May, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
8, rB,6 = virial coefficients in Beattie-Bridgeman equation of state Ao, Bo,a , b, c = constants of Beattie-Bridgeman equation of state a1 an, as, a4 = virial coefficients in Equation 25 al, an . . . , bl, bz . . . , etc., = constants in Equations 25A to 2 5 0 AI, An, . , B1,B,, . . ., etc. = constants in Equation 28 as defined by 27 Literature Cited (1) Bartlett, Cupples, and Tremearne, J . Am. Chem. SOC.,50, 1275
..
(1928).
Beattie, Proc. Natl. Acad. Sci., 16, 14 (1930). Brown, Lewis, and Weber, IND.ENQ.CHEIM., 26, 825 (1934) Brown, Souders, and Smith, Ibid., 24, 513 (1932). Cope, Lewis, and Weber, I W . , 23, 887 (1931). Deming and Deming, Phys. Rev., 45, 111 (1934). Ibid., 48,448 (1935). Deming and Shupe, J. Am. Chem. SOC.,52,1382 (1930). Deming and Shupe, Phys. Rev., 37, 638 (1930). Ibid., 38, 2245 (1931). (11) Ibid., 40,848 (1932). (2) (3) (4) (5) (6) (7) (8) (9) (10)
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551
(12) Dodge, IND.ENQ.CHEM.,24, 1353 (1932). (13) Edmister, Ibid., 30,352 (1938). (14) International Critioal Tables, Vol. 111, pp. 3-17, New York, McGraw-Hill Book Co., 1928. (15) Lewis, G. N., and Randall, Merle, “Thermodynamics”, New York, MoGraw-Hill Book Co., 1923. (16) Lewis, W. K., IND.ENQ.CHEM.,28, 257 (1936). (17) Lewis, W. K., and Kay, W. C., Oil Gas J., 32, No. 45,40 (1934). 25,725 (1933). (18) Lewis, W. K., and Luke, C. D., IND.ENQ.CHEM., (19) Lewis, W. K., and Luke, C. D., Trans. Am. Soc. Mech. Engrs. 54, 65 (1932). Maron and Turnbull, IND.ENQ.CHEM., 33, 246 (1941). Zbid., 33,408 (1941). Maron and Turnbull, J . Am. Chem. SOC.,64,44 (1942). Newton, IND.ENQ.CHEM., 27, 302 (1935). Newton and Dodge. Ibid.. 27.577 (1935). Piokering, Bur. Standards, C&c. 279 (1925). Sage, Schaafsma, and Laoey, IND.ENQ.C H ~ M26, . , 1218 (1934) Sage, Webster, and Lacey, Ibid., 29. 658 (1937). Selheimer, Souders, Smith, and Brown, Ibid., 24, 515 (1932). Watson and Nelson, Zbid., 25,880 (1933). Watson and Smith, Natl. Petroleum News, July 1. 1936.
Degrees of Freedom in Multicomponent Absorption and Rectification Columns c. R. q
u
e. c. R&
Massachusetts Institute of Technology, Cambridge, Mass.
A general analysis of multicomponent interphase contacting systems i s presented to illustrate the restrictions and limitations inherent in any method of rectifier, absorber, or extractor process design. This type of fundamental analysis should be of value to the design engineer in formulating specifications which are consistent with the limitations imposed b y the general laws governing the system in question. The major difficulties encountered in these calculations arise from the practical necessity of “fixing” more variables than are independent in order to expedite the process design as a whole.
WITHIN
the last ten years many articles on multicomponent absorption and rectification have appeared, each advocating some special system of design (1, 8, 3). The general problem is complex, and many of these contributions have proposed the use of numerous simplifying engineering approximations to arrive a t a practical solution. I n most cases the work has been handicapped by the absence of a rigorously correct analysis of the degrees of freedom for the general multicomponent case of countercurrent multistage interphase contact. The purpose of this article is to develop such a theory for the case of equilibrium contacting equipinent, the results being applicable to any case involving countercurrent physical interaction (mass transfer) between two streams.
