Ind. Eng. Chem. Process Des. Dev., Vol. 17, NO. 4, 1978 505
The Design of Adiabatic Packed Towers for Gas Absorption and Stripping Howard M. Felntuch’ and Robert E. Treybal New York University, New York, New York
A computer solution to the problem of design of adiabatic packed-tower absorbers and strippers for simple systems as well as for multicomponent systems is developed. I n the latter case, distribution of all components between liquid and gas phases is assumed. A procedure is also developed for estimating the minimum liquidlgas ratio. For multicomponent systems, an approximate short-cut method based on Edmister’s “effective” absorption and stripping factors, suitable for hand calculation, is developed.
Introduction Theoretical investigations in the field of adiabatic gas absorption and stripping in packed towers have recently been made. These investigations are significant in that they have led to the development of several new methods which enable one to perform design calculations which previously could not be carried out. Especially significant is the development of a rigorous method of calculation to handle complex, multicomponent, adiabatic systems. This rigorous method is unique in that it allows a packed tower to be described properly as a true differential device without resorting to a stage-wise, tray-tower analogy. In addition to the rigorous method for multicomponent systems, which requires a computer solution, a short-cut method which can be carried out as a hand calculation and which gives reasonably close results to the rigorous method was also developed. Before the study of the problems involved in developing a calculation method to handle complex, multicomponent, adiabatic systems could begin, it was first necessary thoroughly to study simple, three-component, adiabatic systems. This was accomplished through the development of a computer program which was used to study these systems. The program was based upon a previous developed method but contained certain improvements to that method. Using published experimental data for several simple, three-component, adiabatic systems, a comparison was made between the experimental results and the results calculated through the use of the computer program. From this comparison, it was observed that the calculated height of packing is quite sensitive to temperature and that any nonadiabatic conditions will greatly affect the calculated height of packing. The calculation method did not yield a very close check for any of the systems studied due to the fact that the experimental data proved to be taken under nonadiabatic conditions. The fact that the calculation method did, however, yield approximately the same height of packing for each case under each system shows that the calculation method is properly describing the system under conditions of different flows and recoveries. For simple, three-component, adiabatic systems a method was developed which allows one to calculate the *To whom inquiries should be addressed. H.M.F. is presently with Foster Wheeler Energy Corporation, Livingston, N. J. 07039, and R.E.T. is with the University of Rhode Island, Kingston, R.I. 02881. 0019-7882/78/1117-0505$01.00/0
minimum liquid-to-gas ratio required to meet a given specification on a “key” component. Background A thorough survey of the literature indicated that a good start had been made in the field of adiabatic gas absorption and stripping in packed towers, but that there was still much need for further research in the field. Sherwood and Pigford (1952) had made the first attempt to account for heat effects in packed tower design, yet their approach contained too many simplifying assumptions. Holland and his students (1969,1970) did not advance the classical theory but instead attempted to draw analogies to tray-type towers. Treybal (1969), by accounting for the mass-and heat-transfer resistances of the liquid phase added a new dimension to the field. Nevertheless, there were many important questions concerning his method left unanswered. How could the method, which was suitable for simple, three-component systems, be extended for use on multicomponent systems? How could the method be used to calculate the minimum L / G ratio for simple, three-component systems? How well does the method compare to experimental data? If the method could be extended to multicomponent systems, could it then be somehow simplified into a shortcut calculation method? Because it seemed so essential that the questions posed above be answered, the research described below was initiated. Theory 1. Simple, Three-Component, Adiabatic Systems. A simple system which may be easily visualized and for which the heat- and mass-transfer calculations can be performed with some degree of simplification is that which we shall refer to here as a “simple, three-component, adiabatic system.” In general, a three-component system can be visualized as being either complex or simple. In a complex system, the simplifications described below do not apply and the three-component system must be handled the same way as any complex, multicomponent system. A simple, three-component system consists of an insoluble carrier gas, a volatile solvent, and a singlecomponent solute. The simplification lies in the fact that the carrier gas is considered to be insoluble and that both the volatile solvent and the solute component consist only of a single species. Yet, even for this type of system, the calculations are so extraordinarily complicated and tedious that it was only recently (Treybal, 1969) that one was able to solve, for the first time, a problem in which the mass-and heat-transfer resistances for both the liquid and the gas-phases are taken into account. All previous work had 0 1978 American Chemical Society
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Ind. Eng. Chem. Process
Des. Dev., Vol.
