Miniature Pipe Bundle Heat Exchanger for Thermophoretic Deposition of Ultrafine Soot Aerosol Particles at High Flow Velocities

Abstract

The deposition of submicrometer soot aerosol particles in a miniature pipe bundle heat exchanger system has been investigated under conditions characteristic for combustion exhaust from diesel engines and oil or biomass burning processes. The system has been characterized for a wide range of aerosol inlet temperatures (390–510 K) and flow velocities (1–4 m s⁻¹), and particle deposition efficiencies up to 45% have been achieved over an effective deposition length of 27 cm. Thermophoresis was the dominant deposition mechanism, and its decoupling from isothermal deposition was consistent with the assumption of independently acting processes. The measured deposition efficiencies can be described by simple linear parameterizations based on an approximation formula for thermophoretic plate precipitators. The results of this study support the development of modified heat exchanger systems with enhanced capability for filterless removal of combustion aerosol particles.

INTRODUCTION

Ultrafine combustion aerosol particles pose a potential threat to human health since they can enter the alveolar system of human lungs. Therefore, public authorities and industry aim to reduce particle emissions by means of combustion process development and exhaust gas treatment. In many applications, conventional filter systems lead to an undesired increase of pressure drop and are prone to clogging by soot and oil ashes (Neeft et al. 1996). Electrostatic filtration has found wide application to remove combustion particles; however, this method imposes high investment and running costs. Due to the limitations of particle loading by their size, small particles between 10 and 30 nm cannot be removed reliably. Furthermore, electrostatic precipitators can generate undesired gaseous components, especially when operated with negative voltage (Yehia et al. 2000). Turbulent precipitators showed particle deposition efficiencies of up to 35% with significantly reduced removal of particles in the range between 80 and 300 nm (van Gulijk et al. 2001). Shi and Harrison (2001) report on thermophoretic deposition of diesel soot in a water-cooled fluidized bed. In diesel processes (Shi et al. 1999) nearly all particles are in the submicrometer size range based on particle mass; in the case of biomass burning (Baumbach 1993) it is more than 80%. Messerer et al. (2003) could show that the thermophoretic coefficient of ultrafine soot agglomerate particles exhibits no significant dependency on agglomerate size. Therefore, thermophoretic soot particle removal can provide a reliable solution for filterless combustion aerosol deposition in the submicrometer size range. Sufficient temperature gradients for particulate removal can be established in heat exchangers, so that future engineering could focus on the parallel optimization of heat transfer and submicrometer particulate removal in one device.

Byers and Calvert (1969) were the first to perform experiments and analyses of thermophoretic deposition of particles in a hot turbulent gas stream from the aspect of air cleaning. Nishio et al. (1974) investigated the thermophoretic deposition of aerosol particles in a heat exchanger pipe, in particular the influence of fouling on the long-term heat exchange behavior of the tube. Further studies of thermophoretic particle deposition in externally cooled tubes include Stratmann and Fissan (1989), Chang et al. (1990), Montassier et al. (1991), Sasse and Nazaroff (1994), Chang et al. (1995), and Lin and Tsai (2003). Sasse and Nazaroff (1994) performed a numerical simulation of a tobacco smoke particle filter based on thermophoretic deposition from natural convection flow. They emphasize the necessity for a proof-of-principle experiment for the validation of their simulation results. Up to now little information about the thermophoretic deposition of submicrometer aerosol particles at high flow velocities in the temperature gradient field around internally cooled tubes is available.

