Centrifugal Differential Mobility Analyzer - Validation and First Two-Dimensional Measurements

1. Introduction

Advances in technology nowadays often require the availability of increasingly sophisticated methodology. Regarding the characterization of nanoparticles, the small dimensions drastically limit their direct investigation. Therefore, in most cases, an equivalent size is established. An equivalent diameter for example, is the diameter, a spherical particle would have regarding the obtained feature (e.g. electrical mobility). Naturally, this results in manifold potential equivalence values. To gain a more comprehensive understanding of a particle collective, it is possible to simultaneously determine multiple quantities. This can be achieved, for instance, using scanning electron microscopy (SEM) or tandem configurations, where one equivalent size can be initially measured and subsequently another one. However, these methods are complex in terms of the equipment which makes them expensive, require a high degree of expertise, or have poor statistical properties regarding sample size. To facilitate the combined measurement of multiple quantities, the CDMA (Centrifugal Differential Mobility Analyzer) was developed. The CDMA is a combination of an Aerodynamic Aerosol Classifier (AAC) ([1] and a Dynamic Mobility Analyzer (DMA) [2], which enables the simultaneous measurement of the equivalent mobility diameter and the equivalent Stokes diameter, thus yielding a two-dimensional distribution of these two variables. Subsequently, additional parameters, such as the effective density or the fractal dimension, can be determined, giving information on particle shape as well [3,4,5].

2. Theoretical Fundamentals

Like the DMA and AAC, the CDMA is composed of two concentric cylinders arranged concentrically, guiding a sheath air flow (c.f. cross-section of the CDMA in Figure 1). When an aerosol volume flow is applied to the inner cylinder (c.f. Figure 1, detail A), the particles are moved toward the outer cylinder by centrifugal and electrical forces. At the end of the classification chamber, (c.f. Figure 1, detail B), the particles are sampled, so that particles matching the predetermined properties are counted. Assuming Stokes’ drag force, the force equilibrium results in:

$$Q_P·E+m_P·a_c=3\pi \eta d_m{w}_{Dr}·{\displaystyle \frac{1}{Cu}}$$

(1)

$Q_P$ represents the particle charge, E denotes the electric field magnitude, $m_P$ is the particle mass, $a_c$ represents the centrifugal acceleration, $\eta$ denotes the dynamic viscosity, $d_m$ is the mobility diameter, $w_{Dr}$ denotes the particle drift velocity, and Cu is the Cunningham slip correction factor. The deterministic description of the particle path is achieved by rearranging and integrating equation (1).

$$r\left(y\right)=\sqrt{{\displaystyle \frac{\left(\tau ·{\omega }^2·r_{in}^2+\frac{Z·U}{ln\left(\frac{r_o}{r_i}\right)}\right)·exp\left\{2·\tau ·{\omega }^2·y·\frac{\pi ·\left(r_o^2-r_i^2\right)}{Q_{sh}+Q_a}\right\}-\frac{Z·U}{ln\left(\frac{r_o}{r_i}\right)}}{\tau ·{\omega }^2}}}$$

(2)

U represents the voltage, $\omega$ denotes the rotational velocity, $r_o$ and $r_i$ are the outer and inner radii, $r_{in}$ is the radius at which the particle enters, y is the position of the particle in the direction of the length L, $Q_{sh}$ is the sheath air volume flow, $Q_a$ is the aerosol volume flow, $\tau$ is the particle relaxation time and Z is the particle mobility. With

$$\tau ={\displaystyle \frac{\rho ·d_v^3·Cu\left(d_m\right)}{18·d_m·\eta }}={\displaystyle \frac{\rho ·d_{st}^2·Cu\left(d_m\right)}{18\eta }}$$

