Flow in Tubes with Negligible Diffusion

Consider isothermal, laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is 

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There is no change in volume, temperature, or viscosity. The use of the simplifying assumption that there is plug flow in the tube leads to an estimated degree of conversion of 86.5%. What would be the actually attained degree of conversion if the real state of flow is laminar, with negligible diffusion ... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

As shown in the previous section, scaling with geometric similarity, Sr = Sl = gives constant pressure drop when the flow is laminar and remains laminar upon scaleup. This is true for both liquids and gases. The Reynolds number and the external area increase as. Piston flow is a poor assumption for laminar flow in anyfhing but small tubes. Conversion and selectivity of the reaction is likely to worsen upon scaleup unless the pilot reactor is already so large that molecular and thermal diffusion are negligible on the pilot scale. Ways to avoid unpleasant surprises are discussed in Chapter 8... 

This approach is different than if we were to neglect the influence of molecular diffusion as done in Section 4.10.2.4 to derive the equation for the conversion in a tube with laminar flow and negligible molecular diffusion, Eq. (4.10.29). 

The diffusion coefficient for a gas can be experimentally measured in an Arnold diffusion cell. The device is shown in Figure 3.6 consisting of a narrow tube partially filled with pure liquid A. The system is maintained under constant pressure and gas B flows across the open end of the tube. Component A vaporizes and diffuses into the gas phase, hence the rate of vaporization can be physically measured. Develop a general steady-state expression to describe the diffusion of one gas through a second stagnant gas. Assume that the gas has negligible solubility in liquid A and is also chemically inert in A. 

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