Flow and diffusion

Both diffusion and convection are modes of mass transfer. Typically, large-scale mass transfer is accomplished by convection, and small-scale mass transfer is accomplished by diffusion. Similarly, large-scale heat transfer in the Earth is through convection (mantle convection), and small-scale heat transfer is through heat conduction (e.g., through the lithosphere). To treat the complicated convection pattern and diffusion requires a large computational effort. Some simple problems can be treated analytically. 

For one-dimensional diffusion and laminar flow with constant velocity along the direction, the above diffusion-flow equation can be written as 

When we discussed moving boundary problems, we transformed the problem into boundary-fixed reference frame and converted the moving boundary to a 

Diffusion coefficients vary widely, depending on temperature, pressure, the type of the phase, and the composition of the phase. The dependence on temperature and pressure can be described well by the Arrhenius relation including a pressure term (Equation 1-88)  

In the gas phase, typical D values at room temperature and pressure are of the order 10 to 10 m /s. To the first-order approximation, D is inversely proportional to pressure and is proportional to the absolute temperature raised to the 1.5 to 1.8 power (Cussler, 1997). There is not much of an activation energy for diffusion in the gas phase. Interdiffusivity in the gas phase may be found in Cussler (1997). 

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For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

Transport of i due to the flow of the surrounding liquid travelling at a velocity v amounts to c, v so that the total flux. /, resulting from diffusion and flow comes to ... 

Feng, C. and W. E. Stewart, 1973. Practical models for isothermal diffusion and flow of gases in porous solids. Ind. Eng. Chem. Fundam. 12(2) A3-A1. 

Wakao, N., J. Otani and J. M. Smith. 1965. Significance of pressure gradients in porous materials Part I diffusion and flow in fine capillaries. A.I.Ch.E. J. 11 435-39. 

The transport of pollutant in river or in ground water both diffusion and flow... 

So much for the background. The question now is whether we can visualize the solution before going to the computer. This is particularly necessary as we have a delta function as initial condition. However, the equation looks rather like the equation for diffusion and flow in a straight tube, which, if to were the concentration of a solute, D( ) its spatially varying diffusion coefficient, and V(x) its spatially varying convective velocity, would be... 

THE GENERAL EQUATIONS OF DIFFUSION AND FLOW IN A STRAIGHT TUBE... 

The standard equations for diffusion and flow with the boundary conditions referred to above are now ... 

Qian H, Sheetz MP, Elson EL (1991) Single particle tracking. Analysis of diffusion and flow in two-dimensional systems. Biophys J 60 910-921... 

For mono-disperse pore size distributions a combination of steady state diffusion and flow permeability measurements can be used to characterize the structural parameters which enable consistent values for tortuosity to be defined. These results can be used to predict the dynamic response of a Wicke-Kallenbach cell to short pulses of a tracer gas having a comparatively high diffusivity and enable a reasonable estimate of the effective diffusion coefficient to be obtained. 

Transfer through the mobile phase is more complicated because diffusion and flow both act to shuttle molecules to and from the stationary phase (see Chapter 11). The effective coefficient for transfer is DT = Derr + D, Eq. 5.34, where Derr arises from random flow currents. Equation 9.17 assumes the form... 

In actual fact, flow and diffusion are both acting independently in only one sense to cut short the length of steps. Both terminate steps with each termination a new random step begins. Therefore, the only way in which diffusion and flow are additive is in the number of random steps they initiate... 

Because of the complex central term representing mobile phase band broadening, Eq. 12.1 has a rather awkward form, not simple to use. The awkwardness reflects the complexity of packed-bed processes, specifically the complexity caused by (i) the many types of velocity states and (ii) the competition (coupling) between diffusion and flow in controlling random displacements. 

Transport, Space, Entropy, Diffusion, and Flow Elements Underlying Separation by Electrophoresis, Chromatography, Field-Flow Fractionation and Related Methods, J. C. Giddings, /. Chromatogr., 395, 19 (1987). 

M.E. Kainourgiakis, E.S. Kikkinides and A.K. Stubos, Diffusion and flow in porous domains constructed using process-based and stochastic techniques , J. Porous Mat., in press... 

Bae J.-S. and Do D. D., Study on diffusion and flow of benzene, ra-hexane and CCI4 in activated carbon by a differential permeation method. Chem. Eng. Sci. in press (2002). 

Carman P. C., Diffusion and flow of gases and vapours through micropores I. Slip flow and molecular streaming. Proc.Roy.Soc.(London) A203 (1950) pp. 55-74. 

Do H. D. and Do D. D., A new diffusion and flow theory for activated carbon from low pressure to capillary condensation range. Chemical Engineering Journal 84 (2001) pp. 295-308. 

A. The Conditional Probability Function, Self-Diffusion, and Flow... 

J. M. D. MacElroy, Nonequilibrium Molecular Dynamics Simulation of Diffusion and Flow in Thin Microporous Membranes, J. Chem. Phys. 101... 

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