17, No. 4, 1978
L +dL ITIIAYi
,lOT.L
L,+dL, t.+dL, L,;dL, x.
+ dx,
x,+dxr T,+dT,
INTERFACE SURFACE
G B as used earlier now no longer represents the constant flow of a special insoluble component, but instead represents a typical component in the system which is soluble and which may appear to a large or a small extent in any of the streams which are present. The notation Y as used earlier for a simple, threecomponent system to relate gas composition to the carrier gas can now no longer be used. Instead, each component now has associated with it a mass velocity GJ which will vary for each component as it flows through the tower. Each component will have associated with it a nonzero flux of mass transfer NJ and all components will be present to some extent in all streams as well as at the interface. For each component present within the differential section, we can write an equation for its flux of mass transfer NJ as
NladZ= RjFLj [In -RJ] a- x J RJ - x J i
Figure 1. Differential section of absorber or stripper for a multicomponent, adiabatic system.
neglected the mass- and heat-transfer resistances for the liquid phase. 2. Complex Multicomponent, Adiabatic Systems. We are using the term "complex, multicomponent, adiabatic system" to denote one in which the carrier gas, volatile solvent, and solute may all consist of more than one component. In such a system, we really no longer need to consider the presence of an insoluble carrier gas and, indeed, there is no limit as to how many components are present. It is not necessary to visualize a component as being part of the carrier gas, solvent, or solute since all components are interrelated and w ill appear to some extent throughout the bulk streams as well as the interface. For this type of complex system, no method of solving problems had heretofore been devised. The method developed here, which is described later, was used successfully to deal with complex, multicomponent systems consisting of as many as 14 components; indeed, the method can be used for any number of components. In this method, the mass-and heat-transfer resistances for both the liquid and the gas phases are fully taken into account. 3. Differential Section of Packing for Complex, Multicomponent, Adiabatic System. Figure 1 represents a typical differential packed section of a tower for a complex, multicomponent, adiabatic system consisting of J components. It is similar to the figure developed earlier (Treybal, 1969) for a simple, three-component, adiabatic system, with the exception that there is now no limit as to how many components can be handled in the calculations. In addition, there is no longer a requirement that there be an insoluble carrier gas. Thus, the notation
dZ
( J = A , B , C, ..., J) (2)
The quantities a (specific interfacial area) and FGj and FLI (gas- and liquid-phase mass-transfer coefficients) must be known. Typically they may be estimated by methods previously described (Treybal, 1968, 1969). We are now confronted with a major problem. R j which appears in each equation is used to represent the ratio of Nj, the flux of mass transfer for a particular component, to NA + NB + Nc, ..., N J , the sum of the fluxes of mass transfer of all of the components present in the system. For a simple, three-component system, a method was devised whereby the two unknown values for Rj, RA for the solute and Rs for the solvent, could be found by a trial-and-error procedure. This procedure took advantage of the fact that RA + Rs = 1.0, so that if a value of RA were selected, the corresponding value of Rs would then simply be equal t9 1.0 - RA. The correct value of RA and its corresponding value of Rs was then later determined as the value for which the sum of the liquid mole fractions xAi and xsi were equal to 1.0. If we now consider our present system of J components, the problem of finding the correct individual values of R j is much more difficult. The sum of the individual values of R j is still equal to 1.0; thus RA + RB + Rc, ..., R j = 1.0. In this case, however, if a value of RA is selected, it is not obvious how to select the values of RB, Rc, ..., Rj. If we simply select a set of values arbitrarily and then we find does not add to 1.0, how do we proceed to that zJ"=ixJi select new values of RJ so that C S I ~ Xdoes J ~add to 1.0? The solution to this problem, for a multicomponent system, is certainly not obvious. A solution, however, was found and is described later. Development of Computer Program for Simple, Three-Component, Adiabatic System A computer program was developed, based upon Treybal's (1969) method, which could handle simple, three-component, adiabatic systems. In developing this computer program, certain improvements to the original calculation method were made. A consistent enthalpy basis was established for both the overall heat balance on the bulk fluid streams and for the heat balance between the bulk fluid streams and the interface. The original method depended upon integral heat of solution for the overall heat balance and partial molar enthalpies for the interface heat balance. The computer program was set up
