812
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 612-617
1 = dimensionless axial coordinate = z / L R 0 = cylindrical or spherical angular coordinate, rad 9 = auxiliary function defined by eq 12-17, dimensionless K = inner radius to outer radius ratio = rin/rou, dimensionless v = frequency, s-l
= spherical radial coordinate, cm 9 = spherical angular coordinate, rad Q = primary quantum yield, gmol einstein-' = dimensionless concentration = C/Ca, 7 = dimensionless reactor mean residence time = V / Q 0 = dimensionless reaction rate = LRRi/ (( v z )Cj)
p
+
Subscripts h = relative to species h i = relative to species i in = relative to the inner reactor wall L = relative to the lamp m = relative to the inert species ou = relative to the outer reactor wall R = relative to the reactor z = relative to the axial direction Y = denotes wavelength dependence Superscripts O = relative to the inlet section or inlet conditions ' = denotes reverse reaction Special Symbols ( ) = flow rate averaged property - = frequency-averaged property Registry No. CH,, 74-82-8; CH,CI, 74-87-3; CC14, 56-23-5.
Literature Cited Benson. S. W. "The Foundations of Chemical Kinetics"; McGraw-Hili: New York, 1960. Cassano, A. E.; Smith, J. M. AICM J . 1866. 72, 1124. Verbeke, G. Chem. CMltz, G.; Goldfinger, P.; Huybrechts, 0.;Martens, 0.; Rev. 1983, 4 0 , 355. Ciaria, M. A. Doctoral Dissertation, Universidad Nacional del Litoral, Santa Fe, Argentina, 1984. Clyne, M. A.; Stedman. D. H. Trans. F8fad~ySoc. 1868. 64, 2698. Dalnton, F. S.; Ayscough, P. B. I n "Photochemistry and Reaction Kinetics"; Ashmore, P. G., Dainton, F. S., Sudgen, T. M.,Eds.; Cambridge University Press: New York, 1967. Hlrschklnd, W. I n d . Eng. Chem. 1849. 47, 2749. Irazoqui, H. A.; Cerda, J.; Cassano, A. E. AIChE J . 1873, 79, 480. Kurtz, B. E. Ind. Eng. Chem. Prmess Des. D e v . 1972, 7 7 , 332. L h , C. S.; Ho, S. Y. Paper presented at the Proceedings of the NSC, China, 1978. p 4. Lowenheim, F. A.; Morand, M. K. "Faith, Keyes and Clark's Industrial Chemicals", 4th ed.;Wiley: New York, 1975. Mc Bee, E. T.; Hass, H. B.: Neher, C. M.; Strickland, H. Ind. Eng. Chem. 1842, 34, 298. Mc Ketta, J. J. "Encyclopedia of Chemical Processing and Design"; MarcelDekker: New York, 1979; Vol. 8. Noyes, R. M. J . Am. Chem. SOC. 1851, 73, 3039. Noyes, W. A.; Leighton, P. A. "The Photochemistry of Gases"; Reinhold: New York, 1941. Pease, R. N.; Walz, G. F. J . Am. Chem. SOC. 1831, 53, 3728. Romero, R. L.; Alfano. 0. M.; Marchetti, J. L.; Cassano, A. E. Chem. Eng. Sci. 1883, 38, 1593. Stramigioli, C.; Santarelli, F.; Foraboschi, F. P. Ing. Chim. Itai. 1975, 7 7 , 143. Stramigioll, C.; Santarelli, F.; Foraboschi, F. P. Appl. Sci. Res. 1977, 33, 23. Temkin. M. 1. Int. J . Chem. Eng. 1871, 1 7 , 709. Yuster, S.; Reyerson, L. H. J . Phys. Chem. 1935, 39, 859.
Received for review August 10, 1984 Revised manuscript received March 25, 1985 Accepted April 19, 1985
Structural Model for the Optimal Geometrical Design of Packed Distillation Towers Tom68 R. Yelll,t Zvonko Spekuljak,' and R a m h L. Cerro't Instltuto de Desarrollo y Diseiro-INQAR, Avellenede 3657, 3000-Santa
Fe, Argentina
Using a simplified structural model, a criterion is developed to optimize the design of a packed distitation tower from a geometrical point of view. A structural model, SM, Is a digraph where nodes represent the elements of the system and the arcs the information transmitted from one vertex to another. Emphasis is on the descriptive nature of the relationships, rather than on the quantitative aspects of it. Analysis Is focused on two cases: a constant diameter case where floodlng conditions vary along the column, and a constant flooding percentage rate, of ideal performance but of very difficuk realization. An example is developed to show a typical situation of tower diameter optimization.
