The Residue Curve Equation

It is important to note that the residue curve equation (Equation 2.8) is simply a mass balance. In deriving it, there were no pre-defined notions regarding how the vapor phase is related to that of the liquid. The equation is merely providing a mathematical means of determining the composition of the residual liquid phase after an amount of material has transferred to the vapor phase. Solving the residue curve equation, however, requires knowledge about how the two compositions are related to each other. Such a relationship depends on the manner in which the boiling is conducted. 

As mentioned previously, numerical integration of the residue curve equation can be done with Runge Kutta type methods. Formnately, mathematical software packages... 

In the preceding sections, the concept of a residue curve map was introduced. While these maps provide information about the separation of a particular mixture via boUing, they also yield some very interesting mathematical and topological behavior. Analysis of the residue curve equation (Equation 2.8) gives insight into the properties of these maps. 

In terms of the residue curve equation, this means algebraically solving for points where x = y. Generally speaking, these stationary points may be classified into three main types a stable node, an unstable node, and a saddle point, depicted in... 

This is the residue curve equation, since it is mathematically analogous to Equation 2.8. It can also be seen that the dimensionless time variable, in the batch set-up is replaced with an equivalent position variable in Equation 2.20 in the column. 

As discussed in Section 2.4, the residue curve equation is merely a mass balance, and can be used for any relationship between x and y. In this scenario, however, it is logical and common practice to assume that the streams emerging from a tray, or packing segment, are in equilibrium with each other. This then allows y to be defined by the appropriate thermodynamic phase equilibrium model. 

In Section 2.5.1, the separation vector was defined by the right-hand side of the residue curve equation. Examining the right-hand side of the DPE, it can be seen that two vector fields now exist. The separation vector is still present, as one would expect... 

The smaller the refluxes become, the larger the differences between L and V become and simultaneously the greater the departure from RCM conditions become. As shown in Section 3.6.1, the other extreme reflux condition, infinite reflux, results in the residue curve equation and at these conditions the TT and the MET will coincide with each other exactly. 

It is interesting to note that Equation 9.14 is mathematically similar to the residue curve equation for distillation processes (refer to Equation 2.8). As one may recall from preceding chapters, this simplification of the MDPE under total reflux conditions is comparable to the simplification of the distillation-based DPE under infinite reflux conditions. 

Equation 9.14 is the membrane residue curve equation as identified by Peters et al. [14]. It is mathematically analogous to the residue curve equation derived in Chapter 2 for distillation processes. While the residue curve equation was derived... 

Fig. 4.1. Newton s method for solving a nonlinear equation with one unknown variable. The solution, or root, is the value of x at which the residual function R(x) crosses zero. In (a), given an initial guess. vl0,), projecting the tangent to the residual curve to zero gives an improved guess v( l ). By repeating this operation (b), the iteration approaches the root.

We extract Kt and K2 from an experimental titration by constructing a difference plot with Equation 13-59. This plot is a graph of nH(measured) versus pH. We then fit the theoretical curve (Equation 13-60) to the experimental curve by the method of least squares to find the values of Kt and K2 that minimize the sum of the squares of the residuals ... 

Similarly, when a chemical reaction takes place the residue curves can be found by the equation [2] ... 

Using the transformation in Equation 2.7, the familiar residue curve equation may be obtained... 

With that said, it was traditionally believed that residue curve equation (and the resulting maps) were only suitable for equilibrium-based separations and could not be used for the representation of kinetically based processes [3]. However, the differential equations which describe a residue curve are merely a combination of mass balance equations. Because of this, the inherent nature of residue curves is such that they can be used for equilibrium- as well as nonequilibrium-based processes. 

From Equations 3.24 and 3.25, it can be seen that at a pinch point, the vectors S and M point in opposite directions to each other. Furthermore, since S is tangent to a residue curve at x, it then follows that at a pinch point, M is also tangent to the residue curve at x. This can be seen in the illustration shown in Figure 3.14. Equation 3.27 further tells us that this common tangent point, x, on the residue curve is in fact a pinch point in the CPM. 

For the remaining CSs, the first instinct one may have when specifying that all CSs operate at infinite reflux is that the operating profiles of these CSs will also merely lie on the residue curves as A = 0 (V=L). This is, however, not necessarily the case. Consider expanding the DPE shown in Chapter 3 (Equation 3.9) as follows ... 

The term 8 = (y —x ) is known as the difference vector. 8 is the difference between vapor and liquid compositions in a CS and remains constant down the length of the CS. For CSs terminated by a reboiler or cond ser (CSi and CS in the Petlyuk), Equation 5.19 reduces to the classic residue curve equation because y = x. 8 can be und stood in much the same way as X a negative value implies that a component is moving downward in the CS, and vice versa. However, unlike it is important to note that... 

Notice that Equation 8.16 is also a linear transform of the residue curve eqjuation, as the difference vector (S ) is not a function of the column s internal liquid composition (x). Hence, the mathematics describing the zero-order reactive distillation system is in fact identical to the infinite reflux Petlyuk column, even though the phenomena and equipment involved in the process are very different. 

DODS-ProPlot. This is a comprehensive profile plotting package which allows the user to plot single profiles, entire CPMs and ROMs, and their associated pinch points. There are 13 systems to choose from, each of which may be modeled either with a modified Raoult s law and the NRTL activity coefficient model, or with the ideal Raoult s law (does not model azeotropes), or with a constant relative volatility approximation where the software automatically determines the relative volatilities between components (this model also cannot account for azeotropic behavior). One also has the option to insert one s own constant volatilities. It is possible to plot the full DPE, the shortened DPE at infinite reflux (shown in Chapter 7) or the classic residue curve equation. Depending on the equation chosen, the user is free to specify any relevant parameters such as an R value, difference points, system... 

Specify the Equation. In this panel, the user is required to select an equation (i) the entire DPE, (ii) a shortened DPE with the difference vector (see Chapter 7 for use and application), or (iii) a residue curve equation. 

Figure 2.4. Phase diagram for methanol in the synthesis of MTBE expressed in terms of transformed compositions (equation 2.18). Remrirks the location of the reactive eizeotrope is tracked down at the intersection of the residue curve and the line X =Y reacting mixtures of various compositions are depicted. System features operating pressure is 11-10 Pa inert nC4 is present in the mixtme.

Combining the previous n-l ] independent expressions 5.3 with and appropriate vapor-hquid equilibrium model enables to define the set of simultaneous equations that describe the residue curves, dxi... 

The residue curves can also be calculated by numerical integration of the differential equation for open evaporation (Rayleigh equation) if the vapor-liquid equilibrium behavior is known (see Figure 11.11) [8]. Starting from the material balance of component i ... 

The residue curve map graphs the liquid composition paths that are solutions to the following set of ordinary differential equations ... 

Residue curve maps would be limited usebilness if they could only be generated experimentally. Fortunately that is not the case. The simple distillation process can be described (14) by the set of equations ... 

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