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978 507
to use partial molar enthalpies for both heat balances. The computer program could also evaluate all physical properties a t the average temperature between the bulk vapor or liquid stream and the interface. Care was taken to select the best correlations available for generating these physical properties both for individual components and for mixture properties. Details of these correlations are given elsewhere (Feintuch, 1973). A summary of the various equations used to describe the system is also given. A 20-step calculation procedure involving eight steps of trial and error type calculations was inccirporated into the program. Listed below is the outline of this calculation procedure. In addition to certain physical property data, the following information must be submitted to the program as input data: inlet gas-mass velocity, temperature, composition; inlet liquid-mass velocity, temperature, composition; components-name, type, properties, solute component specification; miscellaneouscolumn pressure, selected physical property options, number of packing increments, maxiimum number of overall iterations, and desired printout option. The procedure is as follows: (1)Read in all known input data. (2) Calculate the enthalpy of the inlet gas and the inlet liquid streams. (3) Calculate the quantity of solute in the outlet gas. (4) Assume the temlperature and the solvent concentration of the outlet gas. (5) Calculate the enthalpy of the outlet gas. (6) Calculate the quantity, composition, and enthalpy of the outlet liquid by material and enthalpy balance. Calculate the outlet liquid temperature by trial and error. (7) Assume an interface temperature. (8) Calculate all the physical properties at the average temperature between the bulk liquid or bulk gas stream and the interface. (9) Assume RA and calculate the corresponding Rs. (10) Calculate D h , , DSm,SCGA,SCGS, F G A , FGS, SCL, FL,h G , and hL. (11) By trial and error, determine the point of intersection between eq 3 and 4 and determine the point of intersection between eq 5 and 6.
ysi = function (xsi, Psi, K , Ti) (6) (12) If xAi + xsi = 1.0, proceed to step 13; otherwise, return to step 9 for new assumption of RA.- (13) Calculate h'Ga, dTG/dZ, dYA/dZ, dYs/dZ, HAi and Hsi. (14) Calculate the interface temperature and compare it to the value assumed in step 7. If it agrees, proceed to step 15; otherwise, return to step 7 for new assumption. (15) Calculate the value of AYA based upon the number of packing increments specified in the input data. (16) Proceed to the next level of packing and calculate the following znext = +A z = AYA/(dYA/dz)
z
z
+ AYs = Ys + AZ(dYs/dZ) TG,next = TG + A z (dTG/dZ) Lnext= L + GB(AYA + AYs)
Ys,next= Ys
XA,next
HL,next
=
= (GBAYA
[(LHL)top
+ LXA)/Lnext
+ GB(HG,next
- HG,top)l/Lnext
(17) Find TL,next by trial and error. (18) Calculate the dew point temperature of the gas entering the next level of
packing by trial and error. (19) Compare YA,next to the specified outlet quantity, to determine if the top of the packing has been reached. If it has not been reached, repeat steps 7-18 for the next level of packing. If it has ' and~ been reached, proceed to step 20. (20) Compare 2 to the assumptions made in step 4 for the outlet gas. YS,next If the assumptions are okay, the calculation is terminated. If the assumptions are not okay, return to step 4 for an improved set of assumptions and renew the calculation. The calculation procedure described above includes many items which must be found by trial and error. For each of these items, the program has built into it a closure technique by which consistently better assumptions of the unknown variable are made, until the correct value is found as the one which is able to pass a test for closure to within a given tolerance. In addition, provision was made within the closure technique to ensure that if, for some reason, the correct value