Calculation of Distillation Columns at the Optimum Feed Plate Location Donald N. Hanson* and John Newman Department of Chemical Engineering, University of California, Berkeley, California 94720
A method is shown employing Underwood’s equations for calculation of distillation columns with the feed plate at the optimum location. Calculation of examples shows that the common rules of thumb for location of the feed plate are often unreliable, particularly at reflux values near minimum reflux. A Fortran computer program for the calculation, general for up to 20 components, is available from the authors.
The equations proposed by Underwood (1945,1946,1948) have been extensively used for preliminary design of distillation columns. However, in most cases the calculations have been restricted to approximate solutions using an estimated feed plate composition. Exact solutions which have been proposed by Alder and Hanson (1950) and by Klein and Hanson (1955) have been too difficult to justify the additional computation. The authors propose a new method for obtaining exact solutions which is easily implemented on a computer and which has the important advantage that it yields the solution corresponding to optimum feed plate location. Since the method requires the assumption of constant relative volatility and constant flows in each column section, the reliance that can be placed on it as a final design calculation is limited. However, the fact that it provides a starting column solution which has optimized the feed plate location can be of considerable utility in implementation of the more accurate iterative methods which are used for final design. In using the iterative methods, the column configuration must be specified as a first step. The total number of stages is estimated, and the relative numbers of rectifying and stripping stages are chosen. While estimation of the total number of stages is reasonably satisfactory, e.g., through the correlations of Gilliland (1940) or Erbar and Maddox (1961), the methods used to decide how to distribute these stages between the rectifying and stripping sections are not satisfactory. The authors believe that the calculations proposed in this paper will provide a better value for the total number of stages, and will also yield a reasonable choice of the feed plate location for the initial estimates. Perhaps equally important, the calculations provide a reliable guide for the optimum ratio of the key components in the liquid leaving the feed plate. Changes in the original estimates of plate numbers are normally made on the basis of results obtained in the iterative calculations, and it is common practice to assume that the feed plate is a t the optimum position when the ratio of the key components corresponds to the value obtained from a rule of thumb. Calculations done have shown that the currently accepted rules of thumb can be seriously wrong. A system to be distilled is written as a set of components A, B, C, D, etc., listed in order of descending volatility. The desired separation is commonly described by setting the separation of two of the components, say B and C, specifying the fraction of each which is to be taken in one of the products. In this work these components will be called the light key component and the heavy key component. All components lighter than the light key will be called light diluents, and all components heavier than the heavy key will be called heavy diluents. Components lying between the keys in volatility will not be considered.
Consideration will be restricted to columns with one feed, two products, and two column sections. For such columns Underwood’s equations are expressed as follows. For the rectifying section ald (XI )d - @KS
VRS=C-
@RS,I
I
ai
--
4
ai
(1)
CYIXl
- @RS,2
(z) c
@iXi
i
ai - 4 R S . I
and for the stripping section
-vss =
ai6 (X,) h
___
i ai - @ S S a1x1
(Gd @SS,l
-
5: a/ -
4SS,l
c ai - @ s s , ~ CYiXi
i
where all summations are taken over the component index, i; cui = relative volatility of component i; d ( x i ) d = flow of component i in the top product, d, mol/unit time; 6 ( X i ) b = flow of component i in the bottom product, b, mol/unit time; V R ~ = vapor flow in the rectifying section, mol/unit time; VSS = vapor flow in the stripping section, mol/unit time; x i = mole fraction of i in the liquid leaving the feed plate; n R S = rectifying stages, including the feed plate; nss = stripping stages, not including the feed plate; @RS = root of eq 1;4ss = root of eq 3; subscripts 1 and 2 on 4 refer to two different roots. For eq 1there are a number of roots ~ R equal S to the number of components in the feed, and similarly a number of roots @ssto eq 3 equal to the number of components. The roots can be conveniently characterized by their values relative to the values of the volatilities of the components. For a system of four components, A, B, C, and D, the values of @ can be shown as
> > > ~ R S , B> W; > @RS,C > CUD; > @RS,D > 0;
~ U A ~ R S , A OB;
LYB CYC UD
> aA > @SS,R > aJ3 CUB > @SS,C > a C aC > @SS,D> a D @SS,A
aA
The value of @RS,A lies between CYA and CYB,but it will be convenient here to consider it as associated with component A, as indicated by the subscript. Thus for every component, i, there is a value, c$Rs,~,lying below ai (and above CY for the next heavier component if one exists). Similarly there is a value, ~ S S lying , ~ above ai (and below CY for the next lighter component if one exists). In describing a typical problem for the design of a column to fractionate a mixture of components, only the separation Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
223