Thermophoretic particle motion can be described by (Talbot et al. 1980)

[1]

where K _th is the thermophoretic coefficient, v _th is the thermophoretic particle velocity, and ∇ T represents the temperature gradient in the vicinity of the particle. μ _g is the gas dynamic viscosity, T _p is the particle temperature, and ρ _g is the gas density. For the free molecular regime (Kn ≫ 1), Waldmann and Schmitt (1966) derived a thermophoretic coefficient that is independent of particle size: K _th = 0.55. For the transition (Kn ≈ 1) and continuum (Kn ≪ 1) regimes, however, the thermophoretic coefficient generally depends on particle size, and different formulae for the calculation of K _th as a function of the Knudsen number have been derived (e.g., Brock 1962; Derjaguin et al. 1976; Kousaka et al. 1976; Talbot et al. 1980). In accordance with a theoretical study by Rosner and Khalil (2000), we demonstrated that the thermophoretic coefficient of agglomerate soot particles in the transition regime can be approximated by K _th≈ 0.55 (Messerer et al. 2003). For the simple case of a constant temperature gradient along a flown-through rectangular channel in thermophoretic plate precipitators, the thermophoretic deposition efficiency ϵ _th can be efficiently approximated by (Tsai and Lu 1995; Messerer et al. 2003)

[2]

where H and Δ T are the distance and temperature difference between the plates, respectively; and μ _g,0, ρ _g,0, and v _x,0 are the gas properties (dynamic viscosity, density, and axial velocity) at the average temperature in the precipitator.

Miniaturized heat exchanger systems provide a high surface-to-volume ratio and therefore lead to high heat transfer rates even under laminar flow conditions. The heat transfer behavior in these devices has been investigated by a number of researchers over the last decade, e.g., Peng et al. (1995). Due to high temperature gradients thermophoresis is expected to be significantly higher than in conventional heat exchangers for particle-loaded flows. To our knowledge this is the first study on the removal of aerosol particles in the external flow around cooling tubes in a miniature heat exchanger under experimental conditions relevant for modern combustion exhaust systems with aerosol flow velocities between 1 and 4 m s^{− 1}.

METHODS

High Temperature Gradient Pipe Bundle Heat Exchanger

Figure 1 shows the cross-sectional area of the miniature pipe bundle heat exchanger that has been designed and used to investigate particle deposition at high gas flow rates and temperature gradients. Twenty seven stainless steel tubes with an inner diameter of 0.99 mm and an outer diameter of 1.59 mm are arranged in a near-quadratic area of 10.4 · 10.5 mm such that the axes of all tubes are near-equidistant. At their ends 5 mm of each tube are fitted into 10 mm stainless steel blocks, which are inserted into a stainless steel channel. The top of the channel can be removed to control the alignment of the tubes as well as the soot deposition (Figure 2). The 2 mm viton sealing in the top was found to be leak proof for the experimental conditions of this study. Along the 300 mm effective length of the tubes between the two blocks the bottom area of the channel as well as the top were formed in a way that reduces the free space towards the tubes, simulating half-perimeter tubes directly mounted at the bottom and top plate.

FIG. 1 Cross-sectional area of the miniature pipe bundle heat exchanger. Pipe inner diameter d _i = 0.99 mm, outer diameter d _o = 1.59 mm. Aerosol flow around the cooling tubes flown through by cooling air.

FIG. 2 Photograph of the pipe bundle heat exchanger inlet section.

The hot aerosol was led into the channel by a 8 mm d _i tube at an angle of 35° as a compromise between engineering requirements and a high fluid impulse in axial direction to minimize turbulences in the entry region of the tube channel (Figure 3). The aerosol outlet was designed symmetrically. The cooling fluid—in the study presented here pressurized air—was added by 8 mm d _i tubes through the top plate and by a smooth 90° curvature directed into the steel blocks supporting the 27 heat exchanger tubes. The pipe inlets were conical to minimize pressure build up by the inflowing cooling gas. The outlet was symmetrical. The whole heat exchanger was thermally insulated to reduce heat transfer to and from the channel. Therefore the aerosol flowing between the outer tubes and the walls of the heat exchanger heated up the channel walls, and so no significant undesired thermal gradients that would increase thermophoretic deposition towards a cooler channel wall were established. The walls of the upper and lower plate were adopted to the void space to reduce the aerosol flow between the channel walls and the outer tubes. At the same time there was still space between the outer tubes and the channel walls to avoid direct thermal contact and conductive heat transfer between the wall and tubes. According to the cross-sectional areas (Figure 1) only about 10% of the aerosol flow passed between the outer tubes and the wall. Therefore, the particle deposition effects observed in this study were not significantly influenced by these boundary phenomena. The internal cooling of the 27 tubes lead to a smaller thermal axial expansion of the tubes in comparison to the channel housing the steel blocks. Therefore the tubes were equidistantly distributed over the whole heat exchanger length. The effective deposition length can be calculated from the tube length between the stainless steel blocks, the distance between the blocks, and the axial position where the inflowing aerosol comes in contact with the cooling tubes: L = 300 − 2 · (15) = 270 mm. The relative error in the determination of the deposition length resulting from the diameter of the inflowing aerosol tube (8 mm) is ±3%.