The design of vacuum distillation systems involves a large number of variables and operating parameters. The size of the distillation system is associated with investment, and energy consumption is associated with operating costa. Thus, the optimum design of a distillation system will be defied as the choice of variables and operating parameters such as to minimize production costs. The purpose of our analysis is to show the links among the relevant parameters which will have an impact upon the total volume of the instalation, and consequently on
* To whom correspondence should be sent. Fellow of CONICET.
0 196-4305/86/1125-06 12$01 5010
the fixed capital investment, once the main variables such as head operating pressure, reflux ratio, etc., have been decided. The particular cases of isotopic or isomeric separation where relative volatilities are small are most affected by vacuum operation due to the need to increase the relative volatilities and to use a large number of theoretical plates. Decreasing pressure is a way to increase relative volatilities which are highly dependent on absolute pressure below atmospheric pressure. However, a large number of theoretical plates means large pressure drops and consequently large pressure variations along the column. As a consequence, a particular design that can be considered geometrically optimum at the top of the column 0 1986 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
613
Table I. List of Functions and Variables
Figure 1. Structural model for packed distillation column design: constant diameter.
drops in efficiency along the intermediate and bottom sections of the equipment. The need to define a geometrical optimum for the entire length or when to change the design for different sections of the distillation tower is addressed in this paper. This problem was first discussed by some of the authors (Melli and Spekuljak, 1983) and focused on an isotope separation problem.
I. Structural Model of a Packed Distillation Tower Structural modeling is a fast growing area of system engineering where qualitative and conceptual relationships are developed between the different variables affecting a problem, by means of a directed, signed, weighted graph. The emphasis is on structure and qualitative dependence rather than on quantitative, numerical responses (Lendaris, 1980). Although parametric simulations could give us precise, quantitative answers, meaning would be obscured by the intrincate structure and the large number of variables. A structural model of a packed distillation tower gives us a clear picture of how the system variables interact and what is the expected behavior for a number of different situations. Thus, the view we are able to get from the problem facilitates the analysis and is useful for the beginner as well as for the expert. From the comprehensive review of Bolles and Fair (1979), we can select the fundamental equations to calculate the height equivalent of a theoretical plate (HETP) for a variety of packings of widespread commercial use. Moreover, we will be using the practical approach to operation conditions at about 70%-80% flow rate from flooding. However, choosing the operating conditions in this range at the top of a vacuum distillation tower means that the intermediate and lower parts of it will be working at rates considerably below flooding. If we choose to have a vacuum distillation tower with a uniform design from top to bottom, we will have a wide
range of operating conditions along it. On the other hand, should we choose to follow the optimum flooding rate, we will have the need to change the diameter and/or the packing characteristics along the tower. Obviously, this last approach will not be a practical one. An added feature is the fact that HETP becomes smaller as pressure increases along the tower, if we manage to keep the flooding rate constant. However, the average relative volatility also decreases as a result of increased pressure. There is not an easy way to imagine how variables and operating conditions interact in order to develop simple, reliable rules for design of vacuum distillation systems unless we seek for the help of a structural model. Such a structural model-or SM-is shown in Figure 1 for a given set of design equations, where a sequence of resolution of the equations has been adopted in order to have a directed graph. For this particular SM, the pressure gradients are computed according to Bremer and Kalis (1978),and the mass-transfer rates are computed according to Cornel1 et al. (1960). The procedure is developed to a greater extent in the Appendix section. Table I shows a list of the equations and relationships taken into account for the construction of the model, with the output set indicated (Lee et al., 1966). The sign of the partial derivative of the function joining two adjacent variables is used to determine the sign of the arcs joining these variables in the structural model as indicated in Figures 3 and 5. Although not explicitly identified, each one of the arcs joining the two variables is generated by a relationship or equation from Table I. In this sense, there is a unique relationship between our structural model and the bipartite graph representing the system of equations. One of the interesting featurs of this structural model is the hierarchical structure ordered in 11different levels (Warfield, 1974). The systematic ordering of the variables
614
