Degrees of freedom, design

The design of a distillation column involves many parameters product compositions, product flow rates, operating pressure, total number of trays, feed-tray location, reflux ratio, reboiler heat input, condenser heat removal, column diameter, and column height. Not all of these variables are independent, so a degrees of freedom analysis is useful in pinning down exactly how many independent variables can (and must) be specified to completely define the system. 

The normal situation in distillation design is that the feed conditions are given flow rate F (mol/h), composition Zj (mole fraction component j, temperature and pressure Pp. The 

Distillation Design and Control Using Aspen Simulation, Second Edition. William L. Luyben. 2013 John Wiley Sons, Inc. PubUshed 2013 by John Wiley Sons, Inc. 

The design problem is to establish the operating pressure P, the total number of trays Nj, and the feed-tray location that produces the desired product purities. All other parameters are then fixed. Therefore, the number of design degrees of freedom is five D,HK. and Therefore, if the desired product purities and the pressure are 

Just to emphasis this point, the five variables that could be specified might be the distillate flow rate D, reflux ratio RR = R/D, P, N, and In this case, the product compositions cannot be specified but depend on the distillate flow rate and reflux ratio selected. 

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Figure 10.9 Combining reactions and separation steps leads to a loss in process design degrees of freedom.

This process has four design degrees of freedom. The four design optimization variables used are the diameter of the tubes Aube, the length of the tubes L,ube, the flowrate of the process gas fed into one tube TUlbe. and the v A/yWB ratio. 

Essential performance variables determining the performance of the device in a system are equal to the design degrees of freedom. In this case the problem is a problem of exchange of variables. 

Essential performance variables are less than the design degrees of freedom. In this case, the cost is minimized with respect to the excess degrees of freedom and the envelope of minimum costs is the costing equation. 

Given the fresh feed flow rate Fq, the fresh feed composition zq, the specific reaction rate k, and the desired product purity xg, this process has 2 design degrees of freedom i.e., setting two parameters completely specifies the system. Therefore, there are two design parameters that can be varied to find the besf plant design. Let us select reactor holdup Vr and number of trays in the stripper Mj as the design... 

The majority of distillation columns are designed to attain a specified separation between the two key components. The two steady-state design degrees of freedom are usually specified to be the impurity of the heavy-key component in the distillate and the impurity of the light-key component in the bottoms. Therefore, in the operation and control of a distillation column, the ideal control structure would measure the compositions of the two products and manipulate two input variables (e.g., reflux flow rate and reboiler heat input) to maintain the desired amounts of the key-component impurities in the two product streams. 

The normal distillation column with either a liquid or a vapor distillate product (but not both) has two steady-state design degrees of freedom once feed conditions, column pressure, total trays, and feed location are fixed. Distillate flow rate and RR are usually manipulated to achieve two product-composition specifications (the heavy-key impurity in the distillate and the light-key impurity in the bottoms). 

A distillation column that is designed to produce both a liquid distillate stream and a vapor distillate stream from the reflux drum has an additional design degree of freedom. This is usually specified to be the reflux-drum temperature. Under these conditions, the split between the flow rates of the vapor and liquid distillates is fixed. The condenser heat duty is also fixed. Specifying a reasonable design minimum temperature differential temperature (at the either the cold or the hot end of the condenser), and a reasonable overall heat-transfer coefficient fixes the heat-transfer area. This also fixes the required flow rate of... 

There are three design degrees of freedom in this partial condenser column. They are selected to be 0.1 mol% water in the liquid distillate, 0.1 mol% methanol in the bottoms, and 120 °F reflux-dmm temperature. The resulting small vapor distillate flow rate is 30.08 lb mol/h, and the larger liquid distillate flow rate is 221.9 lb mol/h. Column diameter is 3.06 ft, and the reflux flow rate is 208 lb mol/h. 

It is important to remember that we have specified four variables (the impurity of toluene in the distillate, the impurity of toluene in the bottoms, the impurity of benzene in the sidestream, and the impurity of xylene in the sidestream). To achieve these four specifications, we have varied four variables (distillate flow rate, sidestream flow rate, vapor flow rate to the prefractionator, and reboiler duty). The one remaining design degree of freedom is the flow rate of liquid to the top of the prefractionator Lp (stream 2). 

So the essence of design problems is that there are design degrees of freedom left such that design choices can be made... 

The recycle structure and rates result from using design degrees of freedom. 

Reactive distillation is also different from conventional distillation in that there are both product compositions and reaction conversion specifications. The many design degrees of freedom in a reactive distillation column must be adjusted to achieve these specifications while optimizing some objective function such as total annual cost (TAC). These design degrees of freedom include pressure, reactive tray holdup, number of reactive trays, location of reactant feedstreams, number of stripping trays, number of rectifying trays, reflux ratio, and reboiler heat input. 

There are 12 design degrees of freedom for this multiunit process. Subtracting the number of specifications and assumed heuristic relationships from the degrees of freedom gives the number of design optimization variables. 

An approximate heuristic design procedure is used for the columns in the conventional process. The optimum number of frays is assumed to be equal to 2 times the minimum, and the optimum reflux ratio is assumed to be 1.2 times the minimum. The assumptions and specifications use up 9 of the 12 design degrees of freedom, leaving 3 that can be used to optimize the process. Three design optimization variables were selected ... 

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