cannot be found, the attempt to find it does not continue on indefinitely. The details of these closure techniques are given elsewhere (Feintuch, 1973). Within the calculation procedure, there are several equations which could, under certain conditions, become mathematically indeterminate. These discontinuities are usually encountered in the middle of an iterative loop, where a particular assumed value may be a choice which is far from the correct value. By the time the correct value is found, the discontinuity will in most cases disappear. It is important, however, to avoid these discontinuities before they occur, since they will, in all cases, cause the program immediately to abandon the entire problem, without ever clearly indicating where the discontinuity occurred. Thus for every equation in which there is the possibility of a mathematical discontinuity, it is necessary that a test be made prior to each use of the equation to make sure that the current numerical values of the variables being used will not cause a discontinuity. If it is found that a discontinuity will occur, then provision must be made either temporarily to bypass the equation or to change the numerical value of the variables which cause the discontinuity. The problem that was used to test the computer program during its initial stages of development was the same problem used earlier (Treybal, 1969). This problem involves the absorption of ammonia by liquid water from a gas containing ammonia and air in a tower packed with 11/2-in.Berl saddles. To study the effect of the total height of packing vs. the number of packing increments specified in the input data to the problem, the problem was run several times a t a different number of packing increments. The results are plotted in Figure 2. We can see that as we use more packing increments, the calculated total height of packing increases with the curve asymptotically approaching a constant value of packing height equal to approximately 3.5 ft. If one compares the solution obtained by hand as carried out by Treybal (1969) to the computer solution which is given here, it can be seen that the agreement, in terms of calculated packing height, is not very close. In the hand calculation, 5.2 ft of packing was obtained by computer. The difference can be accounted for, however, by the following reasons: (1)The physical data correlations and interpolation methods used by the computer program were much more rigorously determined than it is possible to do in a hand solution. Especially important is the interpolation of vapor-liquid equilibrium data. (2) The method used in the computer program for doing heat balances between the interface and the bulk fluid streams, on a
~
~
~
508
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Table I. Calculated vs. Experimental Data. Acetone-Air-Water ~~
case Othmer-Schiebel run no. gas temperature in, O F gas temperature out, "F liquid temperature in, F liquid temperature out, F acetone recovery, % packing height, ft
2b 10
la
5 exptl 58.8 72.0 54.5 69.3 92.55 16.29
3= 21
calcd
exptl
calcd
exptl
calcd
58.8 55.6 54.5 62.9 92.55 8.96
61.7 12.0 51.6 68.3 94.85 16.29
61.7 57.9 57.6 60.3 95.85 8.81
56.8 71.6 55.6 66.4 97.79 16.29
56.8 55.7 55.6 58.6 97.79 8.05
Inlet gas: 15.65 mol/h ft2, 3.00 mol % acetone; inlet liquid: 38.50 mol/h ft'. Inlet gas: 11.90 mol/h ft2, 1.29 mol % acetone; inlet liquid: 34.50 mol/h ft2. Inlet gas: 10.05 mol/h ft', 1.56 mol % acetone; inlet liquid: 36.00 mol/h ft'. 5.0
4.0
L L
1'0
b
1'0
1.0
0
10
la
m
4a
sa
m
NUMBER 01 P A C K I N G INCREMENTS
Figure 2. Total height of packing vs. number of packing increments.
consistent basis of partial molar enthalpy, is an improvement, which is different from the method used in the hand calculation. (3) The closure tolerance used by the computer for trial and error solutions is much more accurate than it is possible to achieve in a hand calculation. (4) The computer program solved the problem to complete closure, as compared to the hand solution which was completed for only one iteration. It seems apparent that if the hand-calculated solution could be repeated using the same accuracy as was used by the computer solution, the same answer would be obtained. Comparison with Experimental D a t a Using the computer program which had been developed for simple, three-component, adiabatic systems, a comparison between calculated results and experimental data was made for the following systems: (1)HC1-air-water, (2) acetone-air-water, and (3) ammonia-air-water. Each of these systems is discussed