FIG. 3 Longitudinal section of the pipe bundle heat exchanger. Note the symmetrical design of the deposition device.

Temperatures of the system were measured with four K-type thermocouples (accuracy ±0.1 K). The thermocouples for the aerosol inlet and outlet were placed in the center of the 8 mm tubes about 2 mm before the tube bundle. The thermocouples for the cooling gas inlet and outlet were mounted in the center of the tube blocks at a distance of about 3 mm.

Experimental Setup and Measurement Procedure

The complete experimental setup applied in this study is outlined in Figure 4. The test aerosol particles were produced by a spark discharge between graphite electrodes (Alfa Aesar, Karlsruhe, Germany, purity 99.9995%) in a 3.7 l min^{− 1} argon flow (Messer Griesheim, Krefeld, Germany, purity 4.6). The primary carbon particles produced with the applied spark discharge aerosol generator are known to have a diameter of ∼5 nm (PALAS GfG 1000, Karlsruhe, Germany; Evans et al. 2003). After passing through an agglomeration reservoir (2 l glass flask), the aerosol flow was diluted with 4.4 l min^{− 1}of filtered nitrogen and led through a Kr 85 aerosol neutralizer (10 mCi). The aerosol flow through the pipe bundle heat exchanger was controlled by venting the excess through an outlet valve into the exhaust line. Before entering the heat exchanger system the aerosol was heated to the desired inlet temperature.

FIG. 4 Schematic flow diagram of the experimental setup for counterflow heat exchanger particle deposition measurements.

The symmetric sampling setup at ambient temperature and equal flow conditions enabled near-identical aerosol sampling conditions before and after the heat exchanger. Thus the significant thermophoretic losses which are known to occur upon cooling of a hot aerosol flow to ambient temperature in a sampling line (Berger et al. 1995) could be avoided. Other potential effects of the sampling process on the aerosol properties can be assumed to cancel out and were neglected in the measurement data analysis.

Particle number size distributions were measured with a scanning mobility particle sizer (SMPS) system consisting of an electrostatic classifier (TSI 3071) and a condensation particle counter (TSI 3025) operated with a sample flow of 1 l min^{− 1}, a sheath air flow of 10 l min^{− 1}, and a scan time of 120 s (measurement range 20–300 nm). Before the SMPS, the particle concentration in the sample flow was reduced by a ratio of 1:20 in a dynamic dilution system (TOPAS DDS 560) to minimize soiling and transient particle deposition in the measurement device. The particle size spectra were analyzed with a resolution of 16 particle size classes i per decade, providing sufficient information on the size distribution of the sparkdischarge soot aerosol and maintaining a high signal-to-noise level at the same time.

The parameters describing flow and temperature properties of the different experiments are summarized in Table 1. After every experiment the pipe bundle heat exchanger was flushed with pressurized air at flow velocities of about 40 m s^{− 1} to reliably remove soot deposits on the tubes, which influence particle deposition and heat transfer properties.