Before design can proceed on any chemical engineering system, it is necessary to conduct an analysis to determine how many of the design variables may be arbitrarily h e d before the system as a whole becomes physically fixedthat is, before the remaining design variables can be computed from the equations controlling the operation of the system. For example, in a simple rectifying column operating on a given binary mixture a t atmospheric pressure, the composition and condition of the feed, the composition of the distillate and bottoms, and the reflux ratio above the feed plate are fixed ; then the usual simplifying assumptions made on an ordinary McCabe and Thiele diagram make i t possible to calculate the optimum’ number of theoretical plates required to effect the separation. The simplifying assumptions (7,8) in this case lead to so few and simple equations that there is no doubt regarding the number of variables which must be fixed to fix the system. Introduction of a third component results in considerable complication, even though the usual simplifying assumptions be retained. Experience shows, for instance, that having specified feed composition and condition, column pressure, and reflux ratio, it is no longer possible to specify the complete composition of distillate and bottoms; the concentration of one component in the overhead and in the bottoms is sufficient to fix the optimum number of plates absolutely. Specification of one more terminal concentration will fix the number of plates a t some value not necessarily the optimum, and specification of a fourth terminal concentration would in general be inconsistent with the first three concentrations 1 “Optimum” is not here used in the sense of an eoonomic optimum resulting from oomplete eoonomic balance but in the sense of the minimum number of plates resulting ahen feed is introduced on the proper plate.
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already specified. The reasons underlying this situation will be developed here. The general theory underlying relations among the many variables in a complex countercurrent system may be derived rigorously, unencumbered by any kind of simplifying assumptions; and the results of this theory may be employed with confidence in evaluating the consistency of simplifying assumptions introduced into approximate design methods. The theory also provides a fundamental insight into the basic quantitative relations governing multicomponent absorption, extraction, and rectification, and indicates the basic difficulties facing any design method.
Analysis of a Single Stege The method of analysis to be employed will be illustrated h s t by application t o a simple one-stage contacting unit for gas and liquid. In the unit shown in Figure 1 a liquid and gas stream are mixed together, and the gas and liquid products of the mixing are removed continuously so that there is no acLIQUID IN
GAS OUT 62 ADIABATIC - SEPARATOR
7 Q FIG. I. SINGLE-STAGE
LIQUID OUT Lz
CONTACTING UNIT
cumulation or depletion of matter in the system. I n general, a quantity of heat, &, may be added to or removed from the unit continuously. For adiabatic operation of the entire unit Q = 0. If it is assumed that the gas and liquid leaving the separator are in equilibrium, it is evident that this unit is equivalent to one perfect plate in a distilling column, an absorber, or a stripper. (For the sake of a definite illustration, the explanation is confined to a unit contacting liquid and gas. The general principles developed, however, are applicable to the equilibrium interaction of any two phases under the ordinary phase rule limitations.) Clearly, if liquid and gas of definite composition and physical condition are mixed in a definite ratio with the addition (or removal as the case may be) of a definite quantity of heat, and if a definite pressure is maintained in the separator, the system will automatically produce gas and liquid of definite composition and condition, in a fixed ratio. The efficiency of mixing and the time of contact will determine how nearly the exit gas and liquid have reached equilibrium. Equilibrium conditions will be assumed in the following discussion, and the effect of nonequilibrium conditions will be examined later. The variables describing the operation of the system in Figure 1 may be grouped under two general classes: 1. The thermodynamic intensive variables associated with the several streams-via., concentrations, temperatures, pressures, densities, specific enthalpies, specific entro ies, etc. These variables are independent of both the total and re7ative quantities of the various streams, and will be referred to as intensive variables. 2. The relative quantities of the various streams of matter and energy-via., the ratio of gas feed t o liquid feed, the ratio of net heat supplied t o liquid feed, the ratio of gas product to liquid product, etc. Only a few o€ the many variables which might be mentioned are independent, and when all of the independent variables have been set at fixed values, the remaining or dependent variables may be calculated from equations expressing the