below. (1) HC1-Air-Water. The first system that was investigated was that of HC1-air-water, using the data of Oldershaw et al. (1947). This was thought to be a good system to study since the heat effects involved are very large. Unfortunately, the data given in the paper were very incomplete. Information was missing on certain flow rates, temperature, and compositions. Reasonable assumptions were made for the missing data. It was found, however, that for this system, due to the large heat effects involved, the answers obtained were highly sensitive to the assumptions made. Since it was apparent that the solution obtained for this system is so sensitive to the values which had to be assumed, due to the incompleteness of the published data, it was decided that it would be pointless to continue
further study of this system since it could not possibly lead to any really meaningful comparison between the calculated and experimental values. (2) Acetone-Air-Water. The next system studied was acetone-air-water using the data of Othmer and Schiebel (1941). These data were much more complete than that used for the HC1-air-water system; however, these data, too, had several shortcomings. A heat balance check of the data showed that they were not obtained under truly adiabatic conditions, also noted by the authors in their paper. In addition, the water content of the gas in and out was not measured by the observers and had to be assumed for the purpose of carrying out the calculation. A summary of the results is shown in Table I. It can be seen from Table I that the experimental results, which were not taken under truly adiabatic conditions, give temperatures which are much higher than those calculated adiabatically. The colder calculated temperatures, with their resulting easier absorption capability, can account for the fact that, in each case, the calculated packing height is roughly half of the experimental value. Despite the fact that we did not get a close check, the fact that the calculation method yields approximately the same height of packing for each case shows that the calculation method is properly describing the system under conditions of different flows and recoveries. After the runs summarized above were made, additional computer runs were made to investigate the sensitivity of the calculated height of packing to nonadiabatic heat effects. These runs proved that the calculated height of packing is quite sensitive to temperature and that any nonadiabatic conditions will greatly affect the calculated height of packing. Especially important was a run which was made in which the outlet liquid and the outlet gas temperatures were adjusted to match the experimental data. When this was done, the calculated height of packing of 14.69 f t compared favorably well with the experimental value of 16.29 ft. (3) Ammonia-Air-Water. The last system studied was ammonia-air-water using the data of Dwyer (1941). This system gave a much better comparison between calculated and experimental results than the other systems studied. The material balance data reported by Dwyer (1941) were exceedingly accurate, often in error by less than 1%. For this system too, however, a heat balance check of the data showed that the tower was not operated under truly adiabatic conditions. A summary of the results is shown in Table 11. Table 11, in a manner similar to Table I, shows that the experimental data were not taken under truly adiabatic conditions and that the resulting cooler absorption temperatures result in a calculated height of packing smaller than the experimental value. Recently, Raal and Khurana (1973) ran some experimental tests on the ammonia-air-water system and
Ind. Eng. Chern. Process Des. Dev., Vol. 17, No. 4, 1978 509 Table 11. Calculated vs. Experimental Data. Ammonia-Air-Water
gas temperature in, ’F gas temperature out, F liquid temperature in, “F liquid temperature out, F ammonia recovery, % packing height, ft
26 52
la
case Dwyer run no.
51
3c 53
exptl
calcd
exptl
calcd
exptl
calcd
82.5 87.8 88.0 77.5 79.5 4.00
82.5 85.1 88.0 71.4 79.5 3.01
82.5 88.2 88.0 80.2 86.8 4.00
82.5 86.6 88.0 75.2 86.8 2.85
82.5 89.7 89.0 83.0 89.9 4.00
82.5 88.0 89.0 78.6 89.9 2.43
* Inlet gas: 18.21 mol/h ftz, 1.99 mol % ammonia; inlet liquid: 33.20 mol/h ftz. Inlet gas: 18.18 mol/h ft’, 2.01 mol % ammonia; inlet liquid: 44.80 mol/h ft2. Inlet gas: 18.11 mol/h ftZ,1.96 mol % ammonia; inlet liquid: 58.10 mol/h ft2. showed that their results compared favorably with the values predicted by the method of Treybal (1969).