TABLE 1 Experimental parameters for the investigation of the particle deposition mechanisms in the miniature pipe bundle heat exchanger

Exp.	V h,out [l min^−1]	v x,hot,in [m s^−1]	T h,in [K]	T h,out [K]	T c,in [K]	T c,out [K]	ΔT log,mean [K]	Re	V c,in [l min^−1]
I a	3	1.31	403.5	301.9	300.5	329.5	18.30	64.8	5
I b	3	1.41	436.0	302.8	301.0	338.8	23.92	63.1	5
I c	3	1.49	460.5	303.5	301.4	346.5	28.02	62.0	5
I d	3	1.58	489.0	304.1	301.5	356.0	33.14	60.8	5
II a	4	1.73	400.2	301.1	300.0	333.1	16.04	86.7	5
II b	4	1.79	414.0	303.5	299.5	342.5	23.40	85.4	5
II c	4	2.05	474.8	303.3	300.1	363.2	30.60	81.8	5
II d	4	2.16	499.0	300.4	297.8	373.0	32.80	81.0	5
III a	5	2.17	401.5	301.3	300.5	330.4	15.64	108.2	10
III b	5	2.37	440.1	301.6	300.7	341.0	20.87	105.0	10
III c	5	2.55	473.0	302.8	301.7	352.5	25.42	102.5	10
III d	5	2.74	507.9	304.4	303.0	364.5	30.69	100.1	10
IV a	6	2.55	393.5	302.4	297.6	344.7	18.97	130.5	5
IV b	6	2.64	407.3	303.7	302.3	339.8	17.06	128.8	10
IV c	6	2.86	442.5	304.9	303.2	352.1	22.32	125.2	10
IV d	6	3.03	468.0	300.8	298.9	359.0	26.45	123.8	10
IV e	6	3.2	494.7	301.0	299.3	367.1	29.23	121.6	10
V a	8	3.35	387.5	302.3	300.3	339.9	14.36	174.9	10
V b	8	3.86	447.0	303.8	300.7	366.5	23.77	166.7	10
V c	8	4.16	482.1	304.0	300.2	381.0	29.63	162.8	10
Volumetric flow rates at ambient temperature and pressure: 300 K, 950 mbar

The spark-discharge soot aerosol exhibited an approximated log-normal size distribution with a median particle diameter of about 82 nm, a geometric standard deviation of 1.64, and a number concentration between 6 and 9 · 10⁶ cm^{− 3}. It is similar to those encountered in the exhaust of modern heavy duty diesel engines (Shi et al. 1999). The average particle size distribution before and after the heat exchanger for experiment Id is given in Figure 5.

FIG. 5 Soot-particle size distributions measured before and after the heat exchanger in experiment Ia (arithmetic mean ± standard deviation of 6 measurements each).

For each set of parameters the heat exchanger system was heated up by the aerosol flow until thermal equilibrium was reached. Then 12 consecutive particle size distribution measurements were taken, alternatingly before and after the heat exchanger. For every switching, the particle concentration difference was divided by the particle concentration measured before the precipitator. The arithmetic mean of the 11 values per particle size class i obtained by this procedure was taken as the (size-dependent) total deposition efficiency, ϵ _tot,i . The measured total deposition efficiency can be split into an isothermal contribution ϵ _iso,i caused by diffusion, impaction, and interception under isothermal flow conditions and a thermophoretic contribution ϵ _th , _i . ϵ _iso,i is the deposition efficiency measured when the heat exchanger was operated under isothermal but otherwise unchanged conditions. It was generally in the range of 2–20%.

The small volume of the heat exchanger necessitates a macroscopic description of the thermal gas properties by measuring temperatures at the hot aerosol inlet, T _h,in, the aerosol outlet, T _h,out, the cooling gas inlet, T _c,in, and the cooling gas outlet, T _c,out. The logarithmic mean temperature difference, Δ T _log,mean which is generally used to describe heat transfer characteristics, is given by

[3]

RESULTS AND DISCUSSION

In Figure 6 the measured ϵ_tot,i and ϵ_iso,i are displayed for the conditions of experiment series I with average aerosol flow velocities v _x,0 between 1.15 and 1.28 m s⁻¹. Small particles in the size range of 34 to 70 nm are found to exhibit significant particle deposition up to 20% in the case of small aerosol flows through the heat exchanger when no temperature gradients are established. This observation can be attributed to diffusion processes occuring in the heat exchanger, leading to an increased particle deposition with decreasing d _p . Increasing T _h,in leads to significantly enhanced particle deposition ϵ_tot,i , indicating that thermophoretic deposition occurs.

