Residue curve equation

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

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. 

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. 

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. 

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... 

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... 

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. 

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