Development of Method for Calculating Minimum Liquid-to-Gas Ratio A method of calculating the minimum liquid-to-gas ratio for a simple, three-component system is important because one should always know the minimum L / G ratio before the operating L / G ratio is selected. Before the work described here was begun, however, a rigorous method for calculating the minimum L / G ratio for adiabatic absorption was still unknown. It was at first thought, before preliminary work was begun, that the problem of calculating the minimum L / G ratio depended upon the establishment of two criteria. The first criterion was that of being able to complete the calculation from the bottom to the top of packing without the formation of a “pinch” on the concentration of the solute component. A “pinch” can be visualized as a point a t which the solute concentration in the bulk stream and in the interface are equal and thus there is no longer any driving force present for mass transfer to occur. The second criterion was that of being able to avoid the problem of fog formation throughout the packed tower. After several calculations were made, however, at very high inlet liquid or lean oil rates, using Treybal’s (1969) sample problem, it became clear that the second criterion, that of avoiding fog formation, was not suitable for determining the minimum L / G ratio. It was observed that even at high-lean oil rates, the packed tower would always have sections which were very close to fog formation. It thus became clear that the only meaningful criterion for determining the minimum L / G ratio was that of being able to complete the calculation without the formation of a “pinch” on the solute concentration. The question then became exactly how to locate the proper L / G ratio for which a “pinch” is just avoided. The simple way of visualizing the problem is to say that the rate of flow, composition, and temperature of the inlet gas is fixed and that in determining the minimum L / G ratio we have been given the composition and temperature of the inlet liquid, and we are looking for the minimum rate of flow of the inlet liquid to meet a given specification on the outlet gas. At a point just slightly above the minimum liquid rate, it would still be possible to carry out the calculation without the formation of a “pinch”. Below it, however, one would recognize the formation of a “pinch” by the fact that the interface solute vapor concentration would be greater than the bulk solute vapor concentration. Once this occurs, if the calculations were still continued, one would calculate a negative value of dY,/dZ indicating stripping rather than absorption, and this would in turn lead to a calculated negative height of packing. Thus, it can be seen that if the inlet liquid or lean oil rate is systematically varied until the rate is found at which the solute concentration in the bulk gas still slightly ex-
ceeds that of the solute concentration at the vapor interface, then the minimum L / G ratio has been established. This procedure was carried out by making suitable modifications to the computer program which was described above. A techique of interval halving was developed by which the lean oil rate was systematically varied until the answer was bracketed between two possible lean oil rates, and then the interval in which the answer lay was narrowed further by each time halving the possible range in which it could lie. The problem that was used to test the program was the same one, involving ammonia absorption, that was used by Treybal (1968). In the test problem, the initial value assumed for the lean oil rate was 200 mol/h ft2. This value was then reduced each time by 20% until a value below the minimum was found. In trial number 4, the lean oil rate of 102.4 mol/h ft2 was found to be the first assumed value which was below the minimum. This bracketed the answer as lying between the value of 102.4 assumed for trial number 4 and the value of 128.0 assumed for trial number 3. The subsequent trials then halved the interval until the answer of 122.4 mol/h ft2 was found. The calculated minimum liquid rate of 122.4 mol/h ft2, which corresponds to a minimum L / G ratio of 4.90, must be considered to be much more accurate than the value of 106.5 mol/h ft2corresponding to a minimum L / G ratio of 4.27 which was calculated by Sherwood and Pigford (1952) using a much less rigorous approach to the same problem. Development of Computer Program for Rigorously Calculating Complex Multicomponent Adiabatic Systems Although a good deal of the work done on simple, three-component systems was also applicable to complex, multicomponent systems, most of the equations and many of the computational techniques which were suitable for simple, three-component systems had to be modified or extended for use in complex, multicomponent systems. In addition, several new iterations and closure techniques had to be developed. Although there is no limit on the number of components which can be handled in the calculation procedure, the program was set up to handle a maximum of 14 components. Several test problems were successfully run and as the ultimate test of the computational method, a 14-component problem was successfully run. Details of the various physical property correlations that were used as well as a summary of the equations needed are given elsewhere (Feintuch, 1973). Listed below is the outline of the calculation procedure followed by the computer program in solving a complex, multicomponent, adiabatic problem. In addition to certain physical property data, the following information must be submitted to the program as input data: inlet gas-mass velocity, temperature, composition; inlet liquid-mass velocity, temperature, composition; components-name,
510
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
type, properties, “key” component specification; miscellaneous-column pressure, selected physical property options, number of packing increments, maximum number of overall iterations, desired printout option, tolerances and averaging factors required for certain closure routines, assumed componential flow rates, and assumed temperature for outlet gas. The procedure is as follows: (1)Read in all known input data. (2) Calculate the enthalpy of the inlet gas and the inlet liquid streams. (3) Calculate the quantity of the “key” component in the outlet gas. (4) Assume the temperature and the flow rate of each component in the outlet gas. For the initial iteration, this assumed information is read in as part of the input data. ( 5 ) Calculate the enthalpy of the outlet gas. (6) Calculate the quantity, composition, and enthalpy of the outlet liquid by material and enthalpy balance. Calculate the outlet liquid temperature by trial and error. (7) Assume an interface temperature. (8) Calculate all the physical properties at the average temperature between the bulk liquid or bulk gas stream and the interface. (9) Assume Rj for components 1 to n. (10) Calculate Dj,, Dbm,SCG, FGj, ScLj, and F L j for components 1to n. Also calculate L G and hL. (11) By trial and error, for each component from 1 to n, determine the point of intersection between eq 7 and 8.