(3)

and

$$Z={\displaystyle \frac{n·e·Cu\left(d_m\right)}{3\pi ·\eta {·d}_m}}$$

(4)

the particle path for each variable can be calculated from the volume flow ratio and the applied rotational velocity and voltage. $Z^{*}$ is the mobility required for a particle entering at the center of the aerosol inlet to be sampled exactly at the center of the outlet [6]. ${\tau }^{*}$ describes the same behavior, but for the relaxation time [1].

$$Z^{*}={\displaystyle \frac{Q_{sh}+Q_{ex}}{4·\pi LU}}·ln\left({\displaystyle \frac{r_o}{r_i}}\right)$$

(5)

$${\tau }^{*}={\displaystyle \frac{Q_{sh}+Q_{ex}}{\pi {\omega }^2{(r_i+r_o)}^{2}L}}$$

(6)

$\rho$ is the particle density, $d_{st}$ is the Stokes equivalent diameter, n is the number of charges carried by a particle, e is an elementary charge and $Q_{ex}$ is the excess gas volume flow. Relating the particle properties Z and $\tau$ to the characteristic mean values of the actual operation point $Z^{*}$ and ${\tau }^{*}$ lead to the normalized mobility or the normalized particle relaxation time, respectively.

$$\tilde{Z}={Z/Z}^{*};\tilde{\tau }={\tau /\tau }^{*}$$

(7)

For further information, see [7].

3. Design

Figure 1 shows a cross-section of the CDMA. The aerosol is fed from the left side, enters through the center bore, and flows through a small gap to the inlet of the classification channel A. The sheath air is fed between two ferrofluidic seals on the left side. Then it passes through 8 axial holes into the CDMA, where it mingles with the aerosol volume flow at A. Together the flows pass through the classification zone. At point B, the excess volume flow continues the inner cylinder, flows to the center and exits through a center bore on the right side. The sample air is extracted in the radial direction at point B. Then, the sample flows through small channels until it leaves the CDMA through a small hole between the ferrofluidic seals on the right side. Voltage can be applied through a carbon-sliding contact on the outer cylinder, the speed is applied by a belt drive mounted on the left side [7].

4. Numerical Flow Simulations

CFD simulations were performed under the assumption of incompressible flow using the k-$\omega$ SST-model with OpenFOAM. Since there are eight symmetrically arraanged inlet pipes for the sheath air, the flow domain can be reduced to an eighth of the whole geometry, as shown in Figure 2.

The symmetry interfaces were defined as periodic interfaces, so fluxes across the interfaces were admitted. The boundary condition at the walls is a zero-slip boundary condition (Dirichlet boundary condition for the velocity). The aerosol inlet, sheath air, and excess air are also defined as Dirichlet boundary conditions. In this case, the velocity is not set to zero but to values that correspond to the desired flow rate in or out of the domain at the respective in- or outlet. The aerosol inlet and the excess air outlet have a plug-flow velocity profile, while the inlet of the sheath air is modeled with a fully developed laminar flow. The sample outlet is defined by the Neumann condition; thus the mass balance is always maintained.

The rotation is implemented by a single rotating frame approach, using the ’SRFSimpleFoam’ solver to calculate the velocity fields. The flow ratio $\beta =0.2$ ($Q_a=Q_s=0.3l/min$, $Q_{sh}=Q_{ex}=1.5l/min$) is considered in the following simulation results.

4.1. Flow Behavior Without Rotation

To investigate the radial symmetry of the flow, Figure 3 shows the flow pattern at different positions. Here, x is the absolute position in axial direction, where $x=0$ is at the inlet of the sheath air. For reasons of clarity the velocities in axial direction in the highlighted planes are color coded. On the right side the calculated vectors are shown exemplary to emphasize that there is only flow in the axial direction. At the axial coordinate $x=0.14m$, the deflection of the sheath air ends. The second plane from the left is located directly before the aerosol joins the sheath air. It can be observed that the flow profile is symmetrical across the gap width, with no significant deviations in the circumferential direction. Additionally, there is only a slight change in the axial direction. However, due to the limitations of this representation, it is challenging to accurately quantify the data. Figure 4 provides a more detailed illustration of the axial velocity at the center of the gap, which allows for a more precise analysis of the circumferential direction (white lines of Figure 3).