FIG. 6 Total particle deposition efficiencies ϵ_tot,i and isothermal losses ϵ_iso,i in the pipe bundle heat exchanger for experiments Ia–Id. Data points with error bars represent the arithmetic mean ± standard deviation of 11 differential measurement values.

To investigate the contribution of thermophoresis to the total particulate deposition ϵ_tot,i , the different deposition mechanisms have to be decoupled. The first step in decoupling the thermophoretic deposition component ϵ_th,i is to determine how the mechanisms couple together. A general expression for combining the mechanisms occuring in this study is given by

[4]

where f _iso,th is a function of the mechanisms involved in the deposition process describing the interaction between the different deposition mechanisms. A detailed description of the different methods of decoupling thermophoresis from other deposition mechanisms is given in Romay et al. (1998). Still, the determination of f _iso,th is a matter worthy of discussion. For their studies on the contribution of thermophoresis to particle deposition in tubes, Nishio et al. (1974) and Romay et al. (1998) simply added ϵ_iso,i and ϵ_th,i with f _iso,th = 0. On the other hand, Brockmann (2001) proposed f _iso,th = −ϵ_iso,i · ϵ_th,i for independently acting deposition processes. The experimental data of this study allow the testing different coupling approaches. Figure 7 shows ϵ_tot,i and ϵ_iso,i measured in experiment Ia and thermophoretic contribution ϵ_th,i calculated from Equation (4) for the different assumptions outlined above. With f _iso,th = −ϵ_iso,i · ϵ_th,i , ϵ_th,i exhibits no significant size dependency, as expected from earlier experimental results and theory calculations (Messerer et al. 2003). With f _iso,th = 0, on the other hand, ϵ_th,i exhibits a pronounced decrease towards smaller particle size at d _p < 100 nm, which is not consistent with earlier investigations (Messerer et al. 2003). Similar effects were observed for all experiments performed in this study (more pronounced at low and less pronounced at high aerosol flow rates). Therefore, all further values of ϵ_th,i presented in this study have been calculated from

[5]

Particle removal efficiencies representative for the investigated size range were calculated by concentration-weighted averaging,

[6]

[7]

where c _i represents the particle number concentration per size class averaged over the 12 consecutive SMPS measurements, and c _int is the particle number concentration integrated over the investigated size range.

FIG. 7 Total particle deposition efficiencies ϵ_tot,i , isothermal losses ϵ_iso,i , and thermophoretic deposition efficiencies ϵ_th,i calculated with f(iso,th) = −ϵ_th,i · ϵ_iso,I and f(iso,th) = 0, respectively, for experiment Ia. Data points with error bars represent the arithmetic mean ± standard deviation of 11 differential measurement values; dashed line illustrates ϵ_th,avg calculated with f(iso,th) = −ϵ_th,i · ϵ_iso,i .

A detailed mathematical model of particle deposition in the pipe bundle heat exchanger would require solving a complex set of differential equations describing fluid flow, heat transfer, and particle motion by extensive numerical calculations for every relevant set of experimental conditions. To find a simple semiempirical parameterization of deposition efficiency as a function of temperature and flow conditions, we tested the applicability of the plate precipitator formula in Equation (2).

In Figure 8 the average total particle deposition efficiencies ϵ_tot,avg from all experiments are plotted against the dimensionless “precipitator number” (Lμ₀Δ T _log,mean)/(v _x,0ρ₀ TH _ch ²). μ₀, v _x,0 and ρ₀ are the arithmetic mean values of the gas properties calculated for the temperatures of the aerosol flow at the inlet and outlet of the heat exchanger (T _h,in, T _h,out), respectively. L and Δ T _log,mean are the effective deposition length and logarithmic mean temperature difference as defined above. H _ch is the characteristic distance for particle deposition along the temperature gradient perpendicular to the gas flow. In a plate precipitator it is the distance between the hot and cold plates. For the miniature pipe bundle heat exchanger it was calculated as the arithmetic mean of vertical and horizontal half-distances between the outer surfaces of the cooling pipes: H _ch = 0.25 · (0.51 mm + 1.88 mm) = 0.60 mm (Figure 1). This is assumed to be a characteristic average for the varying distances between maximum temperatures in the middle of the flow channels and minimum temperatures at the cooling pipe surfaces.

