Countercurrent differential flow with

The use of differential permeation in countercurrent flow with recycle is developed in Chapter 7 and Appendix 7 of Hoffman (2003). 

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. 1.27 shows a differential type of contactor as in, say a countercurrent flow packed gas absorption column, together with the resulting approximate continuous concentration profiles. 

The steady-state mass balances for countercurrent flow can be formulated by considering changes in the component mass rates over a differential length of column. In this case, the length is measured from the top of the column (Z = 0) to the bottom, as shown in Fig. 1. The column is assumed to consist of a moist solid or liquid phase with volume Vs, and a gas phase of volume VG. The actual support volume is included in Vs. 

Although the most useful extraction process is with countercurrent flow in a multistage battery, other modes have some application. Calculations may be performed analytically or graphically. On flowsketches like those of Example 14.1 and elsewhere, a single box represents an extraction stage that may be made up of an individual mixer and separator. The performance of differential contactors such as packed or spray towers is commonly described as the height equivalent to a theoretical stage (HETS) in ft or m. 

The Podbielniak contactor is a differential type. It is constructed of several perforated concentric cylinders and is shown schematically in Figure 14.15(b). Input and removal of the phases at each section are accomplished through radial tubes. The flow is countercurrent with alternate mixing and separating occurring respectively at the perforations and between the bands. The position of the interface is controlled by the back pressure applied on the light phase outlet. 

Experiments to measure pressure drop and flooding limits were performed in a set-up accommodating monoliths with diameters of 43 mm (Fig. 8.16), while the length of the monoliths varied up to total length of 1 meter. The liquid was distributed by a nozzle the gas was introduced in countercurrent mode via mass flow controllers in the system. At the outlet of the monolith, a special device was mounted (Fig. 8.17), which improved draining of the monolith. The pressure drop along the column was measured using differential pressure transmitters. All experiments were performed at room temperature and atmospheric pressure. 

With countercurrent flow, most of the differential heat in the inlet-catalyst stream above the average reactor temperature is lost to the effluent vapors, and some of the heat brought in with the feed is immediately carried out of the reactor by the catalyst. 

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... 

At steady-state, with no chemical reaction, a mass balance on the differential section yields (for countercurrent flow)... 

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). 

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