At $x=0.14m$, which is upstream of the aerosol inlet, the velocity distribution is slightly irregular, due to the influence of the deflection arc. However, immediately at the beginning of the transfer domain at $x=0.151m$, the velocity distribution becomes quite regular. Further downstream, when the aerosol is inserted, the velocity in the axial direction increases because the volumetric flow rate is higher while the cross-section remains constant. Because there is no change in the circumferential direction, it is valid to represent the velocity in the axial direction in the transfer domain by using velocity values from the middle plane of the investigated flow domain if no rotation is applied.

Figure 5 shows the streamlines at $0RPM$ for this middle plane. There appears to be no cross mixing, but the streamlines of the particle laden flow widen to a certain extent. There are two reasons. First, the flow has a little radial component since the outlet/inlet is very short. Second, the CDMA is constructed that for $\beta =0.1$, so the two streams would perfectly fill the space. A $\beta$ of $0.2$ means that there is relatively more of $Q_a$ that needs to occupy more space. Additionally, this will be increased through the presence of a flow profile, because if there is a laminar flow profile over the gap, the velocity in the middle is much higher, so there is more fluid transported over time.

4.2. Flow Behavior with Rotation

Figure shows the axial flow profiles in the circumferential direction analogous to Figure 4, but for certain values of rotational velocity of the outer cylinder. It can be observed that, at low speeds, the flow profile distribution over the circumference is homogeneous. At higher rotational speeds, a larger flow length is required until a homogeneous flow profile is achieved over the circumference. The axial velocity also increases with increasing speed, at least during the initial phase of the classification zone (x = 0.151 m). This can be attributed to a higher volume flow at the inner boundary due to the elevated pressure at the outer boundary. Consequently, the resulting velocities are higher. Although an inhomogeneous velocity profile was identified over the circumferential direction, the difference in flow velocity at each location was found to be insignificant (with a maximum deviation of $\pm 10\%$ around the value at $\varphi =0$). Consequently, only the results from the center plane are discussed below.

The flow of the sheath air through the arc ahead of the transfer domain is displayed at a rotational speed of $100RPM$ in Figure 6. The displayed vectors represent the relative velocity and are directed in the opposite direction to the rotation. This is because the flow continuously experiences a higher circumferential velocity while traveling in the positive radial direction. While having only a density of about $1.2kg/m^3$, air experiences mass inertia and cannot instantly adopt its flow direction to the increased circumferential velocity. Thus, the flow appears to move against the direction of rotation based on the relative velocity. At the end of the deflection arc, the flow of the sheath air remains at the respective level of circumferential velocity and eventually adopts its flow direction in the axial direction (right hand side, Figure 6).

The resulting velocity vector needs more traveling distance to point only in the axial direction. This relation is displayed in Figure 7. The vectors of the relative velocity on the middle plane are displayed from a top-down perspective for different rotation numbers.

With increasing rotational speed, the pressure increases at the outer wall when the sheath air arc is left. In Figure 8 axial velocity fields of the cases for 0, 475, and 2000 rpm are compared.

It can be observed that the effective flow cross-section for the flow in the axial direction decreases with increasing rotational speed. This resulted in a shifted velocity profile toward the inner wall of the transfer domain and higher peak velocities. Downstream in the axial direction, the effective cross-section widens up to the geometrically available cross-section. This results in a more symmetric profile for the axial velocity.

