Heat transfer in laminar flow in circular and flat conduits with (constant) surface resistance

Isokinetic sampling, in which a subsample is extracted from the center of laminar aerosol flow, is routinely used to collect representative particles for analysis. Isokinetic sampling minimizes wall effects, including particle loss due to Brownian diffusion to the tube wall. This particle diffusion is analogous to the heat transfer problem originally posed by Graetz in 1883. Analytical solutions to the Graetz problem have been applied to calculate particle loss averaged over the entire main flow. However, these solutions overestimate diffusional particle loss near the center of the main flow. In the present solution, confluent hypergeometric functions are used to solve analytically for particle concentration as a function of radius. The solution is integrated near the center of the main flow to determine particle loss for isokinetically sampled aerosols. Sampling efficiencies valid down to nanometer-sized particles are presented in terms of dimensionless parameters. Diffusional particle loss for isokinetically sampled aerosol can be 1.8 times less than that from the main flow aerosol. The present results can be used to design isokinetic sampling systems and to assess particle loss in these systems. For 5 nm diameter particles sampled isokinetically from a laminar flow tube (0.318 cm tube radius, 10 m length) into an ultrafine condensation particle counter, sampling efficiency is strongly affected by main flow Reynolds number, Re; sampling efficiency increases from 4.9% at Re = 100 to 99% at Re = 1500.

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“…Since the eigenfunctions, Y n (r ), are orthogonal, the eigenconstants, C n , can be determined from (Sideman et al 1965)…”

Section: Graetz Problemmentioning

confidence: 99%

“…For the special case r 1 = 1, the concentration averaged over the main flow can be found analytically using the relationship (Sideman et al 1965…”

Section: Appendixmentioning

confidence: 99%

Diffusional Particle Loss Upstream of Isokinetic Sampling Inlets

Tyree

Allen

2004

Aerosol Science and Technology

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“…The integral is equal to 4/3 times the mean concentration 0, for a parallel-plate channel (Bowen et al, 1976). The mean concentration is also identified as the mixing-cup concentration (Colton et al, 1971) in the mass-transfer context and the mixing-cup temperature (Sideman et al, 1965) in heat transfer. After introducing the mean concentration and integrating over the channel width, we arrive at the total particle flow exiting the channel (5)…”

Section: Aiche Journal February 2001mentioning

confidence: 99%

“…1 and 2 in a parallel-plate geometry, the mean concentration can be written as an infinite series of modes (Sideman et al, 1965). For the present purposes, it is convenient to write the series as where is the residence time/diffusion time: the average residence time for a particle in the channel of length z divided by the time for the particle to diffuse the distance h across the channel.…”

Section: Aiche Journal February 2001mentioning

confidence: 99%

“…The problem then becomes equivalent to mass transfer with a permeable boundary (Colton et al, 1971) or heat transfer with constant wall resistance (Sideman et al, 1965). For special geometries such as parallel-plate and circular cylinder channels, analytical solutions to the governing equations may be obtained using a modified Graetz solution (Bowen et al, 1976).…”

mentioning

confidence: 99%

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