passing through the point (xf,xf). The values of x and y required must satisfy, not only the equation, but also the appropriate equilibrium data. Thus these values may be determined graphically using an x — y diagram as shown in Figure 11.9.
In practice, the quantity vaporised is not fixed directly but it depends upon the enthalpy of the hot incoming feed and the enthalpies of the vapour and liquid leaving the separator. For a given feed condition, the fraction vaporised may be increased by lowering the pressure in the separator.
An equimolar mixture of benzene and toluene is subjected to flash distillation at 100 kN/m2 in the separator. Using the equilibrium data given in Figure 11.9, determine the composition of the liquid and vapour leaving the separator when the feed is 25 per cent vaporised. For this condition, the boiling point diagram in Figure 11.10 may be used to determine the temperature of the exit liquid stream.
Solution
The fractional vaporisation = V/F = f (say) The slope of equation 11.31 is:
When f = 0.25, the slope of equation 11.31 is therefore:
and the construction is made as shown in Figure 11.9 to give x = 0.42 and y = 0.63.
From the boiling point diagram, in Figure 11.10 the liquid temperature when x = 0.42 is seen to be 366.5 K.
11.3.3. Rectification in the two processes considered, the vapour leaving the still at any time is in equilibrium with the liquid remaining, and normally there will be only a small increase in concentration of the more volatile component. The essential merit of rectification is that it enables a vapour to be obtained that is substantially richer in the more volatile component than is the liquid left in the still. This is achieved by an arrangement known as a fractionating column which enables successive vaporisation and condensation to be accomplished in one unit. Detailed consideration of this process is given in Section 11.4.
In batch distillation, which is considered in detail in Section 11.6, the more volatile component is evaporated from the still which therefore becomes progressively richer in the less volatile constituent. Distillation is continued, either until the residue of the still contains a material with an acceptably low content of the volatile material, or until the distillate is no longer sufficiently pure in respect of the volatile content.
11.4. THE FRACTIONATING COLUMN 11.4.1. The fractionating process
The operation of a typical fractionating column may be followed by reference to Figure 11.11. The column consists of a cylindrical structure divided into sections by
a series of perforated trays which permit the upward flow of vapour. The liquid reflux flows across each tray, over a weir and down a downcomer to the tray below. The vapour rising from the top tray passes to a condenser and then through an accumulator or reflux drum and a reflux divider, where part is withdrawn as the overhead product D, and the remainder is returned to the top tray as reflux R.
The liquid in the base of the column is frequently heated, either by condensing steam or by a hot oil stream, and the vapour rises through the perforations to the bottom tray. A more commonly used arrangement with an external reboiler is shown in Figure 11.11 where the liquid from the still passes into the reboiler where it flows over the tubes and weir and leaves as the bottom product by way of a bottoms cooler, which preheats the incoming feed. The vapour generated in the reboiler is returned to the bottom of the column with a composition ys, and enters the bottom tray where it is partially condensed and then revaporised to give vapour of composition y1. This operation of partial condensation of the rising vapour and partial vaporisation of the reflux liquid is repeated on each tray. Vapour of composition yt from the top tray is condensed to give the top product D and the reflux R, both of the same composition yt. The feed stream is introduced on some intermediate tray where the liquid has approximately the same composition as the feed. The part of the column above the feed point is known as the rectifying section and the lower portion is known as the stripping section. The vapour rising from an ideal tray will be in equilibrium with the liquid leaving, although in practice a smaller degree of enrichment will occur.
In analysing the operation on each tray it is important to note that the vapour rising to it, and the reflux flowing down to it, are not in equilibrium, and adequate rates of mass and heat transfer are essential for the proper functioning of the tray.
The tray as described is known as a sieve tray and it has perforations of up to about 12 mm diameter, although there are several alternative arrangements for promoting mass transfer on the tray, such as valve units, bubble caps and other devices described in Section 11.10.1. In all cases the aim is to promote good mixing of vapour and liquid with a low drop in pressure across the tray.
On each tray the system tends to reach equilibrium because:
(a) Some of the less volatile component condenses from the rising vapour into the liquid thus increasing the concentration of the more volatile component (MVC) in the vapour.
(b) Some of the MVC is vaporised from the liquid on the tray thus decreasing the concentration of the MVC in the liquid.
