1.061 / 1.61 Transport Processes in the Environment

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1 MIT OpenCorseWare / 1.61 Transport Processes in the Environment Fall 008 For information abot citing these materials or or Terms of Use, visit:

2 Answer 8.1 Diffsion is a specific term referring to the net flx reslting from zero-mean, random motions. The random motions may be Brownian motion œ reslting in moleclar diffsion œ or may be de to trblent eddies. Strictly, the size of the eddies mst be small compared to the patch of particles in order for impact of the eddies on the patch to be trly random, and ths the net flx to be prely diffsive. Advection is the transport of particles (concentration) by the mean crrent. Dispersion is a broad term sed to describe all processes, except diffsion, that act to disperse a patch, and ths diminish concentration. Typically, dispersion coefficients reflect a combination of advection and diffsion processes that are difficlt to model separately. For example, the copled effect of differential longitdinal advection and cross-channel diffsion creates shear dispersion. The existence of mltiple discrete flow paths, e.g. throgh a poros medim, also creates a dispersion of particles, becase the time to traverse each path differs with its geometry and the degree of bending/trning œ often called the tortosity. Ths, many particles released together bt which take different pore pathways will get separated (dispersed) in the longitdinal direction. Finally, the lateral diffsion of particles into a region of zero velocity (a pore space with no otlet, a side-pool in a river, a wake behind an obstrction) will create longitdinal dispersion. The particles that enter the zones of zero velocity get held back relative to the particles that do not get trap. When the particles are eventally released from the dead-zone, they are separated in space (dispersed) from the rest of the clod. Answer 8. T he expression for longitdinal dispersion in a wide channel is K x = 5.93 * h, where * is the shear velocity and h the water depth. We wish to make the comparison, K X1 K X * 1h1 = (1) * h First consider how * will change between section 1 and. For steady niform flow driven by bed slope, the momentm balance reqires that * = ghs Eqation 3, Chapter 7. S ch that * 1/ * = (h 1 S 1 ) (h S ) () If the channel depth is constant, then from (1) K X1 = * 1 = S 1 K X * S > 1. The dispersion is greater in the steeper channel. If the channel width is constant, then from continity the channel depth mst change inversely with the depth-averaged velocity, U. Specifically, U1h1=U h, or h1/h = U/U1. (3)

3 If the sbstrate is the same, we can infer that U/ *, is constant. More formally, sing the drag coefficient for the bed, C b, we can write τbed=ρc b U =ρ *, which also indicates that for constant C b the ratio U/ * is constant, i.e. U1/ * 1 = U/ *. (4) Combining (3) and (4), h1/h = * / * 1. Using this ratio in (1), K X1 /K X = 1. That is, for a constant channel width, the longitdinal dispersion is the same in both sections. Answer 8.3 Vegetated Floodplain h1 h b When the water depth increases from h1 to h, flow enters the floodplain. Vegetative drag and the shallow depth combine to retard flow on the floodplain relative to the channel. This prodces strong lateral shear that agments dispersion. The lateral profile of depth-averaged velocity is shown below for flow depth h1 (red) and h (black). The velocity flctations, ', are deviations from the channel mean velocity, =Q/A total. Adapting the expression for shear dispersion (eq. 16, chap. 8) to the lateral shear, y y 1 B K x = ' ' dydydy, BD y 0 00 where B is the total channel width with the flood plain. From the sketch below, it is clear that the sm of spatial flctations, ', is greater when the water depth permits flow on the floodplain, and so we expect K X to increase at the greater water depth. This is a greater effect than the increase lateral diffsivity that might occr with increasing width, which according to the above eqation wold decrease K X. In particlar the lateral diffsivity cannot increase significantly throgh the floodplain becase of obstrction by vegetation.

4 Channel B Floodplain y ' ' Answer 8.4 Time t1<< B /4Dy: When the slg is initially released, it is very small compared to the width of the channel. Released in the center of the channel, the variation in velocity (shear) across the patch is negligible, and the entire patch advects at the same speed. Becase the patch is not experiencing differential advection, the spreading of the clod in the longitdinal direction is de to longitdinal diffsion only. The clod's longitdinal length scale is 4 Dt. More specifically, letting the release point be (x, y, z) = (0, 0, 0), the concentration field evolves as, M (x t) y z C(x, y, z, t) = 3/ exp ( 4 π Dt ) 4Dt 4Dt 4Dt

5 Time t>> 0.4 B /Dy. By this time the patch has grown to niformly fill the lateral dimension of the channel. In addition, sfficient time has passed for the longitdinal dispersion de to the lateral shear to reach Fickian behavior. The longitdinal length-scale of the patch is now 4 K x t, i.e. the patch growth rate is dictated by the dispersion coefficient, K X. In the vertical direction the clod contines to grow via vertical diffsion, sch that the concentration field evolves as, M DK x t exp (x - t) z C(x,z, t) = - - B 4π 4K x t 4Dt.

Momentum Equation. Necessary because body is not made up of a fixed assembly of particles Its volume is the same however Imaginary

Momentum Equation. Necessary because body is not made up of a fixed assembly of particles Its volume is the same however Imaginary Momentm Eqation Interest in the momentm eqation: Qantification of proplsion rates esign strctres for power generation esign of pipeline systems to withstand forces at bends and other places where the flow