5. Derivation of Ideal Transfer Functions from the CFD Simulation

To investigate the influence of the flow profile on the classification the transfer function was derived directly from CFD data. The determination of a transfer function for specific CDMA operating parameters (at fixed voltage and velocity) is a stepwise process. First, for a given particle size (e.g. the smallest to be investigated), the entry point of the particles is varied, and the corresponding trajectories are calculated to see which particles will reach the classification zone. Second, the particle size is varied according to a given particle collective. The transfer probability for one particle size is then given by the number of successfully traversed particles $N_{\mathrm{sampled}}$ divided by the number of entry points $N_{\mathrm{tot}}$:

$$\Omega \left(d_p\right)={\displaystyle \frac{N_{\mathrm{sampled}}}{N_{\mathrm{tot}}}}$$

(8)

The particle size can then be converted into the mobility Z, particle relaxation time $\tau$ or their normalized values ($\tilde{Z},\tilde{\tau }$), respectively.

The particle trajectory was calculated using CFD data (c.f. Figure 9). A virtual time step was defined for a trajectory. For every node (black points in Figure 9) the velocity data is known. The algorithm searches the nearest data point to the particle and uses the velocity given there to calculate the way the particle is traveling in axial direction. In radial direction the velocity is computed neglecting diffusion and inertia of the particle, the way of the particle in radial direction could be calculated using the particle size, radial position, and operating parameters of the CDMA (c.f. equation (2)).

Figure 10 shows successfully sampled particle trajectories (green) and deposited ones. It is obvious that the resolution of the simulation depends on the number of streamlines, the duration of the virtual time steps and the number of particle sizes investigated. However, this increases with computational cost, thus a compromise must be made.

5.1. Ideal Transfer Function

To compare the simulated results with theoretical values, we use ideal transfer functions. The ideal two-dimensional transfer function was derived from streamline functions by Rüther (forthcoming2).

$$\begin{array}{c}\hfill \Omega ={\displaystyle \frac{1}{2\beta (1+A)}}·[-|-1-{\displaystyle \frac{\beta +{\kappa }^2}{{\kappa }^2-1}}·A+{\displaystyle \frac{\tilde{Z}}{\tilde{\tau }}}·{\displaystyle \frac{1+\beta }{2\tilde{h}}}·A|+|-1-\beta -{\displaystyle \frac{{\kappa }^2+\beta {\kappa }^2}{{\kappa }^2-1}}·A+{\displaystyle \frac{\tilde{Z}}{\tilde{\tau }}}·{\displaystyle \frac{1+\beta }{2\tilde{h}}}·A|\\ \hfill +|-1+\beta -{\displaystyle \frac{\beta +{\kappa }^2}{{\kappa }^2-1}}·A+{\displaystyle \frac{\tilde{Z}}{\tilde{\tau }}}·{\displaystyle \frac{1+\beta }{2\tilde{h}}}·A|-|-1-{\displaystyle \frac{{\kappa }^2+\beta {\kappa }^2}{{\kappa }^2-1}}·A+{\displaystyle \frac{\tilde{Z}}{\tilde{\tau }}}·{\displaystyle \frac{1+\beta }{2\tilde{h}}}·A\left|\right]\end{array}$$

(9)

$\beta$ is the ratio of aerosol volume flow $Q_a$ to sheath air volume flow $Q_{sh}$, $\tilde{h}$ is the ratio of the gap width to the mean radius, $\kappa$ is the ratio of the inner radius $r_i$ to the outer radius $r_o$ and A is a substitution.

$$\beta =Q_a/Q_{sh}$$

(10)

$$\tilde{h}=2·{\displaystyle \frac{r_o-r_i}{r_o+r_i}}$$

(11)

$$\kappa ={\displaystyle \frac{r_i}{r_o}}={\displaystyle \frac{1-\tilde{h}/2}{1+\tilde{h}/2}}$$

(12)

$$A=\exp \left\{\tilde{\tau }·{\displaystyle \frac{2\tilde{h}}{1+\beta }}\right\}-1$$

(13)

This results in a two-dimensional distribution, displayed in Figure 11. As the computing power would be too immense for a complete two-dimensional simulation as described in Section 5 only the transfer functions for $\tilde{\tau }=0$ and $\tilde{Z}=0$ (red, dashed lines) and the transfer function for spherical particles (red solid line) are investigated.