FIG. 8 Total particle deposition efficiency ϵ_tot,avg plotted against the dimensionless precipitator number calculated from average flow parameters, (Lμ₀Δ T _log,mean)/(v ₀ρ₀ T ₀ H _ch ²). The lines are linear least-squares fits to the data sets with different cooling air flow rates (5 l min^{− 1} dotted; 10 l min^{− 1} dashed). Error bars represent the standard deviation (±1 s.d.) of the averaged values ϵ_tot,i .

For each of the investigated cooling air flow rates (5 L min⁻¹ and 10 L min⁻¹), ϵ_tot,avg exhibits a near-linear increase with the precipitator number. The increase is steeper for the higher cooling air flow, but the linear least-squares fits to both measurement data sets intercept the y axis (Δ T = 0 K) at ϵ_tot≈ 4%, which is consistent with the particle deposition observed under isothermal conditions averaged over all experiments (ϵ_iso,avg = 4.5 ± 2.5%).

Figure 9 shows the average thermophoretic deposition efficiencies, ϵ_th,avg, plotted against the precipitator number. Again, a near-linear increase is observed for each of the cooling air flows. The linear least-squares fits intercept the y axis (Δ T = 0 K) at ϵ_th,avg≈ 0, as expected for isothermal conditions. The slopes of the fit lines (0.42 and 0.34, respectively) are fairly close to the value of 0.55, which can be observed in plate precipitators and equals the thermophoretic coefficient applicable for soot agglomerates in the investigated range of particle size and experimental conditions (Tsai and Lu 1995; Messerer et al. 2003).

FIG. 9 Thermophoretic particle deposition efficiency ϵ_th,avg plotted against the dimensionless precipitator number calculated from average flow parameters, (Lμ₀Δ T _log,mean)/(v ₀ρ₀ T ₀ H _ch ²). The lines are linear least-squares fits to the data sets with different cooling air flow rates (5 l min^{− 1} dotted; 10 l min^{− 1} dashed) and the theoretical relation for a plate precipitator (solid). Error bars represent the standard deviation (±1 s.d.) of the averaged values ϵ_th,i .

If the dimensionless precipitator numbers were calculated with characteristic distances of H _ch,1* = H _ch sqrt (0.55/0.32) = 0.79 mm and H _ch,2* = H _ch sqrt (0.55/0.42) = 0.69 mm, which are well within the range of different half-distances occurring between the outer surfaces of the cooling pipes in heat exchanger (0.26–0.94 mm), the slopes of the linear least-squares fit would be identical to K _th≈ 0.55. The results confirm that plate precipitator formula can be regarded as a physically reasonable and a consistent basis for simple semiempirical parameterizations of ϵ_th,avg as a function of temperature and flow conditions in miniature pipebundle heat exchangers. For a given geometry and cooling gas flow the characteristic distance, H _ch* can be determined from a few measurements and used to estimate deposition efficiencies for varying thermal conditions.

To investigate the influence of deposited soot on the deposition efficiency, experiment Ic was extended over a time span of 13 h. ϵ_tot,avg was near-constant at (37 ± 1)%. After the long-term experiment the isothermal deposition efficiencies ϵ_iso,i were found to be 5–10% higher than at the beginning. Apparently the deposited soot led to enhanced nonthermophoretic deposition and reduced the contribution of thermophoresis, while the overall deposition efficiency remained near-constant. Substantial blackening of the heat exchanger tubes was observed upon visual inspection after the long-term experiment. The blackening was most intense at the beginning of the heat exchanger and strongly decreased along the flow direction, indicating that the particle removal was dominated by deposition over the first few centimeters of the heat exchanger, where the highest temperature gradients and thermophoretic forces occur.