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conditions imposed by the fundamental laws controlling the system. Granting attainment ol equilibrium, the laws controlling the present case are:
I. Law of conservation of matter. 11. Law of conservation of energy (first law of thermodynamics). 111. Second law of thermodynamics. If there are C independent components in the streams being contacted, law I indicates that there are C independent material balances applying over the contacting unit. For the flow systems under consideration, law I1 reduces t o an enthalpy or heat balance over the contacting unit, and law I11 includes the multitude of equilibrium relations governing the system. The number of independent variables in a contacting unit -i. e, the number of variables which must be ‘%xed” before the operation of the unit is completely determined-is given by the difference between the total number of variables and the total number of equations relating these variables. The number of independent thermodynamic intensive variables in a system is identical with the degrees of freedom (sometimes called “variance”) of the system as predicted by the phase rule: F=C-P+2
where F C P
=
= =
(1)
degrees of freedom number of independent components number of phases
Gas and liquid leaving the separator in Figure 1 may be considered a two-phase equilibrium system of C components, in which case Equation 1 indicates that F = C. It is worth while to recall at this point that the degrees of freedom as predicted by the phase rule represent the difference between the total number of thermodynamic intensive variables in a system in thermodynamic equilibrium and the number of equilibrium relations existing among these variables (6)that ‘is,the independent thermodynamic intensive variables. I n general, liquid and gas entering streams are not in equilibrium with each other and therefore are treated as separate systems. The liquid and gas feeds to the mixer in Figure 1 may be considered as two independent, one-phase, C component, thermodynamic systems having, according t o the phase rule, C 1 degrees of freedom. The total number of independent intensive variables associated with the two entering single-phase systems and the two-phase system leaving is:
+
Z(C
+ 1) + c = 3c + 2
(2)
The remaining independent variables are of the second type mentioned above-i. e., the relative quantities of the various streams of matter and energy. Selecting the entering liquid stream as the basis for computation, there are four quantity , L1/L1. If there are p streams ratios: G2/L1,G1/LI,L 2 / L I and entering and leaving the system, there will be p quantity ratios. Since one of these ratios will always be fixed a t unity, due to the necessity of choosing a basis for calculation there will be ( p - I) variable quantity ratios. The use of quantity ratios is convenient since it renders the argument independent of the scale of operation. Finally, there is the energy ratio-heat added or subtracted per unit of reference stream, &/L1, referred t o here as a heat ratio variable. Since there are C independent material balances and one enthalpy balance, the total number of independent variables or degrees of freedom in the apparatus of Figure 1 will be:
+
+
(3C 2). independent thermodynamic intensive variables 4 quantity ratios + 1 heat ratio - 1 quantity ratio fixed at unity - C material balances - 1 enthalpy balance = 2c 5 (3)
+
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553
obvious limitations on these values if the system is to operate physically. For example, if C = 3, the components being propane, butane, and pentane, it would be absurd to specify that the ratio of pentane to propane in the exit gas should be greater than in either of the feeds. The foregoing analysis is summarized in Table I. Analysis of n Contacting Units in Series Figure 2 illustrates n single stages operating in series on a system of C components.
