In Situ Detection of Altered Particle Size Distributions during Simulated

Sep 16, 2010 - Eric Monsu Lee* and Herek L. Clack. Department of Mechanical, Materials, and Aerospace Engineering, Illinois Institute of Technology, ...
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Energy Fuels 2010, 24, 5410–5417 Published on Web 09/16/2010

: DOI:10.1021/ef100322b

In Situ Detection of Altered Particle Size Distributions during Simulated Powdered Sorbent Injection for Mercury Emissions Control Eric Monsu Lee* and Herek L. Clack Department of Mechanical, Materials, and Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois 60616 Received March 17, 2010. Revised Manuscript Received August 16, 2010

During full-scale sorbent injection tests for mercury removal at coal-fired power plants (CFPPs), a common trend has been observed in which the overall mercury removal efficiency reaches a performance plateau as the sorbent injection rate increases. Two common explanations for this trend are (1) the increased sorbent injection rate induces increased rates of particle agglomeration within sorbent feeding lines, which eventually shifts the particle size distribution to larger sizes and reduces the available particle surface area for heterogeneous oxidation and adsorption, and (2) the increased sorbent injection rate limits the available acid gas concentration for heterogeneous oxidation of Hg0 to Hg2þ. In this experimental study, an in situ measurement technique based on the principles of light extinction is used to assess the degree of particle agglomeration. The present in situ measurement technique applies a simple inline HeNe laser, and a sorbent feeder fluidizes powdered activated carbon (PAC) and drives the suspension through a length of rubber tubing to simulate the sorbent feeding process. Each experimental trial yields a time-dependent function of the light extinction ratio, I(t)/I0, for the time-dependent evolution of the PAC suspensions. Observed differences in I(t)/I0 for suspensions ejected directly from the feeder as compared to those directed through the tubing indicate changes in the overall particle size distribution. A numerical model of the experiment provides excellent agreement and facilitates interpretation of the experimental results.

performance predicted by a model of gas-particle mass transfer within ESPs for such large sorbent concentrations (greater than 99.9999% removal efficiency predicted at PAC injection rates of 32 lb/MMacf ).1,2 Researchers have proposed two possible, nonexclusive explanations. The first centers on a heterogeneous oxidation mechanism for chemical adsorption of Hg0, which requires acid gases (e.g., HCl) within the flue gas for the oxidation of Hg0 to Hg2þ on surfaces of sorbent particles. When sorbents are injected into flue gas, the interactions between sorbent particles and HCl allow the oxidation of Hg0 on sorbent surfaces and the subsequent capture of Hg2þ. However, as the sorbent injection rate increases, the increased scavenging rate of HCl by sorbent particles inhibits this Hg0 oxidation step, increasing the fraction of Hg0, which is far less readily adsorbed by sorbent particles and thus results in lower-thanexpected mercury-capture efficiencies. Because the oxidation of Hg0 by HCl on sorbent surfaces and the scavenging of HCl by sorbent particles both occur in-flight, they are transient in nature and therefore difficult to experimentally verify; as a result, no experimental data or kinetic modeling results have been shown proving this mechanism and its impact on mercury capture within an ESP during sorbent injection. The second possible explanation, the subject of the present investigation, relates to the effect of particle agglomeration, caused by particle collisions and/or external forces during sorbent injection. As the sorbent injection rate increases, the resulting higher particle concentration likely leads to an increased rate of particle agglomeration when sorbent particles are conveyed through sorbent feeding lines to the injection lances. During agglomeration, smaller particles collide with each other or with larger particles. This effectively reduces the