5.2. Simulated Transfer Functions for $\tilde{\tau }=0$ and $\tilde{Z}=0$

The transfer functions illustrated in Figure were calculated at different densities. $\tilde{\tau }$ and $\tilde{Z}$ are unity at a particle size of $100nm$. It is evident that the simulation results exhibit several discernible stages, which can be attributed to the limited resolution, particularly the restricted number of particle trajectories per particle size. An investigation of the pure electrical mode (see Figure ) revealed a high degree of agreement between the transfer functions. The minor discrepancies can be attributed to the flow field. Figures and illustrate the transfer functions at different speeds in purely rotational mode. The transfer functions exhibit a high degree of agreement for the case of $475RPM$ (see Figure ). The transfer function is slightly lower, indicating a slightly higher separation. As the speed increases (cf. Figure ), the relative position of the transfer function remains consistent, although the initially distorted flow profile increases the likelihood of particle separation.

5.3. Simulated Transfer Functions for Different CDMA Operating Parameters

To extend the investigation of the transfer functions not only for the transfer functions at $\tilde{Z}=0$ and $\tilde{\tau }=0$, the combination of applying both, voltage and rotational velocity is examined, too. To simply the two-dimensional transfer function to a one-dimensional transfer function, which then can be investigated, perfectly spherical particles are used. Here, following dependency emerges (see red solid line in Figure 11):

$$\tilde{Z}\propto \sqrt{1/\tilde{\tau }}$$

(14)

Varying the operating parameters results in a shift of that line, resulting in the transfer functions shown in Figure . The simulated transfer functions are again very close to the ideal transfer functions but are smaller in height.

This indicates that the transfer functions derived from the simulation are accurately represented by the theoretical calculation. Furthermore, the results demonstrate that a laminar flow profile within the CDMA has no significant impact on the transfer behavior. At higher speeds, distortion of the transfer function occurs when the flow profile is not yet fully developed.

A significantly greater influence is observed when the boundary condition of a constant particle flow density at the inlet is assumed by assuming a constant particle flow density and a laminar flow profile at the aerosol inlet. This behavior is already described in (Rüther, forthcoming2), but could also be supported by a simulation in a subsequent study.

6. Measurement of Transfer Functions

Consequently, after the simulative validation of the ideal transfer functions, the next step must be experimental validation.

6.1. Theory

Like shown in Rüther et al. ([7]) tandem setup comprising a DMA and a CDMA is employed to determine the transfer functions (c.f. Figure 12). Following the assumption of a broad distribution of the test aerosol, the ratio of the number concentrations ($n_1$ and $n_2$ after the CDMA classification) can be described as follows [8]:

$$n_2/n_1={\displaystyle \frac{\underset{-\infty }{\overset{+\infty }{\int }}{\Omega }_1\left(\tilde{Z}\right)·{\Omega }_2\left(\tilde{Z}\right)\mathrm{d}\tilde{Z}}{\underset{-\infty }{\overset{+\infty }{\int }}{\Omega }_1\left(\tilde{Z}\right)\mathrm{d}\tilde{Z}}}$$

(15)

The transfer functions in general are well described by Gaussian functions (Rüther, forthcoming2). $a,c,d,e,{\tilde{\mu }}_1,{\tilde{\mu }}_2$ being the parameters of the Gaussian function, the transfer function of the pre-classifying DMA ${\Omega }_1$, and the transfer function of the CDMA ${\Omega }_2$, can be described as follows:

$${\Omega }_1=a·\exp \left\{-{\displaystyle \frac{{(\tilde{Z}-{\tilde{\mu }}_1)}^2}{c^2}}\right\};{\Omega }_2=d·\exp \left\{-{\displaystyle \frac{{(\tilde{Z}-{\tilde{\mu }}_2)}^2}{e^2}}\right\}$$