of the light key component and heavy key component would be set. While the separation of all diluent components could be made dependent variables in the solution, experience has shown that it is expeditious to assume these separations, returning a t a later point in the calculation to verify or correct the separations. Typically the calculation starts with the assumption that the light diluent components go completely into the top product and that the heavy diluents go completely into the bottom product. Other estimates, such as those given by total reflux, can be used, but the procedure of calculation is not substantially affected by the values estimated. Values of the vapor flow in both sections are estimated from the set value of a major flow, usually the reflux. The fact that all separations are not exactly known normally causes only minor errors in these flows. Consider a four-component system for illustration, with components B and C as the components whose separation is set. Four values of 4 ~ can s readily be found from eq 1,but it will be noted that C#JRS,Dis very close to OD. The difference a ] ) - d l i s , ~will in general be the same order of magnitude as d ( X D ) d , and as d ( X D ) d 0 , ~ R S , D CYD.Since the value of d ( X D ) d is not known accurately, the difference (YD - ~ R S , I )is ~ not known accurately. Calculation of the values of 4 s will show the same behavior for ~ S S , A .As b ( x ~ ) i , 0, 4 ~ s . O(A, ~ and the value of the difference a~ - $SS,A will be quite inaccurate. Equation 2 can now be written with various pairs of values of 4~s.;and eq 4 with pairs of values of ~ss.,. The choice of pairs is not of great, significance, but in the rectifying section equation, eq 2, let us arbitrarily use 4RS.lk as the value of ~ R S , ? , and in the st,ripping equation, eq 4, let us use @SS.hk as the value of 4 ~ s . ~These ) . equations then become
-
-
-
-
(12)
Solution at a S e t Feed Plate Key Component Ratio Examination of the equations to determine their usefulness in solving for rectifying and stripping stages shows that eq 9 is not useful. If it has been assumed that d ( x ~ ) ) , l = 0, then 413s.~)= and in the denominator of the right-hand side, one term in the summation is infinite NDXD
= m
-~RS.D The left-hand side must then be zero, and the only answer is that n R S = m, which simply says that infinite rectifying plates are indeed required to reduce d ( X D ) d to zero. A better estimate of d(x& does not improve the usefulness of eq 9 in calculating nRS, since the value of nRS is completely dependent on the estimate. A similar statement can be made about eq 10. We are left with eq 7, 8, 11, and 12 to determine six unknowns, X A , X H , xc, XD, nRS, and n S S . However, we have the option of setting one additional variable in the problem, and it can be exercised to choose a value for the ratio of the key components in the liquid leaving the feed plate x l k / x h k . This equation together with CXi
=1
enables solution for the stages a t the chosen value of
XI^/
x hk.
Since the equations are nonlinear in the unknowns n K S and
nss, an iterative solution is required. A Newton-Raphson
and
zmi - ’
.
@SS.hk
where the simplifying nomenclature has been adopted that @I
PI,, = ai
- 4RS.j
a1
and G I , = @!
- 4SSJ
The summations are taken over all components. Subscript denotes any component other than the light key in the rectifying section and other than the heavy key in the stripping section, and the subscripts lk and hk denote light and heavy key. In the example then, we have available the following equations
j
(9)
224
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
approach can be used, but a simpler alternative iteration works well. Estimates can be made of the concentrations of the diluent components in the liquid leaving the feed plate. As starting values, mole fractions a t the end of an infinite rectifying section are accurate for light diluents, and mole fractions a t the end of an infinite stripping section are accurate for heavy diluents. With these assumptions, since the ratio of key components is set, the entire feed composition can be calculated and inserted directly into eq 8 and 11 for the calculation of n R S and nss. Using the calculated values of stages, eq 7 and 12 can be solved together with eq 13 and the set value of X i k / X h k to correct the mole fractions of the diluents on the feed plate. The calculation of TZRSand nss is repeated with the new concentrations of diluents, and the diluent concentrations are again corrected. A few iterations normally suffice to establish all mole fractions within 1 X 10-6. At this point, the column has been solved under the original assumptions made for separations of the diluent components. Equations 9 and 10 which were previously discarded can now be used to correct these distributions. For the heavy diluent D, and writing out eq 9, it is apparent that there is one dominant term in the summation in the denominator of the right side of eq 9, the term fori = D. The difference ai - ~ R S , I )can be readily obtained and used in eq 1written in C#JRS.Dto solve for d(x& In general any other heavy diluents will not affect the solution for the distribution of a particular heavy diluent, and the new distributions of these components can be calculated one a t a time. In the same way, the distribution of the light diluent A can be solved for by use of eq 10 to obtain CYA - ~ S S . Afollowed , by the use of this quantity in eq 3. With the corrected values for the distribution of the diluents, the whole problem can be solved once more, repeating the iteration to convergence. In most problems only a few major iterations are necessary, and an exact solution to the problem is thus ob-
tained for the set ratio of the key components on the feed plate. Whether the column solution obtained thus far represents the optimum solution depends on whether the choice of X l k / X h k on the feed plate is a t optimum. The values of the ratio given by either the ratio of the intersections of the operating lines for the two key components or the ratio of the key components in the liquid portion of the feed have been widely used in the past as estimates of the optimum ratio. However, a small amount of additional calculation will provide the true optimum.
