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Chapter 7: Binary Distillation.
Degree of Freedom Analysis
Consider the case of the distillation of a binary (two component) mixture is a multistage distillation column. The number
of degrees of freedom (DOF) is determined by the discription rule, which states that DOF = number of variables set
during construction or controlled during operation by independent means.
In particular, we have the following:
Variables set during construction:
- n (number of plates above the feed plate)
- m (number of plates below the feed plate)
Variables controlled during operation:
- z_A (Feed mole fraction)
- T_F (Feed temperature)
- P_F (Feed pressure)
- F (Feed flow rate)
- Q_F (Heat transferred in the feed preheater)
- Q_R (Heat tranferred in the reboiler)
- Q_C (Heat transferred in the overhead condenser)
- P (Column pressure)
Note that the last four variables correspond to valves in the process. There are therefore 10 degrees of freedom for this
binary systems. In general, there are N + 8 degrees of freedom where N is the number of components in the feed. Note that
this number differs by two from the value given in Separation Processes by C. J. King given in the link below since the
development in that link does not consider the feed flow rate and feed pressure as variables.
Design and Simulation Problems
Generally, the feed properties (z_A, T_F, P_F, F, Q_F) and the column pressure (P) are set in the problems under consideration here,
so there are 4 degrees of freedom left. This leads to 2 types of problems commonly considered in this webBook, each
of which considers 4 degrees of freedom as follows:
1. Design problems where the following three degrees of freedom are specified:
- Separation variable #1 (e.g., the composition of the top (distillate) product).
- Separation variable #2 (e.g., the composition of the bottom product).
- The reflux ratio (this is often set above the minimum reflux ratio by a multiplicative
design factor of ~1.5 since this often corresponds to an optimal design as discussed below).
To complete the design problem, the final (fourth) degree of freedom is used as an independent variable in a one
dimensional optimization
problem to minimize the total
number of plates in the column. Most often this fourth degree of freedom is taken as the location of the feed plate in the
column.
A more rigorous approach to a design problem would be to use both the reflux ratio and the location of the feed plate as
independent variables in a two
dimensional optimization problem to minimize the total cost of the process.
2. Simulation problems where the following is specified:
- The number of plates above the feed plate.
- The number of plates below the feed plate.
- One external flow, such as the flowrate of the top product.
- One interal flow, such as the reflux flow from the condenser (or the reflux ratio).
The Operating Line
Solving either of the two types of problems just mentioned involves solving a large number of material balance relations and
equilibrium relations. Fortunately, there is an organized method to solve these problems. One key
concept for this method is the operating line, which is defined to be a line consisting of the liquid (x) and vapor
compositions (y) that pass by each other between plates inside the column. Usually, the more volatile component is
chosen as a composition variable so the graphical construction described below always has the same general
form. An equation for the operating line can be determined by the fact that x-y points on the line, together
with the correponding liquid and vapor flow rates inside the column, satisfy a material balance with one of the
product streams. The assumption of constant molar overflow is also used, which replaces the use of an
enthalpy balance. The assumption of constant molar overflow implies that the liquid and vapor molar
flow rates inside the column do not vary from plate to plate. In practice, two points, or a point and a
slope, are used to locate an operating line, chosen from the following properties:
1. The slope of the operating line is the ratio L/V in the column section under consideration, where L and V are the liquid and vapor flow rates.
2. The intersection of the operating line with the line x = y line (45 degree line) on an x-y diagram occurs at the composition of the product from the column section under consideration.
3. The y-intercept of the operating is given by the relation NUP/V for the column section under consideration. NUP is the "net upward product", i.e., the net flow upward of the component used to make the x-y diagram counting both liquid and vapor flows.
4. The intersection of the operating lines in the upper column section and lower column section takes place on a line which intersects the 45 degree line at the feed composition and whose slope is -L_F/V_F where L_F and V_F are the flow rates of the liquid and vapor components of the feed.
Stage-to-Stage Calculation Method for a Design Problem
To solve a design problem, an overall material balance can first be used to solve for unknown properties of the external streams, such as the compositions and flow rates of the product streams. Then, a balance on total moles can be used to determine the internal total molar flow rates, i.e., the total liquid and vapor flows from each plate. Finally, the compositions on each plate and in the condenser and reboiler can be determined by starting at one end of the column and advancing to the other end and solving just one equation at a time. This is the so-called stage-to-stage method, and the graphical version of this method on a y-x diagram is the McCabe Theile method. Briefly, starting at one end of the column or the other, you alternately solve the
equilibrium relation for streams leaving a particular plate (by moving either horizontally or vertically, as appropriate, to the equilibrium line from the operating line) and then the material balance relation for streams passing each other between plates (by moving either horizontally or vertically, as appropriate, to the operating line from the equilibrium line).
Optimal Feed Plate Location
The optimal feed plate is determined by switching from the operating line for one section of the column (e.g., the lower section below the feed plate) to the operating line for the other section of the column (e.g., the upper section above the feed plate) in the vacinity of the intersection of the operating lines so that the graphical procedure accomplishes the largest change in composition with the fewest number of stages.
Optimal Reflux Ratio
There is a generally a tradeoff between capital costs (the cost of constructing the column) and operating costs (the cost of energy or steam) in distillation. High capital costs correspond to a large number of plates in the column, and therefore low reflux ratios and low energy costs. High energy costs (and therefore high reflux ratios) correspond to a small number of plates (and therefore also low capital costs). A good "rule of thumb" is that an optimal design that minimizes the total cost is where the operating reflux ratio is 1.25 times the minimum possible reflux ratio. An error tolerant design
results if the number of plates in the column is chosen using a reflux ratio equal to 1.25 times the minimum reflux ratio while
the column diameter is selected so it can accommodate, if necessary, a reflux ratio that is about 2 times the minimum reflux ratio. The resulting column has
an oversized diameter so the reflux ratio can be increased from the theoretical optimal value to account for design errors.
Stage-to-Stage Method for a Simulation Problem
A strategy similar to that just described above can be used to solve simulation problems.
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