The two arrows associated with the letters Q indicate that heat may either be removed or added to any stage. I n general, Figure 2 might represent an absorber, a stripper, or even a countercurrent liquid-liquid extractor. It will be assumed that the two streams leaving any stage are in equilibrium and therefore constitute a two-phase, thermodynamic equilibrium system. This twophase system and the two one-phase systems entering each stage possess a total of (3C Courtesy, Leeds & Northrup Company 2) independent intensive variables. Also associated with each stage are four quantity Control Panel in an Oil Refinery ratio variables and one heat ratio variable. One of the stages will have only three quantity ratios since one stream in the system must be taken as a basis for calculation. The total number Equation 3 states that (2C 5) variables must be fixed of independent intensive variables, quantity ratios, and heat before the operation of the unit in Figure 1 is completely deratios then becomes: termined. Specification of the complete composition and condition of the two feeds involves fixing 2 (C 1) = (2C n(3C 2) 5n - 1 (4) 2) variables. Specification of LJG1,the liquid gas ratio to the mixer, and Q/L1fixes two additional variables. SpecificaRelating these variables are a total of nC independent mation of the value of one more variable is now sufficient to deterial balances and n independent enthalpy balances. I n termine the operation of the system in the sense that, given addition, certain important equations of condition are introthe value of this variable, the values of all the other variables duced by the series type of operation in Figure 2. Since the can in principle be determined from the available equations. liquid effluent from stage 2 is equal to the liquid feed to stage The choice of variables to be fixed is arbitrary and dictated 1 in quantity, composition, and condition, the result is the largely by convenience. For example, the final variable to equality of (C 1) intensive variables and one quantity ratio be fixed might be the equilibrium pressure or temperature in variable. (All heat quantities concerned may, for purposes the separator, any concentration in one of the exit streams, or of analysis, be associated with a suitable stage and all intera ratio like L2/C2,LI/L2,etc. Although from a mathematical stage flow considered adiabatic.) Since there are a total of 5) variables in the unit of Figure 1 are point of view (2C 2(n - 1) interstage streams, the total number of these condiindependent and may be given any values, there are certain tion equations is 2(n - 1) (C 2). The number of independent variables in the multiple-stage unit of Lo----, Gn UNIT(FIGURE1) TABLE I. SINGLECONTACTING Figure 2 is obtiined from Q" (Independent variables = (2C + 5); any 2C + 5 of the following the equation: common variables may be fixed within appropriate physical ranges.
+
+
+
+
+ +
+
+
+
--,
1 S T A t E j e
Only the most common variables are given; many more might be listed but only 2C 6 of the total may be arbitrarily specified.)
+
Type of Variable Complete oompn. of two onephase feed streams Condition of two feed streams Operating pressure Operating temp. Heat loss or gain Relative quantity of two feed streamsCompn. of product streams
No. of Variables Fixed 2(C-l) compns. 4 (selected from temp., pressure, sp. vol., enthalpy, etc.)
1 1 1 1
2n
(C-1) oompns.; number of compn. variables in these streams limited by assumption of their being in equilibrium
Relative quantity of two-product streams
1
+
2) independent intensive variables 5% quantity ratios plus heat ratios - 1 quantity ratio &xed at unity - nC material balances - n enthalpy balances - 2(n - 1) (C + 2) interstage condition equations =
n(3C
+
*nL
STAGE n-i
(I"-,
+ 2C + 3
One more variable which must be considered in the case of a multiple-stage system is the number of stages, n. The necessity for introduction of this variable is obvious from the
'myQ1 STAGE I
L,
FIG. 2. MULTIPLE STAGE CONTACTING UNIT
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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following example: Consider several absorption columns each with a different number of equilibrium stages or perfect plates. If each column operates adiabatically and is fed with the same liquid and gas in the same ratio, the nature of the liquid and gas products will be definite, but different in each column, depending upon the number of stages. The total number of independent variables in this case then becomes:
2n
+ 2C + 4
(6)
If the number of stages is specified by assigning some value to n,the total number of independent variables is reduced by one, and Equation 6 reduces to Equation 5 . I n the special case when n = 1, Equation 5 reduces t o Equation 3. As another example it would be possible to fix the composition and condition of the liquid and gas feeds t o the system by k i n g 2(C 1) variables. Fixing the heat loss from each stage fixes n variables, leaving, according to Equation 6 (n 2 ) variables still independent. If the pressure in each stage is fixed, only two independent variables will remain, and these may be fixed in many different ways. For example, it would be possible t o ~IX the ratio of liquid feed to gas feed, and the percentage recovery of any component in the liquid effluent from the system. (Specification of a percentage recovery of a given component in some particular stream is equivalent to fixing the product of a concentration in this stream by the quantity of this stream per unit of reference stream; thus a percentage recovery must be treated as a single variable.) Alternatively, any two concentrations in any two streams could be specified. As soon as these variables have been fixed, it is possible in principle to calculate the remaining variables, including the value of n, the required number of equilibrium stages. In still another case the final two variables might be &xed by specification of the value of n, together with the ratio of liquid to gas feed; complete equilibrium and thermal data mould then permit calculation of all dependent variables in the unit, including the complete compositions of the exit streams. Table I1 summarizes the above results.