The condition of optimum feed plate location is expressed by adding the rectifying section derivative and the stripping section derivative.
(""") dXhk
+(""") hk,lk
dXhk
=o Ik,hk
or
Optimization of the Feed Plate Let us reconsider the equations available for solving for stages. In the rectifying section, for example, an equation written with @RS,Ik and @RS,hk is always available. In the example it is eq 8. The equation may be expressed explicitly in n ~ sFor . each light diluent (subscript Id) an equation may also be written using @RS,]kand @RS,Id,expressed explicitly in TLRS, and the two equations subtracted. For each light diluent this yields
There are thus two equations for each light diluent, (14) and (161,and two for each heavy diluent, (15) and (17). Equation 18 plus the equations
(nRS)ld,lk- (nRS)hk,lk = 0
EXl = 1
=O
(13)
or
Similarly, in the stripping section, for each heavy diluent (subscript hd)
In
(-)
[In
ZGi.hdXi
I
- In CGi,hkX,
\@SS,hk/
Equations 14 and 15 may now be differentiated with respect to some significant variable on the feed plate to provide one additional equation for each diluent. The mole fraction of the heavy key has been arbitrarily chosen as the variable of differentiation. Thus for each light diluent
and for each heavy diluent
-
1
(&) @SS.hk
1
J
completes the set and allows calculation of all x i and dXl/dxhk a t the optimum feed plate. The equations are nonlinear, and their solution by the Newton-Raphson procedure presents certain difficulties. A more satisfactory procedure is to accept the composition and solve eq 16 and 17 plus 19 and 20 directly for the derivatives. Equation 18 is then solved for a new estimate of the ratio of the mole fractions of key components a t optimum, using the calculated values of the derivatives and the previous values of the diluent concentrations. The diluent mole fractions are corrected a t this new ratio as detailed previously. The derivatives are then recalculated, and the ratio is again reevaluated. This procedure converges rapidly to the optimum feed plate ratio due to the fact that the derivatives are substantially independent of the ratio. The procedure as outlined will provide an exact solution to the column a t the optimum feed plate. However, the best procedure for the entire calculation appears to be to solve for the optimum feed plate ratio before correcting the separations of the diluents, reoptimize, correct the diluent separations, etc. The separations obtained for the diluent components are quite dependent on the number of stages in each section, whereas the optimum column configuration is not highly dependent on the diluent separations. The authors have prepared a Fortran program for the solution which is available on request. The program is general for any number of components up to 20. I t can be used with different relative volatilities in the two sections, but it is suggested that the appropriate values of relative volatility are those in the region of the feed stage since the optimum location of the feed stage depends on these values rather than the volatilities a t the ends of the column. The program has been extensively tested and has been found to fail on problems in which the amounts of diluents are excessively large compared to the amounts of keys (more than 10 times), particularly when the diluents are very close to the key components in volatility ( a = 1.01) and the separation of the keys is sloppy. Further development of the procedures would probably allow extension of the solution to these problems, but the procedures outlined above are satisfactory for the great majority of Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
225
Table I. Comparison of Optimum Design and Rule-of-Thumb Design Relative volatility
Component
Specified recovery fraction in top product
Feed
A 3.3 0.50 B 3.0 0.05 C 2.0 0.20 D 1.0 0.25 Saturated liquid feed Rule of thumb ratio of key components on feed plate = 0.80 Minimum reflux = 0.440
-
0.99 0.10
Optimum Reflux min. reflux 1.00
1.10 1.15 1.20 1.25 1.30 1.40 1.50 2.00
1.190 1.159 1.144 1.131
1.117 1.105 1.081 1.059 0.971
Rectifying stages
Stripping stages
Total stages
8.25 7.41 6.87 6.47 6.17 5.14 5.44 4.71
13.61 12.83 12.28 11.85 11.50 10.96 10.54 9.31
21.86 20.24 19.14 18.32 17.68 16.70 15.98 14.02
