ENCH 445: Lecture 7 -- Binary Distillation
Consider the case of the distillation of a binary (two component) mixture is a multistage distillation column. The degrees of freedom is determined by the discription rule (i.e., D.O.F. = 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:
zA (Feed mole fraction)
TF (Feed temperature)
F (Feed flow rate)
QF (Heat transferred in the feed preheater)
QR (Heat tranferred in the reboiler)
QC (Heat transferred in the overhead condenser)
P (Column pressure)
Note that the last four variables correspond to valves in the process. There are therefore 9 degrees of freedom for this binary systems. In general, there are N + 7 degrees of freedom where N is the number of components in the feed.
Generally, the feed properties (zA, TF, F, QF) and the column pressure (P) are set in the problems under consideration in this course, so there are 4 degrees of freedom left.. This leads to 2 types of problems, each of which specifies 4 degrees of freedom:
1. Design problems where the following is 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 fact that the optimal feed plate is used that minimizes the total number of plates used.
The reflux ratio.
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).
Solving either of these two types of problems involves solving a large number of material balance relations and equilibrium relations, in fact, one relation of each type on each plate. Fortunately, there is an organized method to solve these problems. One key concept of this method is the operating line, which is defined to be a line consisting of liquid (x) and vapor compositions (y) that pass 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 molal overflow is also used, which replaces the use of an enthalpy balance. This assumption of constant molal 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 -LF/VF where LF and VF are the flow rates of the liquid and vapor components of the feed.
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).
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.
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 cose is where the operating reflux ratio is 1.5 times the minimum possible reflux ratio.
A strategy similar to that just descri bed above can be used to solve simulation problems.
Several tutorials on this course website explain the details of the McCabe Theile method, one of which is at the website http://lorien.ncl.ac.uk/ming/distil/distil0.htm.