(16)

Thus, equation (15) can be simplified analogous to [7]:

$$n_2/n_1=\sqrt{{\displaystyle \frac{1}{c^2+e^2}}}·d·\left|e\right|·\exp \left\{-{\displaystyle \frac{{({\tilde{\mu }}_2-{\tilde{\mu }}_1)}^2}{c^2+e^2}}\right\}$$

(17)

This means that measuring the ratio $n_2/n_1$ while varying the voltage or rotational velocity of the CDMA (i.e. varying the ${\tilde{\mu }}_2$), makes it possible to determine the transfer function by simply fitting a Gaussian function to the measurement data $n_2/n_1$ [7].

However, for this purpose, the transfer function of the pre-classifying DMA must be precisely determined. The exact approach to achieve this is described in Appendix B, so that $c_{\tilde{Z}}$ and ${\tilde{\mu }}_{1,\tilde{Z}}$ are determined as follows:

$$c_{\tilde{Z}}=0.5732·\beta$$

(18)

$${\tilde{\mu }}_{1,\tilde{Z}}=1.0154$$

(19)

This procedure for determining the transfer function of a DMA-DMA setup can be applied to a DMA-AAC setup as well. In the case of spherical particles, the mobility of the first DMA can be converted into the particle relaxation time. Defining the transfer functions as follows:

$${\Omega }_1=a·\exp \left\{-{\displaystyle \frac{{(\tilde{\tau }-{\tilde{\mu }}_1)}^2}{c^2}}\right\};{\Omega }_2=d·\exp \left\{-{\displaystyle \frac{{(\tilde{\tau }-{\tilde{\mu }}_2)}^2}{e^2}}\right\}$$

(20)

The following parameters can then be determined by applying equation A1.

$$c_{\tilde{\tau }}=0.6369·\beta$$

(21)

$${\tilde{\mu }}_{1,\tilde{\tau }}=-0.99477$$

(22)

These values can be used to calculate the transfer functions for the CDMA according to equation 17.

6.2. Production of the Test Aerosol and Measurement Setup

A method for producing an aerosol of a relatively immutable, broad particle size distribution over an extended period was developed using a setup comprising two hot-wall reactors with an agglomeration tube positioned between them (see Figure 12). Silver is is melted and evaporated at a temperature of ${1150}^{\circ }C$ within the front hot-wall reactor. The air saturated with silver is extracted from the hot-wall reactor via an air flow. The flow rate ($2lpm$) was regulated by a mass flow controller (MFC). During the cooling phase, the solution becomes supersaturated, resulting in the formation of silver particles. Subsequently, larger agglomerates are formed in the agglomeration tube, which are ultimately sintered into spherical particles in the second hot-wall reactor at ${750}^{\circ }C$.

Subsequently, the spherical particles are pre-classified in a DMA, which is set to a specific mobility value. The aerosol then passes through valve $V1$. Opening $V1$ and $V2$ simultaneously bypasses the CDMA, allowing the number concentration $n_1$ to be determined directly by the CPC. Conversely, when both valves are closed, the aerosol passes through the CDMA, which was programmed to run sweeps for the voltage and the rotational velocity. This allows the number concentration to be determined as a function of the operating parameters $n_2=f(U,\omega )$.

6.3. Determination of the Transfer Function Parameters for $\tilde{\tau }=0$ and $\tilde{Z}=0$ at Different $\beta$-Values

Figure illustrates the parameters of the CDMA transfer function over a range of operational parameters for $\tilde{\tau }=0$. Figure illustrates the parameters of the CDMA transfer function over a range of operational parameters for $\tilde{\tau }=0$. Considering the influence of diffusion, particularly in the case of the CDMA geometries (where the gap width is small, thereby necessitating a substantial diffusion component), the diffusion-related transfer functions, can be calculated in (Rüther, forthcoming2). Using these influences adjusting the measurement values. Additionally, Rüther et al. ([7]) demonstrated that at specific points in the CDMA, increased separation occurs due to peak values in the electric fields and the centrifugal field at the sample outlet. These values were also calculated and the measured values corrected accordingly.

