I
ALFRED J. SUROWIEC
78 Midland Blvd., Maplewood N. .I,
Relating Theoretical Plates and Transfer Units Convert theoretical plates into transfer units within experimental accuracy by this one-step method
THE
differential process of the packed column and the single contact concept of the equilibrium plate have always presented the designer with a problem when an attempt is made to convert or express one process in terms of the other. The transfer unit of the packed column and the equilibrium plate of the bubble tower are not the same, except under very unlikely conditions. This study presents a one-step computation technique for converting theoretical equilibrium plates into transfer units by adding a correction term which depends only upon terminal concentrations. I n a distillation process, for example, the determination of the number of equilibrium plates follows readily from the equations for the operating line and the equilibrium curve. For actual equilibrium plates, the compositions of vapor and liquid leaving a plate, y n and x,, are defined only for integral values of n, the plate number. However, very few actual plates are equilibrium plates, and it is necessary to employ theoretical equilibrium plates. Since actual plates are integral in number, theoretical plates must permit numbers other than integers. For theoretical plates, the compositions y n and xn are continuous functions of n, since the counting of theoretical equilibrium plates can be started at any point on the operating line even though the counting process itself is restricted at present by increments of unity. The functional dependence upon n is still indicated by a subscript, though n is not now a true index of demarcation. In this study, x,, and yn are assumed to have derivatives of all orders with respect to n, except perhaps at a finite number of points where feed is introduced, sidestreams are withdrawn, or liquid and vapor are interchanged. At these points, one-sided derivatives are assumed to exist. Existence of derivatives with respect to n permits expansion of y. about yn-cl by the Taylor series, in terms of the derivatives at n 1. And, x , + ~ can be expanded about x, in terms of the derivatives at n. Equations for the number of transfer units are obtained by expressing the
+
composition change across a theoretical equilibrium plate in terms of a Taylor series. Neglecting third order and higher terms in the expansion yields a second order approximation with a maximum error less than 4% when the ratio of composition changes across adjacent trays lies between 0.5 and 2. A more accurate result is obtained by using the second order approximation to evaluate third and fourth order terms in the Taylor expansion. This procedure yields satisfactory results, since terms of only a few per cent in magnitude are evaluated with a n error of the same magnitude.
(4b)
By definition, the number of transfer units in over-all liquid and vapor concentrations is given by (7) :
Integration of Equation 4 then yields:
Derivation of Equations Formal application of the Taylor series to the composition change across a theoretical plate yields:
A finite difference is usually taken as a first approximation for a derivative. Correspondingly, a derivative of a finite difference can be taken as an approximation for a second derivative. Differentiation of Equation 1 gives:
Equation 6 is the desired relation for converting theoretical plates into overall transfer units. The equations were originally developed from theoretical considerations and are subsequently shown to have an accuracy of 4% when the ratio of composition changes across adjacent trays lies between 0.5 and 2. The height of a transfer unit, H T U , and the height equivalent to a theoretical plate, HETP, are simply related for a packed tower, since : n(HETP) = NOGHTUOG = NQLHTUOL (7)
The number of transfer units can be eliminated from the above expression with the aid of Equation 6 :
These relations may be combined to eliminate a second order derivative :
L
J
n
(84
L
J
n
(8b)
Equation 8 is useful when calculations are made in terms of theoretical plates, while experimental data are in the form of HTU. Equations 6 and 8 apply separately to enriching or stripping.
Neglecting third order and higher terms in Equation 3 yields a second order approximation which is satisfactory for most purposes. Rearrangement gives:
Evaluation of Remainder Terms The third order and higher terms in Eqi ation 3a can be evaluated by writing Equation 4a with a remainder term:
= dR, VOL. 53, NO. 4
APRIL 1961
(9)
289
rCalculation of NOO,NOL, and HETP The following procedure may be used t o arrive a t the number of over-all transfer units from the number of theoretical equilibrium plates:
b
retical step from the e n d point of the operating line. Calculate the ratio of this composition difference to that a t the terminal point of the operating line. If the ratios computed a t each e n d of the operating line lie between 0.5 a n d 2, then the calculated number of transfer units will have a n error less t h a n 4%.
Calculate the n u m b e r of theoretical plates in the customary fashion over a composition change for which the operating line remains unchanged, though not necessarily straight.
b
D e t e r m i n e the extent of d e p a r t u r e from equilibrium, in liquid o r vapor compositions, a t the terminal points of the operating line. Calculate the logarithm of the ratio of these composition differences, as shown in Equation 6a or 6b, to arrive a t the n u m b e r of over-all transfer units. Concentrations other t h a n mole fractions m a y be used, provided the defining integral for the over-all transfer unit in Equation 5 has meaning in these units of measure.
b
A r r i v e a t a “true” value for the number of transfer units by dividing the logarithm of the ratio computed in the second step by the number of theoretical plates, for use as the quantity In r, in Equation 42, a n d calculate a n average value of the factor C,. Multiply the actual number of theoretical plates by the average value of C,, a n d substitute the corrected number of theoretical plates into Equation 6 to arrive a t the “true” n u m b e r of transfer units.
b
Estimate the accuracy of the result by determining the extent of d e p a r t u r e from equilibrium one theo-
The height of a theoretical plate may readily be found from the height of the transfer unit, either in gas o r liquid concentrations, a s follows:
b
T h e ratio of the “ t r u e ” n u m b e r of transfer units to the actual n u m b e r of theoretical plates, computed above, is identically equal to the ratio of the height of the theoretical plate to that of the transfer unit in Equation 8.
those cases where the ratio of the slope of the operating line to that of the equilibrium curve decreases with increasing plate number. W h e n these slopes are constant, r , is constant a n d the exact relations, Equations 43 a n d 44, should be used. W h e n r , varies, a more accurate result m a y be gained by evaluating the q u a n tity A R , in Equation 27 a n d adding it to the n u m b e r of theoretical plates before substituting in Equation 6.
b
Results calculated by this procedure will be within experimental accuracy for most cases. T h e factor C, calculated above from Equation 42 m a y b e used in
dn - (dn)R=o = dR, (10) I n Equation 10 we distinguish between the actual number of theoretical plates and the number obtained from Equation 4a when the remainder term in Equation 3 is zero. In the enriching section of a distillation tower, dyn+ and d(yn - J. + 1) are negative for positive dn so that Equation 9 becomes : so that
dn
-
IdYn+1l - YnLli
IYn
‘/z
1
(dYJL
lYn
-Y%+
1)
- Ync1
I
The quantity rn is the ratio of the slope of the chord drawn between the points n- 1 and n on the operating line and the slope of the chord drawn between the same points on the equilibrium curve. When the results of Equation 12 are introduced into Equation 11, the following inequalities are obtained:
I
= dR, (11) Consider a system with constant relative volatility and constant molal downflow. The term 4%+ can be taken as an increasing function of (2, - yl&+ J with a n increasing positive slope because of the curvature of the equilibrium curve on a McCabe-Thiele diagram. If a curve is drawn of such a function and the slope of the tangent a t y = y. + is compared with the slopes of the chords drawn between the points yn and and y. + and y. + *,then :
TYhen the operating lines and the equilibrium lines are straight, then a study of similar triangles shows that r n + l = r,. However, because of the curvature of the equilibrium curve, 7, +, 5 r7&. Accordingly, ln(jn - y n + J can be taken as a decreasing function of n, going to minus infinity for large n, and with a negative decreasing slope. As before, comparing slopes of tangents and chords : rn+
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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