+
+
TABLE11. n CONTACTING UNITSIN SERIES
+
+
+
+
(Independent variables = 2 n 2C 4 ; any 2n 2C 4 of ,the following common variables may be fixed within appropriate physmal ranges.) Type of Variable No. of Variables Fixed 2 ( C - l ) compns. Complete compn. of two onephase feed streams Condition of two feed streams 4 (selected from temp., pressure, sp. vol., enthalpy, etc. Operating pressure in each n pressures stage Operating temp. in each stage n temp. Heat loss or gain in each stage n 1 Relative quantity of two feed streams Compn. of product streams 2 ( C - 1 ) compns. Relative quantity of two 1 product streams 1 Number of stages
Rectifying Columns with Intermediate Feed Consider next a rectifying column (Figure 3) consisting of a total condenser, a reboiler, a feed plate, n perfect plates above the feed plate, and m perfect plates below the feed plate. The feed will be taken as a single-phase system of C components. The total number of intensive variables, quantity ratios, and heat ratios associated with the ( m n) plates exclusive
+
+
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+ +
+
of the feed plate is ( m n) (3C 2) 5(m n). Since there are three streams entering the feed plate and two leaving in equilibrium, the number of independent intensive variables and quantity ratios associated with this plate is: 3(C 1) independent intensive variables in feed streams C independent intensive variables in equilibrium exit streams 5 quantity ratios 1 heat ratio (7) Employing any one of the streams as reference stream for the calculation fixes one quantity ratio a t unity, and the total number of intensive variables, quantity ratios, and heat ratios associated with the (m n 1) plates is:
+
+
+
+
+ +
(m
+ n)(3C + 2) + 5(m + n) + 3(C + 1) + C + 5
(8)
A single one-phase stream is entering the reboiler, and two streams in equilibrium with each other are leavine the reboiler. By &e phase rule these streams together possess (C TOTAL CONDENSER 1) c = (2C 1) independent intensive variables. Also associated with the reboiler i o n Wn-i are three q u a n t i t y Qn-i PLATE n-i I ratios (one for each stream) and one heat ratio, making a total of 2c 5. PLATE I The feed to the condenser is a one-phase system p o s s e s s i n g (C 1) degrees of freedom or independent intensive variables. Since the reflux and distillate are each onePLATE 2 phase systems of identical composition and condition in the case of a total condenser, they account together for a total of (C 1) independent intensive FIG 3. variables. Since there RECTIFYING COLUMN CONTAINING are three quantity ratios ( m t n t i ) PERFECT PLATES corresponding to the three steams and one heat ratio, the total number of variables associated with the condenser is 2C 6. With feed entering on some intermediate plate, the number of plates above the feed constitutes a variable, and the number below the feed constitutes a variable. Alternatively, it would be possible to consider the total number of plates as a variable and the feed plate location as a variable. (Th6 necessity for considering feed plate location as a variable in addition to the total number of plates, or considering the plates above and below the feed as constituting two variables, arises from the fact that a column containing a given number of perfect plates may be operated on a given feed a t a given reflux ratio and heat consumption per unit of feed to give a wide variety of products, depending upon the feed plate location.) I n summary, the total number of variables t o be considered is :
+
+
+
4
+
+
+
+
(m
+5n)(3C +3mCn) ++3nC 3(C + 1) + c + 5 + 2c + + 2C++2)6 ++ 5(m 2 + 8C + 7m + 7%+ 21 =
+ +
(9)
Relating these variables are ( m n 3)C independent material balances, ( m n 3) enthalpy balances, and 2(m n 2 ) (C 2 ) equations expressing equality of con-
+ +
+
+ +
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the total number of plates shall be a minimum, and it effectively fixes one variable. There are only two remaining variables which can be fixed, and the choice of these is dictated by the essential nature of the operation t o be performed (C- 1) compns. Complete compn. of feed Condition of feed 2 in the column. Operating pressure over each m+n+3 When a separation is t o be made between two key compoplate and in still and connents, the two variables may be conveniently fixed by specifydenser Operating temp. on each plate m f n f 3 ing the concentration of the light key in the bottoms and the and in still and condenser concentration of the heavy key in the distillate; or the conm+n+2 Heat gain or loss to or from centration of heavy key in the distillate and the percentage each plate and condenser recovery of heavy key in the bottoms might be specified. It 1 Heat supplied to still Compn. of product streams 2(C-I) compns. is important to emphasize once more that, from the present Relative quantity of two prod1 point of view, the unit operational design (specification of uct streams the condition, composition, and