Fractionating columns enthalpy balances

If we count the equations listed, we will find that there are 2n + 4 equations per stage. However, only 2 n + 3 of these equations are independent. These independent equations are generally taken to be the n component mass balance equations, the n equilibrium relations, the enthalpy balance, and two more equations. These two equations can be the two summation equations or the total mass balance and one of the summation equations (or an equivalent form). The 2n + 3 unknown variables determined by the equations are the n vapor mole fractions the n liquid mole fractions, the stage temperature 7 and the vapor and liquid flow rates LJ and Ly. Thus, for a column of 5 stages, we must solve s 2n + 3) equations. 

The equation system is therefore defined by the mass balance, the enthalpy balance, the phase equilibrium relationships and the summation equation for the mole fractions. It contains n m mass balances, n enthalpy balances, n m phase equilibrium relationships and 2 n summation conditions. Overall this gives n - (2/m -l- 3) equations. Since the enthalpy hj and the equilibrium constant K j are not linearly dependent on the state variables, particularly pressure and temperature, the equation system is nonlinear. In absorption columns, the number of unknown state variables equals the number of equations. In rectification columns three additional variables must be specified ex-... 

Consider the two-feed fractionator of Fig. 9.32. The construction on the Hxy diagram for the sections of the column above F, and below Fj is the same as for a single-feed column, with the and A points located in the usual manner. For the middle section between the feeds, the difference point can be located by consideration of material and enthalpy balances either toward the top, as indicated by the envelope shown on Fig. 9.32, or toward the bottom the net result will be the same. Consider the envelope shown in the figure, with A , representing a fictitious stream of quantity equal to the net flow upward and out... 

Since (H y- Htf) represents the amount of heat needed to vaporize one mole of saturated liquid under feed plate conditions, q also represents the fraction of that heat required to vaporize completely the feed as it is introduced into the column. If the feed introduced into the column is a saturated liquid at feed plate conditions, Le. Hf= H, (/ = 1. If the feed is introduced as saturated vapor at feed plate condition, then Hf= = 0. If the feed introduced is a mixture of liquid and vapor, then 0 < < 1. If the feed is introduced as a subcooled liquid, then Hf < the molar enthalpy of the saturated liquid at the temperature and pressure of the feed plate correspondingly, q >1. If the feed is introduced as a superheated vapor vis-a-vis the feed plate conditions, then Hf > H aaA q <0. Note at this time the overall material and component i balance for the column ... 

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