Introduction Sorbent injection is the most mature and cost-effective retrofit technology for reducing mercury emissions from coal fired power plants (CFPPs). Sorbent injection is the injection of a powdered sorbent into the flue gas to adsorb gas-phase mercury (i.e., Hg2þ and Hg0). However, results from selected full-scale tests of sorbent injection upstream of an electrostatic precipitator (ESP) diverge from model predictions,1,2 examples of which include the Pleasant Prairie,3 Brayton Point,4 and Salem Harbor5 power plants, where Norit Darco FGD PAC was injected. During these tests, mercury removal efficiency increased as the sorbent injection rate increased; however, removal efficiencies exhibited plateaus at higher sorbent injection rates. For example, measured mercury removal efficiency at Pleasant Prairie Power Plant (PPPP) approached a plateau of approximately 65% at sorbent injection rates greater than 10 lb/MMacf, up to a maximum of 32 lb/MMacf. These field results fall well below the *To whom correspondence should be addressed. E-mail: eric.monsu@ gmail.com. (1) Clack, H. L. Environ. Sci. Technol. 2006, 40, 3617–3622. (2) Clack, H. L. Environ. Sci. Technol. 2006, 40, 3929–3933. (3) Pleasant Prairie Power Plant Unit 2 - Sorbent Injection into a coldside ESP for mercury control; ADA-ES: Littleton, CO, 2003. http://www. netl.doe.gov/technologies/coalpower/ewr/mercury/control-tech/pubs/ FinalReportPleasantPrairie.pdf (accessed Aug 2010). (4) Brayton Point Station Unit 1: Sorbent Injection into a ColdSide ESP for Mercury Control; ADA-ES: Littleton, CO, 2005. http://www. netl.doe.gov/technologies/coalpower/ewr/mercury/control-tech/pubs/ FinalReportBraytonPoint.pdf (accessed Aug 2010). (5) PG&E NEG Salem Harbor Station Unit 1 - Sorbent Injection into a Cold-Side ESP for Mercury Control; ADA-ES: Littleton, CO, 2004. http:// www.netl.doe.gov/technologies/coalpower/ewr/mercury/control-tech/pubs/ FinalReportSalemHarbor.pdf (accessed Aug 2010). r 2010 American Chemical Society

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Energy Fuels 2010, 24, 5410–5417

: DOI:10.1021/ef100322b

Lee and Clack

Figure 1. Schematic of the in situ measurement technique for studying the effect of particle agglomeration (agglomeration sensor).

particle number concentration, increasing the mean distance between particles and the diffusion distance of trace gas-phase species to a particle surface. The present study is based on our previous work involving in situ measurement of particle mass concentration within PAC suspensions.6 This study also represents the first systematic test of the potential for agglomeration to occur during sorbent feeding. The study demonstrates a method for in situ sensing of changes in sorbent particle size distribution (PSD) without requiring extractive sampling and/or post facto separation and analysis of injected sorbent from the native fly ash. Such features make this in situ sensing technique attractive for ensuring optimal sorbent utilization for mercury emissions control. Experimental Section Figure 1 shows the experimental setup of the in situ measurement technique for studying sorbent particle agglomeration. This setup represents an extension of a previous study of in situ measurements of mass concentration within PAC suspensions6 by the addition of a 16-ft length of tubing to simulate a portion of the sorbent feed lines, which convey sorbent to injection lances during full-scale sorbent injection tests. The experimental setup is composed of a sorbent feeding system, an optical bench and data acquisition system, and a particle collection system. A sorbent feeder fluidizes and mobilizes the powdered sorbents (Figure 1). The sorbent feeder consists of a Plexiglas cylinder (400 o.d., 3.7500 i.d., 700 h), divided into two separate chambers that are connected by a 1/2-in. orifice. A function generator (BK Precision 4001A, 0-4 MHz) and an amplifier (Crown Power Base 3, 12.5 A,

Figure 2. Schematic representation of the sorbent feeder and relevant quantities used in the 1-D fluid numerical model.

700 W max output) drive an acoustic transducer (Morel, model MW113, 150W rms/210W Max, 70-6000 Hz), mounted to the bottom of the lower chamber. After adding a known mass of powdered sorbent to the lower chamber, the acoustic transducer fluidizes the powdered sorbent into a suspension, which is then driven into the upper chamber and ejected from the feeder altogether by two metered compressed air streams. Although these compressed air streams slightly pressurize the chambers, back pressure provided beneath the transducer provides a counterbalance and maintains actuator performance and consistent powder fluidization. The optical bench (Figure 1) is a simple inline HeNe laser beam (Coherent, model 31-2017-000, 0.46 mm diameter, λ = 633 nm,

(6) Lee, E. M. M.Sc. thesis, Illinois Institute of Technology, Chicago, IL, 2009.

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Figure 3. Experimental I(t)/I0 compared with numerical I(t)/I0 at a specified PAC mass.6

pinhole aperture (d = 0.46 mm) before impinging upon an unbiased photodiode (Thorlabs, model SM05PD1A), powered by a DC power supply (TEK Power, model HY3003D).