The dominant influence of the conditions at the aerosol inlet was confirmed by test calculations in which the inlet temperature and flow parameters (Δ T _h,in, μ _h,in, v _x,h,in, T _h,in) instead of the average parameters (Δ T _h,0, μ _h,0, v _x,h,0, T _h,0) and an effective deposition length L* = 0.1· L = 30 mm were inserted in Equation (2). The assumption L* = 0.1· L = 30 mm is based on observed deposition patterns and is consistent with basic temperature profile calculations. With these parameters the data points plotted against the precipitator number converged towards the theoretical line for a simple plate precipitator with K _th≈ 0.55 (Figure 10).

FIG. 10 Thermophoretic particle deposition efficiency ϵ_th,i plotted against the dimensionless precipitator number calculated from effective flow parameters at the hot inlet, (L ^*μ _h,inΔ T _h,in)/(v _h,inρ _h,in T _h,in H _ch ²). The line is the theoretical relation for a plate precipitator. Error bars represent the standard deviation (±1 s.d.) of the averaged values ϵ_th,i .

CONCLUSIONS

The results of this study demonstrate the applicability of miniature pipe-bundle heat exchangers for efficient particle deposition under typical combustion exhaust conditions. At aerosol inlet temperatures of 390–510 K, flow velocities of 1–4 m s^{− 1}, and cooling air inlet temperatures around 300 K, deposition efficiencies of up to 45% have been achieved for submicrometer spark discharge soot aerosol particles.

Thermophoresis was the dominating particle deposition mechanism, and its decoupling from isothermal mechanisms was consistent with the assumption of independently acting processes (Brockmann 2001).

Simple parameterizations derived from plate-precipitator theory and experiments (Tsai and Lu 1995; Messerer et al. 2003) were found to be suitable for the description of particle deposition efficiencies as a function of inlet temperatures, flow rates, and geometric parameters without solving the complex set of differential equations describing fluid flow, heat transfer, and particle motion in the pipe-bundle heat exchanger.

Overall, the results of our study confirm the high potential of miniature pipe bundle heat exchangers for combustion exhaust treatment systems combining efficient heat recovery and aerosol particle deposition.

NOMENCLATURE

A HE	=	Cross-sectional area of heat exchanger (m2)
c p	=	Particle number concentration (cm− 3)
d p	=	Particle diameter (nm)
D h	=	Hydraulic diameter, 4A HE/P HE
fiso,th	=	Coupling term of deposition mechanisms
H ch	=	Characteristic distance (m)
H ch*	=	Modified characteristic distance (m)
H P	=	Plate distance (m)
Kn	=	Knudsen number, 2λ/d p
K th	=	Thermophoretic coefficient
L	=	Deposition length (m)
L*	=	Modified deposition length (m)
P HE	=	Perimeter of heat exchanger (m)
Re	=	Heat exchanger Reynolds number, v h,0 D h ρ g,h,0/μ g,h,0
T	=	Gas temperature (K)
T p	=	Particle temperature (K)
v th	=	Thermophoretic velocity (m s− 1)
v x	=	Axial velocity (m s− 1)

Greek Lettters

ϵiso	=	Isothermal particle deposition
ϵth	=	Thermophoretic particle deposition efficiency
ϵtot	=	Measured particle deposition efficiency
Δ Tlog,mean	=	Mean logarithmic temperature difference (K)
λ	=	Mean free path of gas molecules (m)
μ	=	Dynamic viscosity of the gas (kg m− 1 s− 1)
ρ g	=	Density of the gas (kg m− 3)
∇ T	=	Temperature gradient in the heat exchanger (K m− 1)

Additional Subscripts

i	=	Particle size class
avg	=	Weighted average over particle spectrum
0	=	Arithmetic mean of inlet and outlet aerosol properties
h	=	Hot aerosol
c	=	Cooling air

Acknowledgments

This work is part of the project “Katalytisches System zur filterlosen kontinuierlichen Ruβpartikelverminderung für Fahrzeugdieselmotoren,” supported by the Bavarian Research Foundation, Munich. Additional funding from the Max-Buchner-Forschungsstiftung and technical support by Sebastian Wiesemann are gratefully acknowledged.