problems encountered. A typical four-component system requires 0.3 s and a typical eight-component system requires 0.7 s on a CDC 6400 computer. Results a n d Discussion Calculations have been made on over 70 problems in order to establish the benefits which might be obtained through use of the optimum feed plate ratio in the design of columns. The results of the calculations lead to several conclusions. (1)Neither of the rules of thumb for establishing the optimum feed plate ratio is sufficiently trustworthy. The ratio given by the intersections of the operating lines for the key components is generally better for systems with a preponderance of heavy diluent. The ratio given by the ratio of the keys in the liquid portion of the feed is generally better for systems with a preponderance of light diluents. However, both criteria often yield values significantly different from the optimum ratio. ( 2 ) The total stage requirements given by the optimum ratio or by either of the rules of thumb are very close for many cases, even though the ratios of the keys on the feed plate are quite different. The minimum in the curve of total plates vs. feed plate location is apparently broad for these cases. However, cases frequently arise for which this is not true. Some of these cases cannot even be solved a t the ratios given by the rules of thumb; even infinite stages will not produce the required ratio. (3) Systems which contain large amounts of light diluents or large amounts of heavy diluents, for example, amounts equal to the amount of the nearest key component, are most likely to have an optimum ratio which deviates substantially from the rule-of-thumb ratio. If the system contains both kinds of diluents, it is more likely to have an optimum ratio close to the rule-of-thumb ratio. Diluents very close to the corresponding key in volatility or widely different from the key have less effect in shifting the optimum ratio away from the rule-of-thumb ratio than do diluents moderately removed from the key. (4)Light diluents raise the optimum ratio (of light to heavy key). Heavy diluents lower the ratio. The state of the feed is relatively unimportant. 226
Rule-of-Thumb
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
Rectifying stages
Stripping stages
Total stages
9.61 9.42 9.25 8.66
22.64 18.37 16.86 14.14
a+ W +
m+
m+
13.04 8.95 7.61 5.48
The conclusions above apply to multicomponent systems. The optimum ratio for binary systems has been accepted for many years as the ratio given by the intersection of the operating lines, a judgement reached from consideration of the McCabe-Thiele diagram. It is interesting that the calculations show that this is not correct. For all practical design purposes the rule of thumb is sufficiently accurate, but it is not correct. For multicomponent systems, the most substantial deviations from either rule of thumb occur as minimum reflux is approached. At high multiples of reflux, of the order of 2, the rule-of-thumb ratio is satisfactory. However, a t the currently accepted design reflux of 1.25 times minimum reflux, the rule-of-thumb ratio is often not satisfactory. As the reflux used approaches minimum reflux even more closely, which appears to be the trend, it will be more and more important to use the optimum feed plate. Table I shows the results for a typical case where the rule of thumb fails badly. The separation specified cannot be achieved with a feed plate ratio determined by the rule of thumb, here xc/xn = 0.8, for any reflux below about 1.3 times minimum reflux. At these flows, even a column which contains an infinite number of rectifying stages is incapable of reaching a feed plate liquid composition in which x(./xn is as low as 0.8. A t higher reflux values the rule-of-thumb ratio is applicable but the increase in reflux over that required for operation with the optimum feed plate ratio is substantial. A satisfactory column can be designed at a reflux of 1.25 times minimum using the optimum feed plate; 18.3 theoretical stages are required. The same total number of stages using the rule-ofthumb feed plate ratio will produce the specified separation, but at the expense of a 15% increase in reflux. Since Underwood’s equations require the assumptions of constant relative volatility and constant molal flows in each column section, the results of the calculations should not be taken as design values. However, if the relative volatilities used are those pertaining to the general region of the feed stage, the optimum ratio calculated by Underwood’s equations should be reasonably correct, since it is relatively independent of flows in the region near minimum flows. I t does provide a reliable basis for a rapid search of the stage requirements a t
various flows, or vice versa, by more exact column calculations. The final location of the optimum feed plate should be determined by a search of the neighboring plates.