The number of molecules passing in each direction from vapour to liquid and in reverse is approximately the same since the heat given out by one mole of the vapour on condensing is approximately equal to the heat required to vaporise one mole of the liquid. The problem is thus one of equimolecular counterdiffusion, described in Volume 1, Chapter 10. If the molar heats of vaporisation are approximately constant, the flows of liquid and vapour in each part of the column will not vary from tray to tray. This is the concept of constant molar overflow which is discussed under the heat balance heading in Section 11.4.2. Conditions of varying molar overflow, arising from unequal molar latent heats of the components, are discussed in Section 11.5.
In the arrangement discussed, the feed is introduced continuously to the column and two product streams are obtained, one at the top much richer than the feed in the MVC and the second from the base of the column weaker in the MVC. For the separation of small quantities of mixtures, a batch still may be used. Here the column rises directly from a large drum which acts as the still and reboiler and holds the charge of feed. The trays in the column form a rectifying column and distillation is continued until it is no longer possible to obtain the desired product quality from the column. The concentration of the MVC steadily falls in the liquid remaining in the still so that enrichment to the desired level of the MVC is not possible. This problem is discussed in more detail in Section 11.6.
A complete unit will normally consist of a feed tank, a feed heater, a column with boiler, a condenser, an arrangement for returning part of the condensed liquid as reflux, and coolers to cool the two products before passing them to storage. The reflux liquor may be allowed to flow back by gravity to the top plate of the column or, as in larger units, it is run back to a drum from which it is pumped to the top of the column. The control of the reflux on very small units is conveniently effected by hand-operated valves, and with the larger units by adjusting the delivery from a pump. In many cases the reflux is divided by means of an electromagnetically operated device which diverts the top product either to the product line or to the reflux line for controlled time intervals.
11.4.2. Number of plates required in a distillation column
In order to develop a method for the design of distillation units to give the desired fraction-ation, it is necessary, in the first instance, to develop an analytical approach which enables the necessary number of trays to be calculated. First the heat and material flows over the trays, the condenser, and the reboiler must be established. Thermodynamic data are required to establish how much mass transfer is needed to establish equilibrium between the streams leaving each tray. The required diameter of the column will be dictated by the necessity to accommodate the desired flowrates, to operate within the available drop in pressure, while at the same time effecting the desired degree of mixing of the streams on each tray.
Four streams are involved in the transfer of heat and material across a plate, as shown in Figure 11.12 in which plate n receives liquid Ln+1 from plate n + 1 above, and vapour Vn-1 from plate n - 1 below. Plate n supplies liquid Ln to plate n - 1, and vapour Vn to plate n + 1.
The action of the plate is to bring about mixing so that the vapour Vn, of composition yn, approaches equilibrium with the liquid Ln, of composition xn. The streams Ln+1 and Vn-1 cannot be in equilibrium and, during the interchange process on the plate, some of the more volatile component is vaporised from the liquid Ln+1, decreasing its concentration to xn, and some of the less volatile component is condensed from Vn-1, increasing the vapour concentration to yn. The heat required to vaporise the more volatile component from the liquid is supplied by partial condensation of the vapour Vn-1. Thus the resulting effect is that the more volatile component is passed from the liquid running down the column to the vapour rising up, whilst the less volatile component is transferred in the opposite direction.
Heat balance over a plate
A heat balance across plate n may be written as:
Ln+1 HL+1 + Vn—1 HnV—1 = Vn HV + Ln HL + losses + heat of mixing (11.32)
where: HnL is the enthalpy per mole of the liquid on plate n, and
HnV is the enthalpy per mole of the vapour rising from plate n.
This equation is difficult to handle for the majority of mixtures, and some simplifying assumptions are usually made. Thus, with good lagging, the heat losses will be small and may be neglected, and for an ideal system the heat of mixing is zero. For such mixtures, the molar heat of vaporisation may be taken as constant and independent of the composition. Thus, one mole of vapour Vn—1 on condensing releases sufficient heat to liberate one mole of vapour Vn. It follows that Vn = Vn—1, so that the molar vapour flow is constant up the column unless material enters or is withdrawn from the section. The temperature change from one plate to the next will be small, and HnL may be taken as equal to HnL+1. Applying these simplifications to equation 11.32, it is seen that Ln = Ln+1, so that the moles of liquid reflux are also constant in this section of the column. Thus Vn and Ln are constant over the rectifying section, and Vm and Lm are constant over the stripping section.
For these conditions there are two basic methods for determining the number of plates required. The first is due to Sorel(25) and later modified by Lewis(26), and the second is due to McCabe and Thiele(27). The Lewis method is used here for binary systems, and also in Section 11.7.4 for calculations involving multicomponent mixtures. This method is also the basis of modern computerised methods. The McCabe-Thiele method is particularly important since it introduces the idea of the operating line which is an important common concept in multistage operations. The best assessment of these methods and their various applications is given by Underwood(28) .
