Underwoods plate number

For an overview approximation and dimensioning, the relationships of Fenske and Underwood can be used to determine the minimum plate number N j and the minimum reflux ratio (Equations 2.3.2-26 and 2.3.2-27) ... 

The Underwood and Fenske equations may be used to find the minimum number of plates and the minimum reflux ratio for a binary system. For a multicomponent system nm may be found by using the two key components in place of the binary system and the relative volatility between those components in equation 11.56 enables the minimum reflux ratio Rm to be found. Using the feed and top compositions of component A ... 

The minimum reflux ratio, Rm is calculated using Underwood s method (Example 11.16) as 0.83 and, using Fenske s method, Example 11.17, the number of plates at total reflux is nm = 8. The following data have been taken from Figure 11.42, attributable to Gilliland(30) ... 

For single separation duty, Diwekar et al. (1989) considered the multiperiod optimisation problem and for each individual mixture selected the column size (number of plates) and the optimal amounts of each fraction by maximising a profit function, with a predefined conventional reflux policy. For multicomponent mixtures, both single and multiple product options were considered. The authors used a simple model with the assumptions of equimolal overflow, constant relative volatility and negligible column holdup, then applied an extended shortcut method commonly used for continuous distillation and based on the assumption that the batch distillation column can be considered as a continuous column with changing feed (see Type II model in Chapter 4). In other words, the bottom product of one time step forms the feed of the next time step. The pseudo-continuous distillation model thus obtained was then solved using a modified Fenske-Underwood-Gilliland method (see Type II model in Chapter 4) with no plate-to-plate calculations. The... 

In this approach, Fenske s equation [Ind. Eng. Chem., 24, 482 (1932)] is used to calculate which is the number of plates required to make a specified separation at total reflux, i.e., the minimum value of N. Underwood s equations [/. Inst. Pet., 31, 111 (1945) 32,598 (1946) ... 

Underwood s method can be extended to calculate the number of plates analytically when the equal molal overflow assumption and constant relativity are appropriate. But even then, the calculations are sufficiently involved that a... 

The spreadsheet will automatically calculate flow rates throughout the column, the number of theoretical plates, the feed plate position, the MR ratio (from the Underwood equation), and the minimum number of theoretical plates (from the Fenske equation). (Microsoft Office Excel, 2007). 

Occasionally there is a need to perform some preliminary but rapid estimates for a specific separation without resorting to the tedious graphical or plate by plate calculations. In such instances one can turn to some of the short-cut methods that have been developed specifically for multicomponent separations in the chemical process industry but which also work reasonably well with binary and multicomponent separations at low temperatures. These are the Fenske-Underwood method for obtaining the minimum number of plates at total reflux, the Underwood method for obtaining the minimum reflux, and the Gilliland correlation to determine the theoretical number of plates based on the information provided by the two prior methods. 

The basic assumption of the Fenske-Underwood relation is that the ratio of the equilibrium constants or the relative volatility, as defined by Eq. (6.19), in a binary mixture or the two key components present in a multicomponent mixture remain constant over the temperatures encountered in the distillation column. If this can be assumed without the introduction of excessive error, the minimum number of plates at total reflux can be determined from... 

Example 6.11. Determine the minimum number of theoretical plates required at total reflux for the nitrogen-oxygen separation specified in Example 6.8 using the short-cut method of Fenske and Underwood. Determine the minimum reflux by the Underwood method. Use the information from these two methods to determine the number of theoretical plates required for an actual reflux ratio of 2.77 with the aid of the Gilliland correlation. 

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