Nomenclature A, B, C, D = specific components b(x,)h = flow of component i in the bottom product, mol/unit time d ( x , ) d = flow of component i in the top product, mol/unit time F L J= @ I / ( ~ I - @RS,]) GI j = f f , / ( a , @SS,,) TZRS = number of stages in the rectifying section nhs = number of stages in the stripping section
VR,
= flow of vapor in the rectifying section, mol/unit
time = flow of vapor in the stripping section, mol/unit time x, = mole fraction of component I in the liquid leaving the feed stage Vbh
dss,] = root of eq 3 lying numerically just above a, Z = summation over all components
Subscripts A,B,C,D = refer to corresponding specific components i = refers to any component j = refers to any component other than the light key component in the rectifying section, any component other than the heavy key component in the stripping section hd = refers to any heavy diluent component hk = refers to the heavy key component Id = refers to any light diluent component lk = refers to the light key component Literature Cited Alder, B. J., Hanson, D. N., Chem. Eng. Prog., 46, 48 (1950). Erbar. J. H., Maddox, R. N., Pet. Refiner, 40 (5), 183 (1961). Gilliland, E. R., lnd. Eng. Chem., 32, 1220 (1940). Klein, Gerhard, Hanson, D.N., Chem. Eng. Sci.. 4, 229 (1955) Underwood, A. J. V., J, Inst. Pet., 31, 111 (1945). Underwood, A . J. V., J. Inst. Pet., 32, 598 (1946). Underwood, A. J. V., Chem. Eng. Prog.. 44, 603 (1948).
Greek Symbols = relative volatility of component i @RS, = root of eq 1 lying numerically just below a,
Received for reuieu. May 14. 1976 A c c e p t e d October 4,1976
Application of a Diffusion Limiting Model to a Tube-Wall Methanation Reactor Richard R. Schehl,' James K. Weber, Mark J. Kuchta, and William P. Haynes Process Engineering Division, Pittsburgh Energy Research Center, U.S. Energy Research and Development Administration, Pittsburgh, Pennsylvania 15213
The applicability of a diffusion limited model in predicting the performance of a large scale annular tube-wall methanation reactor is examined. The model, developed by Senkan et al. (1976) in a recent article is adapted to the reactor geometry used in our study and tested against experimental data. It is demonstrated that, for the geometry and catalyst used in this study, the reactor is not diffusion limited and that kinetic resistance must be included in any realistic model.
Introduction The Pittsburgh Energy Research Center has been investigating, for a number of years, the feasibility of a tube-wall reactor in the catalytic methanation step for upgrading raw synthesis gas to high-Btu pipeline quality substitute natural gas (SNG) (Demeter e t al., 1967; Forney, 1972; Haynes et al., 1973; Ralston et al., 1974; Strakey e t al., 1975). This type of system has been investigated in units ranging from benchscale (8.5 scfh) to pilot plant scale (1800 scfh) and a multi-tube reactor designed for a 13 000 scfh feed has been incorporated into the Synthane prototype plant. The tube-wall methanator has the advantage of being able to accommodate feeds containing as much as 25% CO and yet operate under near isothermal conditions. In a recent article, Senkam et al. (1976) presented a diffusion limiting model for the tube-wall reactor. Film theory was used to arrive a t an expression for mass transfer rates and the model was generalized to include nonisothermal systems. The model was developed for a surface reaction of arbitrary stoichiometry and then applied to the specific case of a hypothetical tube-wall methanation reactor. The purpose of this communication is to appropriately modify the model so as to
describe turbulent flow through an annulus with the reaction catalyzed on the outer wall and test the applicability of the model by comparing model predictions with experimentally measured values.
Reactor Description Studies of the annular tube-wall methanation reactor were carried out in a pilot-plant scale unit depicted in Figure 1.The reactor is constructed of 304 stainless steel 2-in. schedule 40 pipe, 14 ft in length. Surrounding the 2-in. reactor tube was a 4-in. diameter pipe jacket containing boiling Dowtherm in the annular space to remove the heat of reaction. A 1.5-in. inch tube coaxially located in the reactor confines the reacting gas to the annular region defined by the 2-in. pipe and 1.5-in. tube. The temperature of the coolant, and hence, the catalyst temperature is regulated by controlling the pressure in the cooling system. Nucleate boiling is assumed to take place on the outer surface of the reactor tube, thereby providing a natural convective circulation of the Dowtherm. Dowtherm vapor is condensed via cooling water and returned to the reservoir. The inside surface of the 2-in. reactor tube was coated with Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
227