0.8 mW average output), which traverses the particle-laden jet ejected from the sorbent feeder. After traversing the particle suspensions, the attenuated HeNe laser beam passes through a 5412

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The photodiode registers a voltage V corresponding to the attenuated incident HeNe laser beam intensity I. A data acquisition card (National Instruments (NI) PCI-6013) acquires the photodiode output through an electronic circuit and a noise-reducing BNC connector block (NI BNC-2110). A PC data logger (NI Virtual Bench Logger v2.5) records the photodiode voltages, presented in the form of a voltage ratio V/V0. Because the output of the photodiode is linear over a range of incident energy flux values (0-1 mW), which is larger than the unattenuated HeNe laser intensity, ratios of V/V0 represent the ratio of the attenuated and unattenuated HeNe beam intensities I/I0. During each experimental trial, the ejected powdered sorbent is collected and weighed to obtain a mass balance and ensure a minimal loss of powder, thereby enabling the direct comparison of experimental and numerically predicted I/I0 ratios. Mass balance (m/m0) is defined as the ratio of particle mass collected downstream of the sorbent feeder, m, as compared to the initial mass used, m0. A vacuum pump (RIDGID, model WD4050, 5.0 HP max) draws the ejected powder into a HEPA filter (Dirt Devil, model F2), and a microbalance (Mettler, model AE 200) measures pre- and post-test mass to determine the change in mass resulting from the collected sorbent. Light extinction is the primary theory supporting the proposed in situ measurement technique, which depends on the properties of the particles along the optical path. Although micrographs show PAC particles to be highly irregular in shape, for simplicity, the present study assumes spherically shaped particles. Light extinction is defined by the Beer-Lambert law (eq 1): I ¼ expð - σe LÞ I0

Figure 4. (a) Assumed log-normal PSDs of mean particle size d p = 2.3, 4.1, and 6.6 μm and SD =1.3, 1.8, and 2.3. (b) Numerically predicted I/I0 as a function of mass concentration for the PSDs.

Qe approaches a constant value of 2.0 for dp greater than 4 μm. A 1-D fluid model predicts the time-resolved mass flow rate of particles between the lower and upper chambers (m_ l-u) and exiting the upper chamber (m_ u-e), as depicted in Figure 2. The model enables the direct comparisons of experimental and numerical I(t)/I0 values and also the estimations of number concentrations for experimental measured I(t)/I0. Because the model neglects the effect of particle agglomeration, the results of using the Beer-Lambert law to predict I(t)/I0 across the feeder exit reflect a fixed particle size distribution (PSD). The model handles transport of sorbent particles using a ventilation approach between the coupled chambers of the feeder. In each chamber, particle suspensions are assumed to be wellmixed, and therefore spatially uniform but varying in time. The time-varying particle mass concentrations in the lower (Cl(t)) and upper (Cu(t)) chambers of the feeder, combined with the volu_ into and out of the lower and upper metric air flow rates (Q) chambers, determine the values of m_ l-u (eq 3) and m_ u-e (eq 4), which together describe the temporal changes in Cl(t) and Cu(t). The numerical integration uses a Eulerian forward difference temporal discretization of the mass balance equations for each chamber as described in eqs 5 and 6. Assumptions used in the 1-D model include low particle mass loading (the mass density of air is much greater than the mass density of suspended particles) and that the air behaves as an ideal gas.   ! _ m air m_ l - u ¼ C0l Q_ l - u exp lt ð3Þ mair

ð1Þ

where I0 and I are, respectively, the unattenuated intensity before and the attenuated intensity after the interaction of light with a particle suspension, L is the optical path length through the suspension, and σe is the extinction coefficient. Generally, both absorption and scattering contribute to light extinction by particle suspensions. For strongly scattering particle suspensions such as marine haze, light absorption can largely be neglected. In contrast, scattering can largely be neglected when particle suspensions are strongly absorbing, such as PAC. The extinction coefficient σe for PAC used in the present study is defined by eq 2. For each particle size (di), Ni is the number concentration (#/m3), Ai is the cross sectional area, Qei is the extinction efficiency, and ri is the particle radius. For a specified initial mass, m0, the σe for a polydisperse suspension (i.e., PAC) is computed by eq 2, knowing the bulk density (Fbulk) and mixing chamber volume (Vc). σe ¼