Yi
-=
K, = function (Ti, K )
(8)
“j
(12) If C:=lxji= 1.0 proceed to step 13; otherwise return to step 9 for new assumptions on R . for each componect. (13) Calculate hfGaand dTG/dZ. +hen dGJdZ and Hji for each component, from 1 to n, is calculated. (14) Calculate the interface temperature and compare it to the value assumed in step 7. If it agrees, proceed to step 15; otherwise return to step 7 for a better assumption. (15) For the “key” component jk, calculate the value of AGjk based upon the number of packing increments specified in the input data. (16) Proceed to the next level of packing and calculate the following. Znext= Z + AZ = Z + AGjk/(dGjk/dZ)
+ AGjk = Gj + AGj = Gj + AZ(dGj/dZ) Gjk,next
Gj!,,,t
TG,next
=
Gjk
= TG+ AZ(dTG/dz)
Lnext= L Xj3next = HL,next
+ j=lAGj
(AGj + Lxj)/Lnext
= [(LHL)top +
(GHG)next
-
(GHG)topl/Lnext
(17) Find TL,next by trial and error. (18)Calculate the dew point temperature of the gas entering the next level of packing by trial and error. (19) Compare Gjk,nextto the specified outlet quantity, to determine if the top of the packing has been reached. If it has not been reached, repeat steps 7-18 for the next level of packing. If it has been reached, proceed to step 20. (20) Compare TGqeXt and the values of GjneXtfor each component to the assumptions made in step 4 for the outlet gas. If all the assumptions are okay, the calculation is terminated. If the assumptions are not okay, return to step 4 for an improved set of assumptions and renew the calculation.
The computer program consists of a main program together with a set of 15 subroutines. The main program coordinates the activities of the various subroutines, as well as prints out the results of the calculations. Given below is a listing of the names of the subroutines together with a brief description of their function. The numbering system used below goes from S1 through S16 and includes the main program and all its subroutines. The numerical sequence follows the order in which the routines were originally written and then numbered for submission to the computer. Although the subroutines, when originally written for simple, three-component systems, were set up so that, wherever possible, they could be later used for complex, multicomponent systems, it was still necessary to revise substantially or rewrite most of the original subroutines in order to make them all suitable and mutually compatible for complex, multicomponent systems. S1. MAIN PROGRAM. Coordinates subroutines and prints out calculation results. S2. READIN. Reads in all input data. S3. DBIN. Calculates binary diffusivities for gases and liquids. S4. XYKAY. Calculates equilibrium values of xj, yj, and K j for components 1 to n. S5. UCON. Calculates viscosity and thermal conductivity for mixtures of gases and liquids. S6. TERPEL. Performs linear interpolation in a two-coordinate system. S7. CPLAM. Calculates componential and mixture heat capacities for liquids and gases, as well as latent heats. S8. TERXYZ. Performs linear interpolation on a three-coordinate system. S9. DENS. Calculates gas and liquid mixture densities. S10. PACK. Calculates packing data. S11. DMULT. Calculates effective diffusivities and mass transfer coefficients for mixtures of gases and liquids. S12. INTERP. Performs linear interpolation, for certain trial and error calculations, to determine the next assumed value to be used for a variable. S13. TINT. Calculates Rj for components 1to n, as well as, the interface temperature with its related variables. S14. ENTH. Calculates the enthalpy of the inlet and outlet gas and liquid streams. S15. NEXT. Calculates the variables at the next level of packing. S16. HEAT. Calculates gas and liquid componential enthalpies by use of a polynomial equation. As discussed previously, one of the major problems in solving complex multicomponent systems is being able to find the correct values of Rj 0‘ = 1,n). To do this, a special closure technique was developed to find the correct values of Rj 0’ = 1, n). In this technique for the first iteration, an initial value of 1.0 is arbitrarily assumed for all components in the system, with the exception of the last component whose Rj value is adjusted so that C,”=,Rj = 1.0. For each assumed set of Rj values, the interface liquid composition x - and the interface gas composition yji are found, by triaf’and error, for each component. If z:jn=lxji is equal to 1.0, to within a tolerance read into the program as input data, the assumed set of Rj values is correct. The provision to read the tolerance into the program as to part of the input data is made so that an opportunity to economize on computer time for slow closing problems is made available. The tolerance normally used is 0.01 mole
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
fraction. If the assumed set of Rj values is found not to be correct, then values of Nj are calculated for each component from eq 9.
