field charging is the dominant charging process.Our aim is to calculate the number of charges a breaking dendrite fragment carries.To achieve this, we apply the field charging equation to the dendrite diameter.This approach serves a sufficient approximation in this scenario, capturing the diverse effects of different sizes at this crucial transition point.The model accounts for particle charges while also incorporating dependencies on various parameters.The consideration of the fragment as carrier particle obviates the need for tracking dendrite populations as suggested by Bilby et al [13].
Furthermore, the model derives the sensor signal in relation to particle diameter, particle size distribution, particle mass concentration, and particle number concentration.With the increasing stringency of regulations governing particulate emissions, modern vehicles emit very low particle concentrations.Concurrently, notable advancements in particle measurement technology have underscored the significance of accurately ascertaining particle numbers, specifically particle number concentration.
The implementation of the segmented case with equation (1) could allow to predict large-scale sensor studies with various field gradient settings.The model is a useful instrument for comprehending and enhancing the operational efficiency of electrostatic particle sensors.Next steps involve a thorough exploration of the segmented case, encompassing variations in electrostatic field gradient along the sensor's x-axis, accompanied by empirical corroboration.
One constraint of the arbitrary case is closely tied to the presumption that the field lines are still parallel to the y-axis and the opening angles of the simple linear wall function are kept small.The detailed implementation of the arbitrary case would in a next step open the model for more complex geometries as considered in this work.For this step, the flux velocity and field strength might deviate from the previously assumed linearity.

Figure 1 .
Figure 1.Sensor description: left midsection of the sensor, right: simplified circuit diagram of the sensor setup.

Figure 2 .
Figure 2. Electrode arrangement in the three cases, A parallel, B segmented and C arbitrary.The electrodes are depicted in brown.The path of the inflowing particles is marked with cyan lines/dots whereas the path of the dendrite fragments is marked with purple lines/dots.B: U1 to U5 are the high voltage electrodes, each on a different electrical potential.C: widening of the electrodes leads to changes in the flow profile and the electric field.

Figure 3 .
Figure 3. Experimental setup.The aerosol stream from the aerosol source (miniCAST) is split and follows two paths: (a) the sensor setup, and (b) ventilation.After entering the dilution bridge, the aerosol combines in a static mixer.Post-mixing, the aerosol splits again.One portion enters a reference instrument's path (d), parallel to the sensor, requiring diluted air from (c).Simultaneously, a mass flow controller draws air via the sensor through path (e), safeguarded by a HEPA filter (in the device under test).

Figure 4 .
Figure 4. Model output as a function of particle size at different.

Figure 5 .
Figure 5. Signal as a function of particle size at different.

Figure 6 .
Figure 6.Sensor response versus captured soot mass for 3 particle size distributions and 2 voltages with a flow of 1 l min −1 .The solid line shows our simulated sensor signal and the dashed line represents the model calculations from[13].The experimental data is also taken from[13].C1-C3 represents CAST 1 soot to CAST 3 soot (see[13]).

Figure 7 .
Figure 7. Simulated sensor response (orange) and experimental sensor response (blue) from the custom-built sensor versus particle mass (MSSPlus) for CAST 1 soot at 1000 V and 1 l min −1 sensor flow.
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Table 2 .
model parameter and sensor geometry.