X

Ni Ai Qei ¼

X 3m0 Qei 4ri Vc Fbulk

ð2Þ

Qe is the total extinction efficiency, defined as the ratio of the total scattered and absorbed radiant power to that geometrically blocked by particles.7 The value of Qe is a function of particle properties, including the refractive index (m), particle size (dp), and incident wavelength (λ), and represents the sum of the scattering efficiency Qs and absorption efficiency Qa. As PAC is highly absorbing in the visible spectrum, specifically λ = 633 nm of the HeNe laser beam, it would be expected that Qe ∼ Qa. Hodkinson7 shows that Qe exhibits a strongly inflected functional dependence on dp for transparent materials (i.e., when the refractive index m is real) but is far less dependent on dp for highly absorptive materials (i.e., when m is complex). Because PAC is black in color and expected to be highly absorptive in the visible wavelength range, the least inflected Qe curve is used in the present analysis, representing a refractive index m = 1.59 0.66i. On this basis, according to Hodkinson,7 for m = 1.59 - 0.66i,

m_ u - e ¼ Cu ðtÞQ_ u - e dmu ½ði - 1ÞΔtΔt, where dt i ¼ 1 to ¥

ð4Þ

mu ðiΔtÞ ¼ mu ½ði - 1ÞΔt þ

dmu ¼ m_ l - u - m_ u - e dt

ð5Þ ð6Þ

Results and Discussion The present in situ measurement technique and the ventilation model were previously compared as a part of a master’s

(7) Hodkinson, J. R. The Optical Measurement of Aerosols. In Aerosol Science; Davies, C. N., Ed.; Academic Press: New York, 1966.

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Lee and Clack Table 1. Proportions of Coarse and Fine PAC Sorbents Used to Create Different PSDs PSD #

Darco FGD (0.3-102 μm)

Darco 80  325 (10-220 μm)

1 2 3 4 5

100% 75% 50% 25% 0%

0% 25% 50% 75% 100%

Figure 5. (a) Truncated log-normal PSDs (= 10% mass loss, fine particle fraction) of mean particle size d p = 2.3, 4.1, and 6.6 μm and SD = 1.3, 1.8, and 2.3. (b) Numerically predicted I/I0 as a function of mass concentration for the truncated PSDs.

Figure 6. (a) Assumed log-normal PSDs of mean particle size d p = 2.3 μm and SD = 1.3, 1.8, and 2.3. (b) Numerically predicted I/I0 as a function of mass concentration for the PSDs.

Figure 7. (a) Measured I(t)/I0 for five admixtures of coarse and fine PACs yielding different PSDs. (b) Predicted I(t)/I0 for three monodispersed PSDs at an identical initial mass of PAC.

thesis.6 Figure 3 shows direct comparisons between experimentally measured and numerically predicted I(t)/I0 for six initial masses (i.e., 0.01, 0.02, 0.04, 0.06, 0.08, and 0.1 g), showing excellent agreement. Therefore, the present study focuses on the further investigation of sensitivity to different particle size distributions (PSD). The sensitivity of the Beer-Lambert law to characteristics of the PSD is shown in terms of predictions of I/I0, as a function of (i) variations in both mean particle size (d p) and standard deviation (SD) (Figure 4), (ii) the potential loss of fine particle mass (Figure 5), and (iii) variations in only geometric SD (Figure 6) of the log-normal PSD. The geometric standard deviation here is abbreviated as SD to differentiate from the light extinction coefficient σe. Figure 4

presents predicted I/I0 over a range of initial sorbent mass concentration for mean particle size d p = 2.3, 4.1, and 6.6 μm and for SD = 1.3, 1.8, and 2.3. Figure 4 indicates that relatively small differences in particle size and/or SD, such as would occur as a result of agglomeration, can produce substantial changes in I/I0. Figure 5 addresses the potential for the loss of fine particles to produce changes in I/I0, given the imperfect mass balances (m/m0 < 1) that are unavoidable in the experimental method. Figure 5 presents truncated versions of the assumed PSDs from Figure 4, intended to simulate a loss of the fine particle fraction. Comparing the trends in I/I0 between Figures 4 and 5, I/I0 varies nearly identically with sorbent mass concentration in both cases, indicating that the loss of fine particles does 5414