Besides these adjustments, a direct relation between particle size and deposition is evident in Figure . Moreover, a reduction in deposition is evident at elevated aerosol volume flows. It is noteworthy that values exceeding 1 were observed, which can be attributed to potential measurement errors or inaccuracies in the determination of the transfer function of the pre-classifying DMA.

Moreover, the width of the transfer function is nearly identical to the calculated ideal width (cf. Figure ). However, the measurement series for $0.3lpm$ exhibits a slight elevation for $200nm$ particles, which can also be attributed to measurement errors. This is because only a negligible number of particles were counted at $200nm$, as this was at the edge of the particle size distribution of the test aerosol.

The width of the transfer function (Figure ) is comparable to the results obtained from the DMA experiments, which are presented in Figures to .

The transfer functions for $\tilde{Z}=0$ are illustrated in Figure . It is important to note that for $\beta <0.1$, there is no measurable transfer function. This is due to particle separation occurring at the outlet due to the prevailing centrifugal forces. Thus, each measurement series comprises only two data points [7].

Figure illustrates the maximum height of the transfer functions. Despite the corrections, the transfer function remains dependent on the particle size and the prevailing flow conditions. This finding is particularly relevant in the context of rotational operation, because even small imperfections, such as deviations in production, a non-concentric alignment of both cylinders, and similar factors have a greater impact on the flow field, generating this dependency. The measurement was conducted over the entire CDMA (including transport to and from the measurement gap), thus precluding the possibility of correcting all undetermined losses. However, no direct correlation could be established, and thus, no equivalent loss length for diffusion, as would be present when flowing through a pipe, could be determined.

Figure illustrates the width of the transfer function. The measured values displayed do not exhibit a correlation with the beta values as the range of variation is too big. This can be attributed to the inaccuracies and the big losses of the CDMA system. In comparison to $\tilde{\tau }=0$, a notable shift in the transfer function (c.f. Figure ) is present. This indicates a dependency to particle size, volume flows or rotational speed.

6.4. Determination of the Transfer Function Parameters for $\tilde{\tau }=0$ and $\tilde{Z}=0$ at $\beta =0.2$

Since a complete investigation of all parameters would be very time-consuming, it is better to focus on a more detailed investigation of the targeted operating conditions ($Q_a=0.3lpm$ at $\beta =0.2$). As part of this investigation, the transfer functions for particles of varying sizes are measured once more under the aforementioned conditions. The resulting data is presented in Figure .

After correction, it can be observed that the height of the transfer functions (see Figure ) is highly stable for all particle sizes investigated and exhibits a similar magnitude for both operating modes. It is notable that the width of the transfer functions exhibits a slight upward deviation for larger particles, which can be attributed to the relatively low particle number for particles exceeding $150nm$. Moreover, the correlation between displacement and particle size, speed, and voltage can be determined.

The results are then transferred to the actual measurement operation and the kernel matrix is calculated (see (Rüther, forthcoming3)). This is achieved by forming an average value from all measured values with regard to the height, using the ideal width as the basis for the width. The shift can be corrected by a straight line drawn through the measured values.

7. Measurement Results and Analysis of Two-Dimensional Size Distributions for Different Sintering Stages of Silver Nanoparticles

The results from Section 6 are now employed in the calculation of the kernel matrix, which is analogous to that described by (Rüther, forthcoming3). Subsequently, the two-dimensional distribution is calculated using a POCS algorithm (Rüther, forthcoming3).

