Chapter_08_gravimetric_settling_and_inertial_separation -...

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8.7 Gravimetric Settling in a Room Consider a room of volume V, height H, and horizontal cross-sectional area A as shown in Figure 8.18, which illustrates both models. (a) H y(D p ) (b) (c) c(D p ) = c(D p ) 0 c(D p ) = 0 c(D p ) < c(D p ) 0 Figure 8.18 Gravimetric settling of a monodisperse aerosol in quiescent room air: (a) initial condition for both cases, t = 0, (b) laminar settling model for t > 0, (c) well-mixed model for t > 0. Let c(D p ) 0 be the initial mass concentration of particles of diameter D p in the room (Figure 8.18a). In the laminar model all particles of the same size fall uniformly at terminal velocity v t such that those near the bottom settle to the floor. At any subsequent time there are no particles of that size above a certain height y(D p ), and the concentration remains at c(D p ) 0 below this height (Figure 8.18b). The same argument applies to particles of other sizes, except the values of y(D p ) are different because they settle at different velocities. For the well-mixed model, on the other hand, some particles settle, but those that remain are mixed throughout the room volume such that c(D p ) decreases with time (Figure 8.18c). 8.7.1 Laminar Settling Model …Some algebra yields … pt y(D ) H v t = Let the average mass concentration of particles be denoted by c (D p ). Thus, ( ) pp 0 t p 0 t p p0 Ay(D )c(D ) A H v t c(D ) vt c (D) c (D) 1 VV  == =   H The average concentration of these particles decreases linearly with time until a time (t c ) called the critical time elapses, where c t H t v = and all particles of the size whose terminal velocity is v t have settled to the floor. Note also that t c varies with terminal velocity v t , which depends on particle size, density, etc. – i.e, in a polydisperse aerosol, particles of different diameters settle at different speeds, and therefore have different critical times.
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8.7.2 Well-Mixed Settling Model On the other hand, suppose the well-mixed settling model is valid. As particles of a particular size fall to the floor with velocity v t , an idealized mixer instantaneously redistributes the remaining particles throughout the room. The rate of change of mass of particles of this size suspended in the room air is equal to the rate of deposition onto the floor, ( ) p p tp d c(D )V dc(D ) Vv A c dt dt == ( D ) which can be integrated to yield pp t p0 c (D) c vt exp c(D ) c(D ) H  Since the well-mixed model presumes that the concentration is the same throughout the enclosure at any instant of time, the term c(D p ) is also equal to the average concentration, c (D p ). The well-mixed model predicts that the average mass concentration decreases exponentially, while the laminar model predicts that it decreases linearly. In the well-mixed model c (D p )/c(D p ) 0 = 0.368 at t = t c , while in the laminar model it is zero. In the well-mixed model, the average mass concentration does not decrease to 0.001 (0.1%) of its initial value until nearly seven of these time constants, i.e. until t 7t c .
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This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Penn State.

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Chapter_08_gravimetric_settling_and_inertial_separation -...

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