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in different levels has two meanings. On one hand, the flow of information proceeds from the bottom to the top of the graph; thus, variations of parameters placed at level ‘‘7do not affect variables and parameters placed at levels “ j + 1”and larger. On the other hand, the adjacency and reachability matrices can be ordered to be lower diagonal or quasi-lower diagonal if maximal cycles arise. This last result should come as no surprise since the output set from the list of equations of Table I has already been chosen in order to generate a computational sequence with no interactions. Another point to be stressed about our structural model is the possibility of taking into account the nature of the relationships between variables. The particular model shown in Figure 1was developed for a given type and size of packing. Moreover, our analysis is focused on the separation of systems with low relative volatility. This type of system demands a large number of separation stages and low-pressure operation, as is the case for the separation of isotopes or chemically similar substances. Finally, since we are concerned about chemically similar substances, an additional hypothesis is introduced that physicochemical and transport properties do not vary widely with compositions along the separation column. Column diameter, D, and absolute pressure at the top of the column, Po,are the design variables to be specified. Feed rates, QG and QL,are the operating parameters and the total column volume, AVTOT,is the design parameter to be computed and also a measure of the investment costs. Two important cases will be analyzed: (a) traditional design procedure keeping constant the column diameter for all the length of the column; (b) Ideal most-efficient design procedure keeping constant the percent of flooding rate, F1 %, along the column. In both cases, the effect of simplifying assumptions, as well as the approximation of some of the parameters on the overall design, will be considered. 11. Constant Column Diameter for All the Length of the Column Without further simplifications, our structural model is the one shown in Figure 1. The 11 levels shown in the figure are connected and cycleless. In this sense, computations can be carried such as to compute first the lower levels and make our way up in the hierarchy. There is no need to compute all the items of a given level before moving up to the next level. The only requirement to be able to compute a variable is to have computed before all its antecessor. Next, we introduce the assumption that for the range of variations of P and consequently T found during our isotopic or isomeric distillations, the values of Schmidt numbers of the liquid and gas phase are nearly constant. Moreover, for the liquid phase, all properties are nearly constant. Under this assumption, all the physicochemical properties of both fluids can be considered nearly Constant, except for the gas density, pr Moreover, 2, the packing height, is a function of column diameter so it can be taken also as a constant. Finally, A, the ratio between the operating line and the equilibrium line which is a function of both gas and liquid superficial mass flow rates and of the relative volatility, remains also without change due to interacting effects as shown in the signed digraph. Under these simplifications, our digraph is reduced to the one shown in Figure 2. The next step is to detect the weaker relationships and assume that these can also be eliminated. The function $, defined by eq 5 of Table I, is a very weak function of F1 %; thus, for our purposes, it can be assumed constant.
W
Figure 2. Structural model for packed distillation column design: constant diameter, D ;liquid flow rate, QL, gas flow rate, Q G ; nearly constant Schmidt numbers, SCGand Sc.,
Moreover, the effect of pressure, P, is much larger than the effect of temperature, T , on the height of an equivalent theoretical plate, HETP; thus, it can also be taken as approximately constant. Under all the above assumptions, the structural model has been reduced to the one shown in Figure 3. From an analysis of the surviving relationships (kl,j = 1,2,etc., indicates a constant), HG k, (1) HL kzC, (2)
k3 k4 + kbHL = k4
(3)
X
HETP
+ k,C,
(4)
but
[
1
= O(1)
(6)
- k,(
Then, AFJ k7 ePG
(7)
We can deduce the following behavior as we increase the operating pressure, P, from the top of the column to its bottom: (a) HETP increases; (b) APlplate may increase or decrease depending on the particular situation; (c) a, the relative volatility, decreases with pressure; (d) VTOT, the specific volume of the column (taken as specific volume for a fixed separation), consequently increases.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 615
Figure 4. Structural model for packed distillation column design: constant flooding percentage, F1 %. Figure 3. Structural model for packed distillation column design: constant diameter, D; nearly constant relative volatility a. The pressure effect is stronger than the temperature dependence.
111. Constant Percentage of Flooding Rate Along the Column For a constant flooding percentage, F1%, Table I shows the output set used to build the structural model shown in Figure 4. Using the same assumptions as before, we can neglect the variations on the Schmidt number and physicochemical properties to further simplify the model. Finally, assuming the weak dependence of the gas flow parameter, 4, with the liquid flow rate, GL, and a stronger dependence of gas density with pressure than with temperature, our model reduces to the one shown in Figure 5. It is of interest now, to rescue the relationships surviving from our simplifications: HG
N
HL
HETP
N
= k1@(m+2n)
ks( $)m/k,G,"
k1zHG
N
i
/
(8)
(9)
kll
+ k13H~
I
+ k15
k14D(m+2n)
(10)
v
On the other hand, according to Melli et al. (1983), G F ~= & - P G O . ~
(11)
Go,= (F1 9O)GFI
(12)
1
- = k1,pG0.5
Figure 5. Structural model for packed distillation column design: constant flooding percentage, F1 70. The pressure effect is stronger than the temperature dependence of 6 on GL.