When the molar heat of vaporisation varies appreciably and the heat of mixing is no longer negligible, these methods have to be modified, and alternative techniques are discussed in Section 11.5.
Calculation of number of plates using the Lewis-Sorel method
If a unit is operating as shown in Figure 11.13, so that a binary feed F is distilled to give a top product D and a bottom product W, with xf, xd, and xw as the corresponding mole fractions of the more volatile component, and the vapour Vt rising from the top plate is condensed, and part is run back as liquid at its boiling point to the column as reflux, the remainder being withdrawn as product, then a material balance above plate n, indicated by the loop I in Figure 11.13 gives:
Figure 11.13. Material balances at top and bottom of column Expressing this balance for the more volatile component gives:
This equation relates the composition of the vapour rising to the plate to the composition of the liquid on any plate above the feed plate. Since the molar liquid overflow is constant, Ln may be replaced by Ln+1 and:
Ln D
Similarly, taking a material balance for the total streams and for the more volatile component from the bottom to above plate m, as indicated by the loop II in Figure 11.13, and noting that Lm = Lm+1 gives:
and: Thus:
ym Vm ym
Lmxm Lm
This equation, which is similar to equation 11.35, gives the corresponding relation between the compositions of the vapour rising to a plate and the liquid on the plate, for the section below the feed plate. These two equations are the equations of the operating lines.
In order to calculate the change in composition from one plate to the next, the equilibrium data are used to find the composition of the vapour above the liquid, and the enrichment line to calculate the composition of the liquid on the next plate. This method may then be repeated up the column, using equation 11.37 for sections below the feed point, and equation 11.35 for sections above the feed point.
A mixture of benzene and toluene containing 40 mole per cent benzene is to be separated to give a product containing 90 mole per cent benzene at the top, and a bottom product containing not more than 10 mole per cent benzene. The feed enters the column at its boiling point, and the vapour leaving the column which is condensed but not cooled, provides reflux and product. It is proposed to operate the unit with a reflux ratio of 3 kmol/kmol product. It is required to find the number of theoretical plates needed and the position of entry for the feed. The equilibrium diagram at 100 kN/m2 is shown in Figure 11.14.
Solution
For 100 kmol of feed, an overall mass balance gives:
A balance on the MVC, benzene, gives:
(100 x 0.4) = 0.9 D + 0.1 W Thus: 40 = 0.9(100 - W) + 0.1 W
Using the notation of Figure 11.13 then:
Figure 11.14. Calculation of the number of plates by the Lewis-Sorel method for Example 11.7
Mole fraction C6H6in liquid (x)
Figure 11.14. Calculation of the number of plates by the Lewis-Sorel method for Example 11.7
Thus, the top operating line from equation 11.35 is:
Since the feed is all liquid at its boiling point, this will all run down as increased reflux to the plate below.
With the two equations (i) and (ii) and the equilibrium curve, the composition on the various plates may be calculated by working either from the still up to the condenser, or in the reverse direction. Since all the vapour from the column is condensed, the composition of the vapour yt from the top plate must equal that of the product xd, and that of the liquid returned as reflux xr. The composition xt of the liquid on the top plate is found from the equilibrium curve and, since it is in equilibrium with vapour of composition, yt = 0.90, xt = 0.79.
The value of yt_i is obtained from equation (i) as:
yt_1 = (0.75 x 0.79) + 0.225 = (0.593 + 0.225) = 0.818
xt—1 is obtained from the equilibrium curve as 0.644 yt_2 = (0.75 x 0.644) + 0.225 = (0.483 + 0.225) = 0.708
xt _2 from equilibrium curve = 0.492 yt_3 = (0.75 x 0.492) + 0.225 = (0.369 + 0.225) = 0.594 xt_3 from the equilibrium curve = 0.382
This last value of composition is sufficiently near to that of the feed for the feed to be introduced on plate (t _ 3). For the lower part of the column, the operating line equation (ii) will be used.
