Distillation Tutorial II: McCabe-Thiele Method of Distillation Design (Main Page)The McCabe-Thiele method of distillation design is a graphical method of design that dates back to the early 20th century. It is used as an introductory method to teach the design of distillation and is still in use today by engineers who design distillation columns. The McCabe-Thiele method considers binary distillations but can also be used for what are called pseudo-binary mixtures (i.e. mixtures that are treated as binary by identifying light and heavy key components). 1. Introductory Concepts. The main features of the McCabe-Thiele method are that it:
Binary Distillation. The easiest way to understand the McCabe-Thiele method is graphically. To do this, we begin with a slightly more detailed schematic of a distillation column than shown in Fig. 1 of Tutorial I. 2. The McCabe-Thiele Method. One of the key features of the McCabe-Thiele method (as well as other distillation design methods) is that it is assumed that vapor entering from the stage below and liquid entering from the tray above come in contact with each other and leave any given stage in equilibrium. Before presenting the equations of the McCabe-Thiele method we illustrate it graphically for a 50-50 % mixture of saturated liquid ethylene and ethane at 100 psia. We will call this column a C2 splitter. Finally, we specify the reflux ratio to be r = 3.9. We suggest that you follow each step by referring to Fig. 2 below. The Graphical Steps of the McCabe-Thiele Method
We can also begin stepping off stages in any distillation column by starting at the point xB on the 45-degree line and moving upward. It is also possible to specify a desired boil-up ratio, s, instead of a desired reflux ratio, r. Practice Exercises. Understanding the graphical nature of the McCabe-Thiele method is an important first step in understanding distillation design. Consider a 30-70 % mixture of ethane and propane at 150 psia where the feed is saturated liquid at its bubble point. Suppose the desired bottoms composition is xB = 0.05 and the desired distillate composition is xD = 0.99. Let the boil-up ratio be specified at s = 2. Answer the following:
Just press play. 3. Equations of the McCabe-Thiele Method. There are a number of important equations that come from the McCabe-Thiele method that are related to mass and energy balancing. These equations include the operating lines (i.e. rectifying and stripping lines), the feed line, the relationship between reflux ratio and boil-up ratio, and the determination of heating and cooling requirements. Mass Balance Considerations. Mass balancing is an important part of distillation column design, analysis, and simulation. Suppose we return to our C2 splitter illustration, which is fed with 100 lbmol/h of a 50-50 % mixture of ethane and ethylene, has a feed that is saturated liquid at its bubble point, and operates at 100 psia. The desired composition of the bottoms product is xB = 0.05 and the composition of the distillate is xD = 0.98. We now illustrate how mass balancing determines the rectifying and stripping lines in the McCabe-Thiele method. Operating Lines. Go back to Fig. 1 and draw an imaginary (mass balance) box that encloses the reboiler and all stages up to and including stage j in the bottom section of the column so that only the bottoms product and the streams entering or leaving stage j cross this imaginary box. This imaginary box is often called a mass balance envelope. If we now write a total mass balance for this imaginary box we get
If we also write a component mass balance for ethylene, which we will call component 1, we get
Let's solve Eq. 2.2 for yj. This gives
Now let's use Eq. 2.1 to eliminate Lj+1 from Eq. 2.3. To do this simply substitute Vj + B for Lj+1 in Eq. 2.3. This gives
If we define s, the stripping or boil-up ratio, to be s = V/B, then Eq. 2.4 can be re-written as
Finally, we can substitute s/s for 1 in the first term on the far right of Eq. 2.5 and get
which is called the stripping line. Now it is important for you to see that the slope of this line is (s+1)/s and that it intersects the 45-degree line at the point xB. First, we consider Eq. 2.6 to define yj as a function of xj+1. Thus Eq. 2.6 can be written in the familiar form y = mx + b. If you then compare Eq. 2.6 to y = mx + b, the slope is (s+1)/s. Understanding why Eq. 2.6 intersects the 45-degree line at xB is a little harder. Here you must realize that intersecting the 45-degree line at xB means yj = xj+1 = xB. So if we set yj = xB in Eq. 2.6 and do some simple algebra we should get that xj+1 = xB. Let's do that.
However, from Eq. 2.5 we know that the last term in Eq. 2.8 can also be written as [1 + 1/s]xj+1. Therefore, Eq. 2.8 can be written as
By dividing both sides of Eq. 2.9 by [1 + 1/s], we see that xj+1 = xB. Practice Exercises. To strengthen your mass balancing skills, answer the following.