(13)
increase or decrease, (c) a,the relative volatility, decreases, and (d) AVTOT, the specific column volume, may increase or decrease.
(14) H As P is increasing from the top to the bottom of the column, (a) HETP diminishes, (b) the APlplate may again
IV. Optimum Geometrical Design of a Packed Distillation Column For a constant diameter case, D constant, while the operating pressure, P, is increased from the top to the
A
Thus,
AP - N k18(1 - k19~G1/3)-5
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Table 11. Distillation of D20/H20 Mixture, with the Same Total Tower Volume (Different Separation). Case 1: D Constant. Case 2: F1 70 Constant variable ~~
A, m 2 case 1 case 2
0.527 0.496
-
HETP, m 0.960 0.685
PT, kPa 20 20
Pe, kPa
P , kPa
XT
XB
x
e
36 40
28 30
0.03 0.03
0.166 0.361
0.098 0.195
1.0460 1.0455
bottom of the column, the HETP increases, but (Y and APIH both decrease. For the constant flooding rate, F1 % constant, the h P / H grows from the top to the bottom of the column, the HETP decreases, and there is a stronger rate of reduction of a. If we use the pressure drop per height of a theoretical plate as a design criterion, both expressions are for the constant diameter case
for the constant Flooding percentage case
Table 111. Comparison of Three Different Cases for Different Total Volume (Same Separation) criterion vol, m3 V/Vrmin 35.58 3.435 case 1 D const 1 F1 7’0 const 10.36 case 2 16.80 1.620 case 3 3 diff. col. sections
column diameter, and u < 1,optimization is not convenient, D = constant. When we are aiming at a defined separation specification, i.e., a fixed S (Sl/S2= l),the criterion will be -VTOT, - --HETPIAl In a2= u (23) ~ T O T ~HETP2A2 1n &I Thus, the following relationship can be used
Comparison of these two functions is difficult due to the difference in geometry of the columns. It will be possible to use the comparison of the values of 3, for the case of a same column with different packings. Consequently, a better criteria can be developed with the insight we got from our structural model. According to Fenske’s equation,
where defining the degree of separation
we can write
The bar above the terms of (19) indicates average values computed by using the following definition:
Comparing now the total volume, VTOT, for both the constant and the constant F1 %, indicated as cases 1 and 2, respectively, In az -VToT1 - - SIHETPIAl (21) VToTz S2HETP2A2In a1 For equal total volume, that is, when the ratio vTOT,/ = 1, the ratio of the separation achieved can be computed as S2 - HETPIAl In a2 _ -= v S1 HETPzA2In al VTOT2
where if Y = 1, the design is indifferent to either case, u > 1, it will be convenient to optimize the design changing
Calculation of parameters included in (22) and (23) is straightforward. For the isotopic separation of heavy water, we can show the use of this simplified criterion as a way to decide beforehand how optimization of a distillation system can be achieved. Table I1 shows the relevant parameters and design variables for the case of a 3% mixture of heavy water in natural water at the top of the column. For the example, a 45 m height and 0.819 m diameter column was taken as the constant diameter case. Using the values from Table 11, we can easily compute the design criterion defined by (22) and (23): u z 1.47 For a very similar case, more accurate computer simulations gave a value of u z 1.54 (Melli and Vecchietti, 1981; Melli, 1981). This result indicates that a more efficient separation could be achieved for the same column total volume, if we optimize the column size by using a constant flooding percentage criterion. As it was shown by Melli and Spekuljak (1983), this is not an easy task, but it is more efficient from an operating standpoint. Thus, we resort to a piecewise constant diameter column with flooding percentage maintained between 70% and 80%. Table I11 shows a comparison of total column volumes for different design criteria for a mixture of heavy water ranging from 3% to 96% in comparison. In this case, different volumes at a constant value of S are compared. For a third case, we could use different column sections as indicated by Melli and Spekuljak (1983). Conclusions The geometrical-shape optimization of packed distillation towers is analyzed, using the typical cases of constant diameter design and constant flooding percentage design to compare the relative performances. A criterion, u, defined by (22) and (23), was developed in order to rate the advantages of each approach. For values of u I 1, there should be no economical incentive to optimize the tower diameter, and a constant D could be found as a satisfying solution. For values of v > 1, the greater the value, the more the incentive to change the diameter from the top to the bottom of the column in order to keep a value of the flooding percentage, F1 % , nearly constant. The value
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986 617
of Y is easy to evaluate and may be used during the first steps of design, in order to decide whether to pursue or not a geometrical optimization of the column diameter. If a geometrical optimization is worthwhile, the final design should be a compromise between the operational optimum, i.e., a column where the diameter changes continuously, and practical considerations for chemical construction (columns of different diameter or in some cases different shells). Moreover, it should be possible to change the packing characteristics in order to achieve the type of results we are looking for without complicating too much of the construction of the tower. What is really important in this approach is to increase the degree of understanding of a very complicated design situation by means of a simple, systematic procedure. The case presented in this paper, due to the simplifications introduced, allows a graphical, visual treatment of the problem. For more complicated cases, the structural model, a graph, will be too complicated even to construct. Thus, in these cases, we can use the image of the graph in a computer memory and work with the resulting binary image relationships. Much can be gained from this approach for every kind of design procedure, helping the novice to understand the ways of the expert and allowing the expert to look for more rational and better ways to design.
