高等輸送二 — 質傳 Lecture 3 Dispersion 郭修伯 助理教授

Dispersion Caused by the coupling of concentration differences and fluid flow A form of mixing because of the flow on a microscopic level, it involves diffusion of molecules the microscopic dispersion is not understood in detail, but it takes place so rapidly that is is rarely the most important feature of the process. Compared to diffusion diffusion is caused by slow Brownian motion

Smoke pouring from a smokestack Affected by wind, weather, and different amounts of smoke Qualitatively similar to diffusion: the concentration profile is a Gaussian profile. From our previous calculation on a sharp pulse of solute: Dispersion coefficient (L2/t) Speed of wind

Dispersion coefficient Dimension [length2/time] Independent of chemistry Not a strong function of molecular weight or structure carbon monoxide, styrene and smoke are of similar value A strong function of position Similar to diffusion coefficient Unlike diffusion coefficient Unlike diffusion coefficient Dispersion Diffusion Fick’s law Heat transfer Although the mechanisms are very different, we will apply diffusion equations to study dispersion

Dispersion in Laminar Flow (Taylor Dispersion) Solute pulse injected z Steady solvent flow Assumption: the solutions are dilute the laminar flow is unchanged by the pulse (i.e., the velocity varies only with radius) mass transfer is by radial diffusion and axial convection. Other mechanisms are neglected. Radial diffusion Axial convection

A mass balance on the washer-shaped element Radial diffusion Axial convection Boundary conditions: Fick’s law Laminar flow

Taylor, 1953; Aris, 1956 One solution (1) Average across the tube (2)

An overall mass balance in terms of the average concentration: Note: the radial variations of concentration are small relative to the axial ones (i.e., ) (3) Combine (1), (2) and (3) Peclet number: relative importance of axial convection and radial diffusion

Dispersion coefficient is inversely proportional to diffusion coefficient !!

The analysis of chromatography The walls of the tube are coated with a thin film of absorbent (the stationary phase). The injected solute is retarded by absorption in that thin layer. Because the solutes are absorbed to different degrees, they are washed out (the mobile phase) by the bed at different times. Boundary conditions: A mass balance on the tube’s contents Radial diffusion Axial convection Axial diffusion

A mass balance on the absorbent Boundary conditions: Thickness of the absorbent layer Solving all these P.D.E.s together, the Golay equation gives: Axial diffusion Retardation in the absorbent layer Axial convection + radial diffusion We want to have good separation: reduce Ez v0 and R0

Dispersion in turbulent flow The mass balance in such a flowing system: By diffusion By convection By reaction In turbulent flow, we assume that both velocity and concentration fluctuate:

Dispersion coefficient: How can we find its value? Changes in reaction rate effected by the fluctuations Fick’s law: We define: Dispersion coefficient: How can we find its value? weak functions of the different chemical properties of various solute must be determined by experiments

Dispersion coefficient, E Dimension [length2/time] Our experience: We assume: (i.e., Peclet number for dispersion~ constant) roughly true from our experimental observation

Dispersion of plumes z, stack height x, wind direction y x, wind direction The dispersion coefficients in the y and z directions are commonly given as standard deviations  of the Gaussian profile: For example, a slightly unstable plume, the dispersion parallel to the ground is:

At Reynolds numbers above 10,000, Dispersion coefficients in a pipeline (containing air, water or other fluid) At Reynolds numbers above 10,000, the axial dispersion coefficient Ez is approximately: the radial dispersion coefficient Er is approximately:

Dispersion in porous media Flow through porous materials: filter cake, chromatographic column, reactor filled with solid catalyst. Dispersion coefficient in packed beds are presented as the sum of the contributions of diffusion and flow (Langer et al., 1978): Common value ~ 2.0, especially for the dispersion of gases in beds of large particles. When the particles are smaller than 0.2 cm, the value of 2 rises. The reciprocal of a tortuosity. Common value ~ 0.7 Peclet number for dispersion Peclet number for diffusion

(1) what is the dispersion coefficient at a flow of 1 cm/sec? Example: A pulse of hydrogen cyanide in water is being dispersed by flow in a 1 cm pipe. Its diffusion coefficient is 1.5x10-5 cm2/sec. (1) what is the dispersion coefficient at a flow of 1 cm/sec? (2) what is it at 1 m/sec? (1) Reynolds number Laminar flow: dispersion is caused by coupled radial diffusion and axial convection: (2) Reynolds number Turbulent flow: dispersion is caused by coupled velocity and concentration fluctuation Laminar flow > Turbulent flow > Diffusion

~ 1% will contain mixed gases Example: We have a 10 cm pipeline 3 km long for moving reagent gases at 500 cm/s from the wharf to the plant. If the pipeline is used for different gases, one after the other. How much will the gases mix? We choose a coordinate system originally located at the initial interface between the gases but moving with the average gas velocity. The mass balance around this moving point gives: ( l >> d well-mixed in radial direction) B.C. The concentration change is significant when: ~ 1% will contain mixed gases