Mole-fraction-averaged relative volatility

A column with partial condenser and reboiler is to be used for the separation of benzene (1) from toluene (2), giving a distillate with 0.95 mole fraction benzene and a bottoms product with 0.10 mole fraction benzene. The column will operate at 105 kPa pressure and a reflux ratio of 4. The feed, at 55°C and a flow rate of 100 kmol/h containing 45 mol% benzene and 55 mol% toluene, enters the column at the fifth theoretical stage from the top. The estimated average relative volatility (benzene relative to toluene) is assumed constant, and estimated at 2.41. Based on the column conditions and thermodynamic properties, the predicted q-value is 1.2. It is required to determine the number of theoretical stages below the feed to complete the separation. 

The second main effect of decreasing the rectification pressure is the increase in average relative volatilities, which is defined as the ratio of the vapour mole fraction over the liquid mole fraction of two components ... 

Xj is the mole fraction of the more volatile component in the overhead distillate is the mole fraction of the more volatile component in the bottom products a g is the average relative volatility of the more volatile component to the less... 

In equation set 12.17, F and/ are known quantities. The relative volatilities a, are assumed constant and known for each component. At a fixed pressure, K, is a function of temperature only. The bottoms total flow rate, B, may be replaced by the sum of bi, the component fiow rates in the bottoms. There are therefore C + 2 variables in Equation 12.17 Z , (/ = 1, 2,..., C), N, and (or an average column temperature). Equation 12.17 is a set of C independent equations for i = 1, 2, C. With C+2 variables and C equations, the column has two degrees of freedom, that is, two variables must be specified in order to define the column performance. The two specifications could be any two of the above variables or functions thereof. For instance, the specifications could be N and B, or fc, and fc, the light and heavy key component fiow rates in the bottoms, or YdIi and Xg the mole fractions of the heavy key component in the distillate and the light key component in the bottoms, and so on. In each case, with two variables specified, all the other variables can be calculated from equation set 12.17. 

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