Acknowledgment We acknowledge the support of the National Research Council of Argentina, CONICET, through the research program for the Development of Regular Packings for Heat and Mass Transfer. Nomenclature A = transversal area of the tower C, = packing parameter (flooding % function) DL, DG = diffusivity, liquid and gas D = tower diameter F,, F,,F3 = correction factors (in eq 2 ) G = mass flow per unit area H = total height of the tower HG = height of the transfer unit, gas phase HL = height of the transfer unit, liquid phase HETP = height equivalent theoretical plate Np = number of theoretical stages P = pressure PM = molecular weight A P / H = pressure drop/unit height Qmol = molar flow S = separation SCG, ScL = Schmidt number, gas and liquid phase T = temperature X = molar fraction 2 = packed bed height Greek Letters a = relative volatility X = ratio operation line to equilibrium line p = viscosity v = ratio &/SI p = density I$ = parameter (GL function) rl, = parameter u = surface tension il = pressure drop per HETP Subscripts A, B = A, B components of the mixture G = referred to gas L = referred to liquid i = referred to i component F1 = flood op = operation
w = referred to water
Superscripts B = bottom T = top
Appendix Equations of Model. (a) Pressure Drop, according to Bremmer and Kalis (1978). The wet pressure drop related the dry pressure drop is 1
where e = porosity of bed, h = liquid holdup, X = constriction factor (0.435 for Raschig rings and 0.485 for Pall rings), 4 = fraction of the packing by which the gas passes (0.6 for Raschig rings and 0.8 for Pall rings). The liquid holdup is calculated by
where a = specific area of packing. The dry pressure drop is calculated by Po 0.29pgu;a
--
-
H 42t3 and, finally, the wet pressure drop is obtained by 0.29pgv2a
(b) Mass Transfer, according to the Monsanto Model (Cornel1 et al., 1960). The height equivalent to a theoretical plate (HETP) is
The height of the vapor-phase transfer unit is given by ~(sC,)0.5(o/i2)m(2/10)1/3 HC = (FlF83GL) where = packing parameter = f (packing type, size F1 %), m = 1.24 for rings and 1.11for saddles, n = 0.6 for rings and 0.5 for saddles, Fl = (uL/uW)O.l6, F2 = ( ~ ~ / p ~ ) - ' , ~ , F3 = ( Q L / u W ) + . ~ . The height of a liquid-phase transfer unit is given by HL = I $ C , ( S C L ) ~ ~ ~ ( Z / ~ O ) ~ . ~ ~
+
where 4 = packing parameter = f (packing, type, size, GL), c p = f[l/(GOP/FFl)] = f(l/Fl Literature Cited Bolles, W.; Fair, J. Inst. Chem. Eng. Symp. Ser. 1979, No. 56, 33-35. Bremer. 0.; Kalis, G. Trans. Ind. Chem. Eng. 1978, 56, 200. Cornell, D.; Knepp. W. 0.;Fair, J. R. Chem. Eng. Prog. 1960, 56, 8. Lee, W. J.; Christensen. H.; Rudd, D. F. AIChE J . 1886, 10, 1104. Lendaris. 0.IEEE Trans. Syst. Man Cyber. 1980, SMC-10.807. Melii, T. "Programa de Dieefio Interactivo de Columnas Rellenas para Destiiacibn DICOLR"; INGAR: Santa Fe. Argentina, 1981; MO/RES/006. Meill, T.; Vecchletti, A. "Balances de Materia y Enerila PDF"; INGAR: Santa Fe, Argentina, April 1981; PAII1025. Melli. T.; Spekuljak, 2. I d . Eng. Chem. Process Des. D e v . 1983, 22, 230-236. Warfield, J. N. IEEE Trans. Syst. Man Cyber. 1974, SMC-4, 405.
Received for review August 28, 1984 Revised manuscript received July 1, 1985 Accepted August 9, 1985