quantity of all important No. plates above feed 1 streams, location of feed plate, number of plates, heat load No. plates below feed 1 1 Relative quantity of liquid reon reboiler and condenser, etc.) of a rectifying column is conturned to top plate to overtrolled in principle by a system of simultaneous equations. head product Some of the equations like material balances are simple and readily expressed analytically; others expressing the essential equilibrium data may be known only in the form of tables or graphs. Frequently insufficient equilibrium data are availdition, composition, and quantity ratios; there is one for each able, and simplifying assumptions are made as to their nastream which, due to the series type of operation, acts simulture. In general, there are more variables appearing in this taneously as effluent from one stage and feed to the next. system of equations than there are equations; and if a soluIn the system as a whole there are 2(m n 2) such streams. tion of the system is to be obtained, enough of the variables The total number of equations is: must be fixed so that the number of variables remaining is 3mC + 3nC 7C 5m 5n 11 (10) exactly equal to the number of equations. When all of the independent variables are fixed, the remaining dependent The degrees of freedom in the column, obtained by subvariables are- determined tracting &e total number by simultaneous solution of equations from the total of the equations. This n u m b e r of v a r i a b l e s operation may be very are: complex and usually must C + 2m + 2n + 10 (11) be performed by successive approximation, toThe variables from which gether with appropriate these degrees of freedom simplifying assumptions. may be chosen are sumFrequently, the method marized in Table 111. of calculation a d o p t e d In the process design of n e c e s s i t a t e s assuming a rectifying column the values for several of the variables of Table I11 dependent variables. At ordinarily fixed are: comthe conclusion of the calposition and condition of culation, checks like overfeed, heat losses from each all material balances are plate (usually taken as applied and may indicate zero), heat removed from the necessity of repeating condenser, pressure over the entire procedure with each plate and in still and the assumption of better condenser, and the reflux values for the dependent ratio. Reference to Table variables. I n this manner I11 indicates that these a solution of any desired variables total t o C degree of accuracy may 2m 2n 7. According be obtained by a series of to Equation 11 this leaves trial-and-error calcula3 variables to fix before tions. It is clear that in the other dependent varia l l m e t h o d s of s u c ables can be calculated. cessive approximation It is wise to specify feed any values arbitrarily location at the optimum assigned to dependent point-i. e., at such a variables must be repoint that the columnshall garded aa strictly tentarequire a minimum numtive and subjeot to revision ber of plates when operatas calculation proceeds. ing according to the other An example of the solution fixed conditions. This of such a set of equations procedure is equivalent to for a single contacting unit Courtesy, E. B. Badeer 8 Sons Compdny specifying that the ratio is given by Carey (6). I n n/m shall be such that multiple-stage systems the S p e c i a l N a p h t h a Fractionator TABLE111. RECTIFYING COLUMNVARIABLES No. of Variables Fixed Type of Variable
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inherent complexity of all composition of the feed solving large numbers of in one component or the simultaneous equations is fraction liquid in the total frequently reduced by such feed, or some equivalent assumptions as constant variable. ] molal reflux and simplified If side cuts are withequilibrium relations. In drawn from the column, the case of simple binary their composition and consystems the assumption of dition will be identical constant molal reflux so with the composition and simplifies enthalpy and condition of the corrematerial balances throughsponding streams within out the column that the the column a t the points entire calculation may be of withdrawal. For each readily performed graphiside cut the new variables cally. I n multiple-comintroduced are (C 1) ponent systems it is independent intensive usually necessary to resort variables, 1quantity ratio, and 1 variable associated to successive approximawith the location of the tion even with the usual simplifying assumptions. side cut. The new equaAs in most systems of tions introduced are: (C 1) condition equaequations describing a tions equating the compophysical operation, there sition and condition of the are in this case limitations side cut to the composition on the range of values for and condition of the corthe independent variables responding streams within if the roots of the equathe column. tions are to be physically Courtesy, E. 8 . Badeer & Sons Cornpdny If the relative