Energy Considerations in the McCabe-Thiele Method. Remember we said that distillation can use large amounts of energy. This energy is supplied in the form of heat to the reboiler and cooling to the condenser. Although it's not obvious, the McCabe-Thiele method does take into account energy considerations in an approximate way through the boil-up and reflux ratios s and r respectively. The definition of s is s = V/B where B is the bottoms product flow rate and V is the flow of vapor from the reboiler. Using a total mass balance around the reboiler (see Fig. 1) we see that the total amount of liquid material entering the reboiler is
Since liquid bottoms B and vapor V both leave the reboiler, V represents the amount of entering liquid L that is vaporized by adding heat. We can express this portion in terms of the boil-up ratio s as follows. Dividing both sides of Eq. 2.10 by V we get
Rearranging Eq. 2.11 we get
which shows that V is just L multiplied by the fraction s/(s+1). Note that s/(s+1) must lie between 0 and 1. For s = 0, there is not vapor flow since Ls = 0 and this is consistent with the fact that s = V/B = 0 means V = 0. For s = 0.5, the amount of vapor is V = [0.5/1.5]L = L/3 . This says the vapor flow is one third that of the liquid flow to the reboiler when s = V/B = 0.5. Note that it also says (by mass balance) that B = 2V. Double checking these results with the overall mass balance for the reboiler shows that
so everything is fine. As s increases, the amount of vapor increases from a minimum of 0 when s = 0 to a maximum of L when s = infinity, s/(s+1) = 1, and B = 0. Now that we have a way to calculate the vapor rate V, we can also compute the heat energy needed to generate that vapor. To calculate the heat required, we need to first determine the heat of vaporization of the bottoms product. To do this, we need the heats of vaporization of both components in our mixture and we must also use a very common engineering approximation of weight averaging. Suppose we denote the heat of vaporization of ethylene in our example by l1 and the heat of vaporization of ethane by l2. The bottoms product has a composition xB, where xB is the mole fraction of ethylene since it is the light component. Thus the mole fraction of ethane in the bottom product is (1 -½ xB). Calculating the heat of vaporization of the bottoms product by weight averaging means that we use the following simple formula:
where λB is used to denote the heat of vaporization of the bottoms product. The amount of heat required in the reboiler, which we denote by QR, can now be calculated using the equation
We can also determine the amount of cooling required in the condenser in much the same way that we just determined the heating requirements for the reboiler. To do this we can simply use the weight averaging formula
where here λD is the heat of vaporization of the distillate product, xD is the composition of ethylene in the distillate product, and (1 -½ xD) is the composition of ethane in the distillate. The amount of cooling can be calculated using the formula
where V is the amount of vapor entering the condenser or the vapor from stage N-1. Note that this assumes that all of the vapor stream entering the condenser is condensed and then split into liquid reflux and liquid distillate. Practice Exercises.
Energy Balances and CMO Behavior. There is a subtle but important energy consideration that is associated with constant molar overflow. Stages Other Than Feed Stages. For any distillation stage, say stage j, without an external feed stream, CMO implies that
These equations result by drawing an imaginary energy balance envelope around any single stage in the rectifying or stripping sections of the distillation column -½ other than a feed stage -½ as shown in Fig. 3 in combination with some physical approximation.
Equations 2.17 and 2.18 can be interpreted physically by saying that CMO means that the heats of vaporization of both components are equal. What this also means is that the amount of vapor condensed at stage j is exactly balanced by the amount of liquid boiled at stage j resulting in no net change in liquid or vapor entering and leaving stage j. Now this is not necessarily a good assumption and can cause students to question the reliability of the McCabe-Thiele method. However, in many cases the CMO assumption provides a reasonable approximation. Quick Exercises. Determine how reasonable the CMO assumption is by answering the following:
Feed Stages. The energy balance considerations around any stage that has an external feed must take into account the thermal quality (or energy content) of the feed stream. By this we mean that the energy content of the feed could be any of the following:
We use the symbol q to denote the feed quality, where q = LF/F and is the fraction of feed that is liquid. Thus immediately we see that for saturated liquid, q = 1. For saturated vapor, q = 0. For a mixture of liquid and vapor feed, 0 < q < 1. For sub-cooled liquid feed, q > 1 and for super-heated vapor feed, q < 0.
Now it is important to understand that Eqs. 2.23, 2.24, and 2.25 actually apply to all five cases of feed quality. This is nice because then only one set of equations can be used to perform approximated energy balance calculations and their impact on the changes in liquid and vapor flow at any stage with an external feed. Practice Exercises. In order to better understand Eqs. 2.23, 2.24, and 2.25, answer the following:
q or the Slope of the Feed Line. To connect the previous discussions of approximate energy balance, thermal quality of the feed and the McCabe-Thiele Method, we need to show that Eq. 2.25 together with the definition of q = LF/F can be placed on a McCabe-Thiele plot and make sense. For this we need to understand that q is related to the slope of the feed line. This calculation of q is very convenient and allows us to draw the feed line properly from the intersection of a vertical line drawn from the feed composition and the 45-degree line. Mathematically, the slope of the feed line is given by the simple equation
If you think about this for a minute and go back to your calculations for q, you will see that
Quick Exercise. For the results from the last set of practice exercises, draw the feed line for saturated liquid, saturated vapor, a mixture of liquid and vapor, sub-cooled liquid, and super-heated vapor feed. Energy Balance Relationship Between Reflux Ratio and Boil-up Ratio. Using an energy balance around the entire distillation column, sometimes called an overall energy balance, it is possible to show that the reflux ratio r and the boil-up ratio s are related by the following equation
Thus the amount of vapor generated in the reboiler is tied to the amount of vapor condensed in the condenser, which makes sense. Summary Practice Exercises. Answer the following questions in order to practice what you have learned in this second tutorial by considering the separation of propylene and propane (a C3 splitter) at 200 psia. Let the feed be a 50-50 mol % mixture of saturated vapor and have a flow rate of 100 lbmol/h, xD = 0.99, and xB = 0.01. Also let the reflux ratio be r = 3.5.
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