Rigorous Variable-Pressure Model

For vacuum columns, where both absolute pressure and tray pressure drops vary significantly, a rigorous vapor-hydraulic model may have to be used. The modeling and simulation are easy. The numerical integration is quite difficult. This is because the ODEs become very, very stilT when vapor hydraulics are included in the model. 

Knowing and h we can go into the physical property data and calculate the temperature T . Note that this is the reverse of the normal procedure where we calculate enthalpy from known temperature and known composition. 

Now using temperature and liquid compositions, we can do a bubblepoint calculation to determine the pressure on the tray P and the vapor composition y . Note that this bubblepoint calculation is usually not iterative since we know the temperature. 

Finally we can now calculate the vapor flow rate through the tray from the pressure drop through the tray (P i - P ) and the liquid height on the tray, which we can get from the weir height fi and the height of liquid over the weir 

The total pressure drop is the sum of the dry bole pressure drop plus the 

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Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

This procedure of lumping all non-idealities into a few adjustable parameters is unsatisfactory for many reasons. Thermodynamic rigor is lost if experimentally determined dissociation constants or vapor pressures are disregarded. Also the parameters determined in this way are accurate only over the range of variables fitted and usually the model cannot be used for extrapolation to other conditions. The attractive feature of these models in the past was their need for little input information and the simple equations could often be solved algebraically. 

Forty years ago these computed variables were calculated using pneumatic devices. Today they are much more easily done in the digital control computer. Much more complex types of computed variables can now be calculated. Several variables of a process can be measured, and all the other variables can be calculated from a rigorous model of the process. For example, the nearness to flooding in distillation columns can be calculated from heat input, feed flow rate, and temperature and pressure data. Another application is the calculation of product purities in a distillation column from measurements of several tray temperatures and flow rates by the use of mass and energy balances, physical property data, and vapor-liquid equilibrium information. Successful applications have been reported in the control of polymerization reactors. 

Figure 3.23 represents the pressure, temperature and internal flow profiles obtained from the results of the simplified model. These profiles are used as initial estimates to enhance the convergence of the rigorous VDU model. We use all major liquid products except for VR, most of the circulation rates and temperature changes of pumparound streams, and flash zone temperature to specify the column model as shown in Figure 3.24. Figures 3.25 to 3.27 show the predictions of the rigorous VDU model for column temperature profile, D1160 curve of VGO and product yields. The results demonstrate that the two-step approach of model development generates accurate predictions on key operation and production variables of VDU.

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