quantity significant. For example, Continuous Distillation Unit a t the Calvert Distillery the volatility relations of side cut is fixed,together with its location, the under a given set of circolumn will be completely cumstances might render physically impossible a specification that the concentration ftxed, provided the other variables have been fixed as menof a given component in the bottoms be greater than in the tioned above. distillate. It is also known that for a given separation under These illustrations should suffice to demonstrate the application of this method of analysis t o the determination of a given set of conditions the number of perfect plates cannot be specified a t less than the number corresponding to total the degrees of freedom in any type of contacting unit. The reflux. If the number of plates is fixed a t a lower value, cereffect on the variance of the system introduced by associated tain of the variables initially fixed have t o be changed. equipment, such as partial condensers, reboilers with mechanical recirculation, interstage recirculation coolers in absorbers, etc., may be determined by the same general methods. From Multiple Feed these broad principles of general application it is possible The method of analysis is readily extended t o the case of a to deduce the degrees of freedom, variance, or independent column operating with several feeds. Each feed plate is variables associated with any type of equilibrium contacting treated exactly like the feed plate described above and hence equipment. NONEQUILIBRIUII COLUMNS. If the vapor and liquid introduces 3(C 1) C 5 1 variables (7). I n addition, the location of this feed becomes a variable, and the total leaving each plate are not in equilibrium, it may be considered additional variables introduced become 4C 10. At the that for the case of a column with one feed plate, a reboiler, n plates above the feed, m plates below the feed, and a total same time there are introduced C independent material balances, one enthalpy balance, and 2(C 2) equations expresscondenser, (m n 2) additional variables are introduced. ing equality of condition and quantity ratios of the streams Specification of the percentage approach to equilibrium in flowing to and from this plate from and to the adjacent plates, each case by statement of (m n 2) “plate efficiencies” will then place this case under Equation 11. Even though respectively. If the composition and condition of the feed are fixed, (C 1) variables are fixed. Fixing the heat losses it is realized that concentration gradients exist on and above from this plate and the pressure over the plate fixes two more the plates of actual columns, it is convenient t o design by variables. Specifying plate location fixes one more variable, means of plate efficiencies based on terminal plate concentrations (4). The case of a packed tower providing continuous leaving an increased variance in the column of one as a result of the additional feed plate. If the quantity of the feed stream nonequilibrium interaction between two multicomponent streams presents greater difficulties; but when more informarelative to the reference stream is fixed, the variance in the tion becomes available concerning the detailed mechanism of column becomes identical with that of the column having multicomponent mass transfer, it may be possible to present one feed. Any number of feeds may be treated by the same a rigorous analysis along the present lines by assumption of type of analysis. A mixed liquid and vapor feed must be treated as one two-phase feed if liquid and vapor are introinterphasal equilibrium, together with introduction of apduced a t the same point in equilibrium with each other, or two propriate mass transfer equations in the two phases. It has been pointed out that in design calculations two of one-phase feeds if nonequilibrium liquid and vapor are introduced a t different points. [Complete specification of a twothe independent variables are often chosen as terminal concentrations or their equivalent. I n the case of a binary mixphase feed possessing C degrees of freedom requires specificature the choice of these two independent terminal concen1) variables, the extra variable being the overtion of (C
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May, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