Thus: yt_4 = (1.415 x 0.382) _ 0.042 = (0.540 _ 0.042) = 0.498
xt_4 from the equilibrium curve = 0.298 yt_5 = (1.415 x 0.298) _ 0.042 = (0.421 _ 0.042) = 0.379
xt_5 from the equilibrium curve = 0.208 yt_6 = (1.415 x 0.208) _ 0.042 = (0.294 _ 0.042) = 0.252
xt_6 from the equilibrium curve = 0.120 yt_7 = (1.415 x 0.120) _ 0.042 = (0.169 _ 0.042) = 0.127 xt_7 from the equilibrium curve = 0.048
This liquid xt _7 is slightly weaker than the minimum required and it may be withdrawn as the bottom product. Thus, xt _7 will correspond to the reboiler, and there will be seven plates in the column.
The simplifying assumptions of constant molar heat of vaporisation, no heat losses, and no heat of mixing, lead to a constant molar vapour flow and a constant molar reflux flow in any section of the column, that is Vn = Vn+1, Ln = Ln+1, and so on. Using these simplifications, the two enrichment equations are obtained:
Ln D
Lm W
These equations are used in the Lewis-Sorel method to calculate the relation between the composition of the liquid on a plate and the composition of the vapour rising to that plate. McCabe and Thiele(27) pointed out that, since these equations represent straight lines connecting yn with xn+1 and ym with xm+1, they can be drawn on the same diagram as the equilibrium curve to give a simple graphical solution for the number of stages required. Thus, the line of equation 11.35 will pass through the points 2, 4 and 6 shown in Figure 11.14, and similarly the line of equation 11.37 will pass through points 8, 10, 12 and 14.
Ln D
and this equation represents a line passing through the point = xn+1 = . If xn+1 is put equal to zero, then = Dxd/Vn, giving a second easily determined point. The top operating line is therefore drawn through two points of coordinates and
For the bottom operating line, equation 11.30, if xm+1 = , then:
Vm Vm
Since Vm = Lm — W, it follows that ym = . Thus the bottom operating line passes through the point C, that is (xw, xw), and has a slope Lm/ Vm. When the two operating lines have been drawn in, the number of stages required may be found by drawing steps between the operating line and the equilibrium curve starting from point A.
This method is one of the most important concepts in chemical engineering and is an invaluable tool for the solution of distillation problems. The assumption of constant molar overflow is not limiting since in very few systems do the molar heats of vaporisation differ by more than 10 per cent. The method does have limitations, however, and should not be employed when the relative volatility is less than 1.3 or greater than 5, when the reflux ratio is less than 1.1 times the minimum, or when more than twenty-five theoretical trays are required(13). In these circumstances, the Ponchon-Savarit method described in Section 11.5 should be used.
Example 11.7 is now worked using this method. Thus, with a feed composition, Xf = 0.4, the top composition, xd is to have a value of 0.9 and the bottom composition, xw is to be 0.10. The reflux ratio, Ln/D = 3.
Solution a) From a material balance for a feed of 100 kmol:
Vn = Vm = 150; Ln = 112.5; Lm = 212.5; D = 37.5 and W = 62.5 kmol b) The equilibrium curve and the diagonal line are drawn in as shown in Figure 11.15.
c) The equation of the top operating line is:
Thus, the line AB is drawn through the two points A (0.9, 0.9) and B (0, 0.225).
d) The equation of the bottom operating line is:
This equation is represented by the line CD drawn through C (0.1, 0.1) at a slope of 1.415.
e) Starting at point A, the horizontal line is drawn to cut the equilibrium line at point 1. The vertical line is dropped through 1 to the operating line at point 2 and this procedure is repeated to obtain points 3-6.
f) A horizontal line is drawn through point 6 to cut the equilibrium line at point 7 and a vertical line is drawn through point 7 to the lower enrichment line at point 8. This procedure is repeated in order to obtain points 9-16.
g) The number of stages are then counted, that is points 2, 4, 6, 8, 10, 12, and 14 which gives the number of plates required as 7.
Point 16 in Figure 11.15 represents the concentration of the liquor in the still. The concentration of the vapour is represented by point 15, so that the enrichment represented by the increment 16-15 is achieved in the boiler or still body. Again, the concentration on the top plate is given by point 2, but the vapour from this plate has a concentration given by point 1, and the condenser by completely condensing the vapour gives a product of equal concentration, represented by point A. The still and condenser together, therefore, provide enrichment (16 — 15) + (1 — A), which is equivalent to one ideal stage. Thus, the actual number of theoretical plates required is one less than the number of stages shown on the diagram. From a liquid in the still, point 16 to the product, point A, there are eight steps, although the column need only contain seven theoretical plates.
Figure 11.15. Determination of number of plates by the McCabe-Thiele method (Example 11.8)
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