Longitudinal Mixing Flashcards
In pipe flow the magnitude of the longitudinal dispersion coefficient varies with Reynolds number. Describe the difference in magnitude of the coefficient between laminar and highly turbulent flow conditions and describe the physical processes which combine to create the variation. (15 marks)
D(laminar) >> D(turbulent) In laminar flow, The effects of differential advection would cause the molecules to move apart. Molecules are fastest in the centre-line of the flow profile and slowest at the boundary. The effect of shear in laminar flow would continue infinitely and the distributions would grow and become infinitely more skewed. In turbulence flow, the molecules move over the complete flow regime through radial and transverse mixing counteracting the differential advection effects. A balance between the effects of differential advection and turbulent diffusion is reached. Once in the equilibrium zone, the skewness of the distribution will decrease . In laminar flow, radial mixing through molecular diffusion (10-9 - 10-10) is much less than radial mixing in turbulent flow through turbulent diffusion (10-3 - 10-1). D is inversely proportional to radial mixing and turbulent diffusion. If there is no transverse or lateral mixing within a flow, then any tracer cloud will experience the maximum longitudinal spread caused by the effects of different longitudinal velocities with the cross-section. This differential advection generates the observed shear dispersion. However, if particles are able to move throughout the cross-section, they will experience a variety of velocities, faster nearer the centre and slower nearer the boundaries. Over a period of time this will cause the average longitudinal velocity experienced by individual particles to become more similar. This tends to reduce the effect of longitudinal differential advection and hence the magnitude of the longitudinal dispersion. There is therefore an inverse relationship between the transverse mixing and the longitudinal mixing.
When performing field tracer studies to quantify the longitudinal dispersion coefficient in rivers it is often necessary to perform a quick quality control check of the data as it is collected. Briefly describe what you would look for in recorded temporal concentration distributions to assess whether the data might be appropriate for further analysis. (10 marks)
Need to obtain a ‘complete’ trace with sufficient accuracy and detail. Data analysis • From continuous in-situ (SCUFA) measurements: – Removal of background concentration – Check temporal variation in turbidity – Non-linear range of concentrations – Identify start and end of “real” trace data •The minimum concentration resolution should be 1/40th of the measured peak concentration (after background removal). Any perceived difficulties in achieving this standard should be resolved with the Project Manager prior to commencing the survey. Much greater resolutions are, however, preferred. •Tracer arrival and peak should be clearly defined. •The tail should recede at least to 10% of the peak concentration. It is expected that unless there are strong reasons for ending the monitoring early, the tail should recede to 5% of the peak and ideally to background. •The temporal resolution of the data should be constant at all stations and should provide a minimum of 40 points to define the distribution.
Control Check for coefficient
Dilution Gauging: Q = V(t)c(t)/(int.t2-t1)(c-c(o))dt Use EA chart to find coefficient
Mechanisms affecting longitudinal mixing
Reviewing the mixing mechanisms promoting longitudinal dispersion:- Molecular diffusion - the spreading due to random Brownian motion (10^-10 to 10-9 m^2/s) Turbulent diffusion - spreading due to the turbulence of the flow regime (10^-3 to 10^-1 m^2/s) Shear Dispersion - the result of depth and/ or width averaging differential advection (1 to 10^3 m^2/s) dominant process
Initial work predicting the effects of longitudinal dispersion on solutes was undertaken by G. I. Taylor (1953 & 1954) and this still forms the basis of many techniques used today to describe longitudinal dispersion. Briefly describe the major processes that are included in this analysis and the limitations when applying it to natural channels.
Early ideas on dispersion of solutes within flows arose from Taylor (1953) who initially was asked to aid the understanding of how drugs are dispersed when injected into blood vessels of animals. His paper describes how in steady laminar flow in a tube, the different longitudinal velocities throughout the cross-section create shear effects and ‘stretch’ a cloud of tracer. The paper shows that the length of an initial cloud of solute increases linearly with the square root of the distance travelled. It also shows that the longitudinal dispersion coefficient, describing the spreading along the axis of the flow, can be determined from a^2.U^2/48D where a is the pipe radius, U is the mean velocity and D is the molecular diffusivity of the solute. The result presented in this early publication, illustrates the often misunderstood concept of the inverse relationship between the longitudinal shear induced spreading and the transverse mixing, in this case molecular diffusion. It shows that when the molecular diffusivity is reduced, the longitudinal dispersion increases. This can be explained by considering the relative position of two parcels of fluid, one in the region of maximum velocity at the centre of the pipe and one in the low velocity near the boundary. If there is no diffusion, they do not move throughout the cross-section, but stay in the same cross-sectional location and move only with the longitudinal velocity. The distance between the parcels during a fixed period is simply a result of their relative velocity, i.e. the difference in their velocities, differential advection. If on the other hand they are subject to any process causing them to move throughout the cross-section, they will experience a variety of velocities. The average velocity of the parcel, previously fixed at the maximum velocity at the centre of the pipe, over the same period of time must decrease. Similarly, the mean velocity of the parcel previously located in the slow flow near the boundary, will increase. This results in a reduction in the distance between the two parcels over the time period, reducing the longitudinal dispersion. If spatial averaging is undertaken, possibly up to the channel dimensions, then shear dispersion coefficients need to be included to represent the effects caused by differential advection over the averaged distance. Depending on the integrated length scales, shear dispersion coefficients in the range 1 to 10^3 m^2/s (Smith, 1992) can be obtained.