trations obviously gives the complete compositions of the distillate and residue, and makes the design calculations easy and straightforward. However, in the case of multicomponent mixtures the problem is more complex, and in selecting these two terminal concentrations it is desirable to choose components that will give a significant control of the separation desired and, at the same time, be components that appear in appreciable amounts in both the bottoms and distillate. Because these controlling components are so important in determining the design calculations, they have been termed the “key Components”. I n other words, they are the key to the design problem. I n the development of design equations, it has been found convenient to pick two key components, the light key component and the heavy key component. The former is the most volatile component whose concentration it is desired to control in the bottoms, and the latter is the least volatile component whose concentration is specified in the distillate. Thus, in the stabilization of gasoline it is often desired to have only a small concentration of propane in the bottoms in order that the vapor pressure of the finished product will meet the desired specifications and also to limit the butane in the distillate so as to retain this component in the gasoline. I n such a case propane would be the light key component and butane the heavy key component. The terminal concentrations of the two key components are important because most of the practical equations which have been developed for the minimum number of theoretical plates a t total reflux, the optimum feed plate location, and the minimum reflux ratio have involved these concentrations. In fact, the real value of picking the key components is for their use in these design equations. However, certain difficulties are involved. First, the design specifications may be such that the key components are not obvious, and secondly, these design equations often ‘require the concentrations of both key components in the distillate and bottoms as well as the concentration of some of the other components; but as demonstrated in the foregoing analysis, only two of these terminal concentrations are independent and can be arbitrarily fixed as design conditions. The difficulties of choosing the key components and estimating the complete distillate aqd bottoms compositions are often the most difficult parts of a multicomponent design calculation. The problem can generally be simplified if the design conditions will be chosen with these difficulties in mind. Thus, in cases in which the separation between adjacent components is essentially complete, the two-independent variables can be chosen as the concentration of the more volatile of these two in the bottoms aqd as the concentration of the less volatile component in the distillate. These adjacent components then become the key components, and the composition of the distillate and bottoms can be determined completely enough for design calculations by simple material balance. Components more volatile than the light key component will be almost negligible in the bottom, and components heavier than the heavy key component will be negligible in the distillate. For rectifications in which there is an appreciable difference in volatility between adjacent components and in which a fairly high degree of separation is being carried out, the design condition can generally be specified in this manner and thereby simplify the problem. If the degree of separation is low, and/or there are several components of nearly the same volatility in the range in which the separation is being made, the selection of the two key terminal concentrations will generally not give enough information to allow the complete terminal compositions to be calculated by simple material balances. I n such a case it is necessary t o estimate the terminal concentration of the other distributed components and then check this estimation by proceeding with the usual
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stepwise plate-to-plate calculations. If such plate-to-plate calculations give a consistent over-all result, the estimated values are satisfactory; if the results are inconsistent, new values must be estimated and the calculation repeated. It should be emphasized that even in this latter case, although a large number of the terminal concentrations may not be known, only two of these are independent after the other usual variables have been fixed. All of the other terminal concentrations are fixed when these two independent ones are chosen and therefore cannot be given values arbitrarily. It is the necessity of having the concentrations of these other components that are fixed but not known, that offers the main difficulty in setting up a multicomponent distillation example.
Literature Cited (1) Brown, G.G.,and Martin, H. G., Trans. Am. Inst. Chem. Engrs., 35, 679 (1939). (2) Gilliland, E. R.,IND. ENQ.CEEIM., 27, 260 (1935). (3) Jenny, F.W.,Trans. Am. Imt. Chem. Enors., 35, 636 (1939). (4) Lewis and Randall, “Thermodynamics”, p, 186, New York. McGraw-Hill Book Co.,1923. (6) Lewis, W. K., Jr., IND. ENO.CHmM., 28, 399 (1936). (6) Perry, Chemical Engineers’ Handbook, 2nd ed., p. 1396, New York, McGraw-Hill Book Co.,1941. (7) Robinson and Gilliland, “Elements of Fractional Distillation”, p. 89, New York, McGraw-Hill Book Co., 1939. (8) Walker, Lewis, McAdams, and Gilliland, “Principles of Chemical Engineering”, p. 557, New York, McGraw-Hill Book Co.,1937.
Courtesy, E. B. Badecr & Sons Company
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