Countercurrent differential flow with equations

Use of Operating Curve Frequently, it is not possible to assume that = 0 as in Example 2, owing to diffusional resistance in the liquid phase or to the accumulation of solute in the hquid stream. When the back pressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-3, a steady-state material balance leads to the fohowing equivalent relations ... 

Fig. 3 illustrates a packed absorption tower with countercurrent flow where flow rates are constant. Making a differential mass balance around the differential cross section in molar units results in Eq. (23). This equation can now be integrated from the top of the column down... 

Calculation of the degree of separation of a binary mixture in a membrane module for cocurrent or countercurrent flow patterns involves the numerical solution of a system of two nonlinear, coupled, ordinary differential equations (Walawender and Stem, 1972). For a given cut, the best separation is achieved with countercurrent flow, followed by crossflow, cocurrent flow, and perfect mixing, in that order. The crossflow case is considered to be a good, conservative estimate of module membrane performance (Seader and Henley, 2006). 

The assumption of plug flow is not always correct. The plug flow assumes that the convective flow (flow by velocity q/A, = v, caused by a compressor or pump) is dominating over any other transport mode. In fact, this is not always correct, and it is sometimes important to include the dispersion of mass and heat driven by concentration and temperature gradients. However, the plug flow assumption is valid for most industrial units because of the high Peclet number. We will discuss this model in some detail, not only because of its importance but also because the techniques used to handle these two-point boundary-value differential equations are similar to that used for other diffusion-reaction problems (e.g., catalyst pellets) as well as countercurrent processes and processes with recycle. The analytical analysis as well as the numerical techniques for these systems are very similar to this axial dispersion model for tubular reactors. 

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