Steady-state equation, countercurrent

For a given two-phase based countercurrent flow system, there will be appropriate boundary/initial conditions. If we neglect any axial/longitudinal dispersion/diffusion in the flowing system, these two steady state equations cUe reduced to... 

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 ... 

Example The equation yx+1 — (a + I )yr + wyx l = 0, yQ = cQ and ym+i = x 1+ i/k represents the steady-state composition of transferable material in the raffinate stream of a staged countercurrent liquid-liquid extraction system. 

To develop the performance equation, we combine the rate equation with the material balance. Thus for steady-state countercurrent operations we have for a differential element of volume... 

Many chemical and biological processes are multistage. Multistage processes include absorption towers, distillation columns, and batteries of continuous stirred tank reactors (CSTRs). These processes may be either cocurrent or countercurrent. The steady state of a multistage process is usually described by a set of linear equations that can be treated via matrices. On the other hand, the unsteady-state dynamic behavior of a multistage process is usually described by a set of ordinary differential equations that gives rise to a matrix differential equation. 

In this section, the equations are presented for the common types of contactors differential contactors and stage-wise contactors. The equations are developed for the case of steady-state, countercurrent contacting of liquid and gas with negligible heat effects, with a single-component absorption. Some discussion of extensions to other situations follows. 

In order to develop a continuous separation process, Kataoka et al. [54] simulated permeation of metal ion in continuous countercurrent column. They developed the material balance equation considering back mixing only in the continuous phase and steady-state diffusion in the dispersed emulsion drops which is similar to the Hquid extraction situation. Bart et al. [55] also modeled the extraction of copper in a continuous countercurrent column. They considered only the continuous phase back mixing in the model and assumed that the reaction between copper ions and carrier is slow, so that the differential mass balance equation for external phase in their model is... 

The total mass of the system is m, n is the total mass flux (mass flow per unit area) relative to the system boundary at any point, and S is the cross-sectional a res normal to flow at that same location. The summations extend over all the mass enuy and exit locations in the system. The mass flux at any point is equal to pv, where p is the mass density and i> is the velocity relative to the boundary at that point. Equation (2-2-1) can be applied equally well to a countercurrent gas absorption column or to a lake with input and output streams such as rainfall, evaporation, streams flowing to or from the lake, deposition of sediment on the lake bottom, or dissolution of minerals from (he sides and bottom of the lake. The steady-state version of Eq. (2.2-1) (dmldt 0) is of use in chemical process analysis because it permits calculation of various flow tales once some have been specified. 

Now, consider a one-dimensional parallel flow of two phases either in co- or countercurrent flow, exchanging mass and heat with each other. Neglecting diffusional (or dispersion) terms, in steady state the balance equations become... 

To illustrate how finite-difference equations arise, consider the countercurrent liquid-liquid extraction battery shown in Fig. 5.1. We first assume the phases are completely immiscible (e.g., water and kerosene). The heavy underflow phase has a continuous mass flow L (water, kg/sec), and the light solvent phase flows at a rate V (kerosene, kg/sec). Under steady-state operation, we wish to extract a solute Xq (e.g., acetic acid) from the heavy phase, and transfer it to the light phase (kerosene), using a nearly pure solvent with composition Since the solvent flows are constant, it is convenient in writing the solute balance to use mass ratios... 

Derive the unsteady-state mass balance equations for a countercurrent absorption column with N plates, where a single component is absorbed from the vapor phase into the liquid phase as shown in Figure P6.3. Assume ideal stages and that the liquid and gas on each plate are perfectly mixed and that the liquid and gas molar flow rate from one plate to the other are constant (dilute system). Assume linear equilibrium relations. Put the resulting equation in a matrix form. Deduce the steady-state matrix equation from the unsteady-state matrix equation. 

Generally, countercurrent flow of two immiscible phases contacting each other in one continuous contacting device (Figures 8.1.2 and 8.1.3) is carried out under steady state conditions. Therefore equations (8.1.1a) and (8.1.1b) are reduced to... 

Continuous countercurrent flow of two phases in a column is, however, normaiiy impiemented in a steady state fashion. Therefore an isothermai nondispersive equilibrium operation of a coiumn wiii iead to the foiiowing balance equation for any species i, from equation (8.1.33) ... 

For ail species moving in a given direction in a coiumn, the solution of this equation wiii indicate that, at that coiumn exit, aii such species wiii appear/exist therefore multicomponent separation is not possibie. Oniy a binary separation is possible with one species moving in the opposite direction in the column and therefore available as a pure species. This is the primary reason why we will see that a countercurrent colutnn used for steady state processes such as distillation, absorption, extraction, crystallization, etc., separates a binary mixture only. For temany mixture separation, two columns are needed. Three columns are employed to separate a four-component mixture (see Chapter 9 for various schematics). However, if a feed sample injection is made, as in elution chromatography, into a mobile phase in countercurrent flow vis-k-vis another mobile phase, transient multicomponent separation would appear to be feasible. If pulse injection of one phctse containing feed is introduced countercurrent to the other phase, it may be possible to achieve a multi-component separation capability (as is tme for cocmrent flow, considered in Section 8.2). 

The steady state conservation equations for a plug-flow reactor with countercurrent cooling are ... 

Modelling both the continuous countercurrent and simulated moving bed processes has been considered by a number of authors. The continuous countercurrent separation process has been addressed by Ching and Ruthven (1984), who assumed axially dispersed flow of fluid and counter-current plug flow of solids in a column. The fundamental differential equation describing the steady state operation of such a system is, for each component. 

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