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Figure 8. (a, c, e, g) Numerically predicted particle number concentrations Nu(t) for PAC exiting the feeder (solid lines) compared to agglomeration threshold (dashed lines). Initial mass m0 of PAC indicated above. (b, d, f, h) Numerical and experimental I(t)/I0 through PAC suspensions at the feeder exit (dash and solid lines) and after flowing through the 16-ft rubber tube (dot lines).

roughly 10% mass loss evident in the experimental results is not likely to be a major factor influencing changes in I/I0. Such sensitivity is an advantage for the current investigation, as the detection of particle agglomeration is the focus. The present experimental setup also demonstrates the sensitivity of the technique and the reasonableness of the assumptions employed without the presence of the tube. Combining different proportions of a coarse (Darco 80  325, dp = 10-220 μm) and a fine (Darco Hg, dp = 0.3102 μm) PAC allows the formulation of different admixtures having different PSDs. Table 1 shows the different admixture proportions, by weight. Figure 7a shows five distinct experimental I(t)/I0 traces representing the five manually mixed

not significantly contribute to the observed attenuation. In addition, the impact of fine particle mass loss on experimental results is likely to be even less than that suggested by Figure 5. Whereas the predictions in Figure 5 assume 10% mass loss occurring exclusively in the fine particle fraction (the worst scenario), actual experimental mass loss occurs across the entire PSD, with the fine particle fraction representing only a relatively small portion. Figure 6 shows predicted I/I0 over a range of initial sorbent mass concentration at a fixed d p = 2.3 μm and for SD = 1.3, 1.8, and 2.3. By comparing Figures 4, 5, and 6, it is clear that Beer-Lambert-predicted light attenuation is more sensitive to variations in d p than to variations in SD, and that the 5415

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PSDs using a fixed mass of 0.1 g. For PSDs with a greater proportion of the fine PAC, greater light attenuation (lower values of I(t)/I0) is shown on the basis of the minimum value of I(t)/I0. As a comparison, Figure 7b shows numerically predicted I(t)/I0 for three monodispersed particle sizes (dp = 1, 10, and 100 μm) with an identical initial mass. As would be expected from light extinction theory, fine particles produce greater light attenuation than coarse particles on an equal mass basis. These results provide validation of the proposed method of in situ sensing of agglomeration by detecting changes in PSD during sorbent feeding. Having demonstrated both experimental measurement and numerical modeling of I(t)/I0, both techniques are used to assess the degree of particle agglomeration occurring after the PAC exits the sorbent feeder and flows through a 16-ft tube. Rubber-lined tubing is frequently used for sorbent delivery and sorbent injection testing for mercury emissions control, and the 16-ft length is roughly an order of magnitude shorter than typical sorbent feed line distances between sorbent storage and the injection lances. Thus, the detection of particle agglomeration in the present study would strongly suggest the occurrence of significant particle agglomeration during fullscale sorbent injection tests for injection rates that produce sorbent loadings (g/m3) equal to or exceeding those considered in the present study. Each experimental trial uses an initial mass m0 of PAC such that the maximum value of number concentration Nu(t) (using the manufacturer-specified particle size distribution) exceeds 1012 m-3, the threshold above which particle agglomeration rates cannot be neglected in particle suspensions.8 Comparing the initial sorbent mass, m0, to the sorbent mass collected in a downstream HEPA filter, m, provides the mass balance m/m0. With the final collected PAC mass m and the measured volumetric flow rates of air (Figure 2), the particle number concentrations Nu(t) and the corresponding I(t)/I0 can be obtained from the model. The sorbent feeder and the rubber tube are cleaned to remove accumulated sorbent particles, because sorbent particles can accumulate on the interior surfaces of the feeder and rubber tube. Each trial at a specified m0 begins with the repeated processing through the sorbent feeder (and rubber tube, if applicable) of PAC samples until a stable mass balance m/m0 occurs, typically no lower than 0.85 and occasionally as high as 0.98. During this conditioning step, m/m0 increases from relatively low initial values, asymptotically approaching a stable value of m/m0. I(t)/I0 is measured only upon reaching a stable value of m/m0 for a given m0. Any differences in I(t)/I0 with and without the rubber tube at a specified m0 would indicate the occurrence of particle agglomeration. Figure 8 shows experimentally measured I(t)/I0 results at four different values of initial PAC mass m0 (from top to bottom, 0.1, 0.2, 0.4, and 0.6 g). The individual subfigures 8a, c, e, and g represent the numerically predicted particle number concentrations exiting the upper chamber (solid line), compared to the threshold number concentration for considering particle agglomeration (dashed line). In Figure 8, the individual subfigures Figure 8b, d, f, and h present measured I(t)/I0 traces with and without the 16-ft-long rubber tube. In addition, the corresponding numerical I(t)/I0 traces without the 16-ft-long tube are presented for comparison against the