7.1. Production of the Aerosol and Measurement Setup for Two-Dimensional Distributions

The first measurements of a two-dimensional particle size distribution are carried out using the same test aerosol from Section 6.2 but with no pre-classifying DMA. Figure 13 shows the setup. In this study, the sintering stage should be influenced by the temperature of the sintering furnace. Here, temperatures of ${20}^{\circ }C$, ${60}^{\circ }C$, ${100}^{\circ }C$, ${175}^{\circ }C$, ${250}^{\circ }C$, and ${400}^{\circ }C$ are used.

7.2. Derivation of Further Properties

From the two-dimensional distribution of the mobility and Stokes diameter, it is possible to derive other quantities like the aerodynamic $d_{ae}$ and volume $d_v$ equivalent diameters, the effective density ${\rho }_{eff}$ or the shape factor $\chi$ [9].

$$d_{ae}=d_{st}·\sqrt{\rho /{\rho }_0}$$

(23)

$$d_v={(d_{st}^2·d_m)}^{1/3}$$

(24)

$${\rho }_{eff}/\rho ={(d_v/d_m)}^3$$

(25)

$$\chi =d_m/d_v$$

(26)

These relations are used to calculate further the two-dimensional distributions which are displayed and discussed.

7.3. Measurement Results

Figure to show the two-dimensional distributions of the forementioned quantities at different sintering temperatures. At high sintering temperatures, the relative effective density ${\rho }_{eff}/\rho$ and shape factor $\chi$ for each particle are close to 1. The $d_{st}/d_m$-distribution shows a bi-sectional distribution as we obtain almost perfectly spherical particles. For lower sintering temperatures, the relative effective density becomes lower, and the shape factor becomes larger, showing more irregularly shaped particles (c.f. Figures to , for more sintering stages in between). The use of this additional information about the particle shape could lead to a better understanding of the process. So the absorption or reaction coefficients can be manipulated trough the particle production process, for generating more effective particles.

8. Conclusions

The proposed methodology for determining two-dimensional property distributions has significant potential for both research and industrial processes. With these additional insights, the behavior of the particle systems can be more comprehensively understood. The Centrifugal Differential Mobility Analyzer can measure two-dimensional distributions regarding the mobility and Stokes diameter and thus enabling the investigation of the spatial distribution of different other characteristic property values. The functionality of the prototype was validated and then meticulously characterized to identify potentials for further improvements. Nevertheless, it is feasible with high accuracy to determine the actual transfer functions, thereby enabling data inversion on this basis. The data inversion is highly robust and yields plausible results (Rüther, forthcoming3), that can be converted into further property distributions. In comparison with one-dimensional measurement techniques, the CDMA offers a substantial increase in the amount of information obtained. Consequently, the next step must be the construction of an enhanced version, which will exhibit significantly fewer losses and therefore provide significantly better measurement values [7], reduce the diffusion by using a lager gap (Rüther, forthcoming2) and enhance the flow field within the CDMA at enhanced rotational speeds. The implementation of the scanning mode enables a substantial reduction in the measurement time, thereby rendering the entire measurement process less susceptible to errors i.e. fluctuations at the test aerosol generation process. Subsequent investigations should also cover the further validation of results, such as the actual particle sizes and effective density, using scanning electron microscopy (SEM) images. If a known particle shape is known, it is feasible to determine the charge distribution and the Cunningham slip correction, thereby establishing a novel methodology for determining these variables, particularly for non-spherical particles. Furthermore, combining the CPC with a Faraday-Cup-electrometer result in more information and thus in the stabilization of the data inversion. The presented study focuses on the determination of particle shape. Therefore, it is essential to determine the density with a high degree of precision. However, in the case of unknown particle systems, no information that allows the actual shape to be determined is available. Integration with additional measurement systems could enable access to a three-dimensional distribution of properties. For instance, it may be feasible to examine the scattered light behavior of particles in conjunction with the CPC or to integrate mass spectrometry.