Figure 9. (a) Minimum I/I0 and (b) ΔI/I0 plotted as a function of the maximum number concentration for indications of particle agglomeration with the presence of the tube.

experimental data. A slight time delay is identified between the I(t)/I0 traces with and without the tube due to the additional time required for PAC to travel through the tube; however, it is assumed that the maximum number concentration is always associated with the minimum I/I0. Therefore, the experimental results (Figure 8) are analyzed and compared on the basis of the maximum light attenuation, i.e., the minimum value of I(t)/I0. It is apparent for all values of m0 and becomes increasingly apparent as m0 increases that the minimum value of I(t)/I0 is greater for PAC suspensions exiting the rubber tube than those measured directly exiting the feeder. Such a difference;reduced light attenuation;is consistent with an increase in mean particle size, d p, such as would occur as a result of particle agglomeration. Denoting the difference between the minimum measured values of I(t)/I0 with and without the rubber tubing as ΔI/I0 (Figure 8f ), increasing values of m0 lead to increasing values of ΔI/I0. Figure 9 with indicated error bars based on two repeated trials shows the trend more clearly in two plots: (a) minimum I /I0 as a function of maximum number concentration with and without the tube and (b) ΔI/I0 as a function of maximum number concentration. Figure 9a shows that measured light

(8) Hinds, W. C. Aerosol Technology: Properties, Behavior, and Measurement of Airborne Particles, 2nd ed.; John Wiley & Sons, Inc.: New York, 1999.

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attenuation by particle suspensions is less when the 16-ft tube is installed and is consistent with the occurrence of agglomeration. In addition, a plateau is reached in Figure 9b as the maximum number concentration is increased. It is possible that this reflects the modulation of gas-phase turbulence intensity at larger initial masses. Gas-phase turbulence intensity strongly influences the evolution of a particle-laden flow and also influences the gas-particle and particle-particle contacts.9 Gore and Growe10 and Hetsroni11 concluded that turbulence intensity in a pipe core was attenuated in the presence of small particles, where the attenuation process was caused by viscous drag force exerted on smaller particles traveling with turbulent eddies. In contrast, turbulence intensity can be enhanced by larger particles that are less responsive to turbulent eddies.10,11 The PAC sorbents used in the present study possesses a log-normally shaped particle size distribution with numbers of smaller particles (0.3 μm e dp e 30 μm) much greater than the numbers of larger particles (30 μm e dp e 103 μm). Therefore, the plateau in Figure 9b with an increasing initial mass of PAC could be caused by the

increasing numbers of smaller particles, which decrease the turbulence intensity and hence decrease the frequency of particle collisions for agglomeration. Conclusion The present investigation uses an approach involving experimental measurement and numerical modeling to demonstrate detection of changes in particle size distributions of powdered mercury sorbents during pneumatic feeding and injection. Experimentally measured changes in the light extinction characteristics of powdered activated carbon as a result of being fed through a length of rubber tubing show excellent agreement with numerical predictions. Such changes likely indicate the effects of particle agglomeration, leading to lower particle number concentrations and larger mean particle sizes at the point of injection, both of which would reduce in-flight mercury capture efficiency. Acknowledgment. Funding for this work was provided by a grant from the National Science Foundation (Grant # BES0607292), and through contractual agreement with BASF Catalysts. The authors gratefully acknowledge the financial support provided by these sponsors, without which this work would not have been possible.

(9) Hadinoto, K.; Sinclair-Curtis, J. Powder Technol. 2009, 195, 119–127. (10) Gore, R. A.; Crowe, C. T. Int. J. Multiph. Flow 1989, 15, 279–285. (11) Hetsroni, G. Int. J. Multiph. Flow 1989, 15, 735–746.

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