Using the newly calculated values of N . a new improved set of assumptions for Rj values is madH by use of eq 10
Rj =
Ni
7
(10)
CNj
j=l
and the entire process is then repeated. In order to avoid the possibility of oscillation, an averaging factor, which is read in as input data, is used to generate the new set of assumptions as a weighted average of the old set of assumptions and the newly calculated values of eq 10. For those problems which exhibit tendencies toward oscillation, an averaging factor of 0.5 is usually effective. The process used is found to be self-improving in that for each iteration the new assumed values of Rj will more and more closely approach the final correct values. The maximum number of iterations allowed is 50. This method proved to be quite powerful in solving complex, multicomponent problems. During the development of the computer program for complex, multicomponent, adiabatic systems, several problems involving four components were run to test the program. The ultimate test of the computational method used by the program was the successful running of a problem involving 14 components. The data used for the 14-component problem were those of McDaniel (1970), who studied and obtained operating data from a full-size commercial plant used to recover propane and heavier hydrocarbon components from a natural gas stream rich in methane. Contained within the plant was a packed tower 3.0 ft in diameter operating at about 800 psia and containing 23.0 ft of 2411. metallic pall rings. Although the data of McDaniel (1970) were suitable for use in testing the computational method of the program, they were by no means complete or accurate enough to be used as a comparison between calculated results and operating data. In fact, a search of the literature has shown that there really are not available any good experimental or plant operating data for complex, multicomponent, adiabatic systems for which such a comparison can be made. Short-Cut Calculation Method for Complex, Multicomponent, Adiabatic Systems Described below is a short-cut calculation method which can be used to obtain solutions for complex, multicomponent, adiabatic systems which are similar to those obtained through the rigorous method described above. It is felt that the method which was developed and is described below will, in most cases, yield a solution which is reasonably close to that obtained rigorously by computer, without necessitating laborious and time-consuming calculations. The method used is based upon the classical concept of transfer units (Treybal, 1968). Under this concept, the height of packing is equal to the overall number of transfer units times the overall height of a transfer unit. For rigorous calculations, overall transfer units are, strictly speaking, only suitable when either the gas or the liquid resistance to mass transfer is negligible or when Henry's law holds (Simon and Rau, 1948). Nevertheless, overall transfer units if discretely used, even when they
511
may not rigorously apply, will in most cases provide a convenient simplified method of doing absorption and stripping calculations for complex, multicomponent systems. This is because in a typical multicomponent system there will be one or more components present that are only slightly soluble and whose mass-transfer resistance is thus significantly present only in the liquid. For these components, an overall liquid transfer unit is applicable. On the other hand, there will also be one or more components present that are very soluble and for which an overall gas transfer unit is applicable. For those remaining components whose solubility is moderate, overall transfer units do not rigorously apply, since there is resistance to mass transfer in both the gas and in the liquid while at the same time Henry's law does not hold because the system is adiabatic rather than isothermal. However, even for these components, if discreet assumptions are made, as to the average values of bulk liquid, bulk gas, and system temperature, the concept of overall transfer units can be effectively used as a short-cut calculation method. Equations to calculate the height of packing using overall transfer units are as follows. = NtOGjHtOG] (11)
z = NtOLJHtOLj
(12)
NtoGj = number of overd gas transfer units for component j ; NtoLl = number of overall liquid transfer units for component j ; HtoGl = height of an overall gas transfer unit for component j , length; and HtoLl = height of an overall liquid transfer unit for component j , length. Equation 11 is usually used when the principal masstransfer resistance for the component in question is within the gas, while eq 12 is usually used when the principal mass-transfer resistance is within the liquid. Values of NtoG, and NtoLJmay be obtained from the following equations.
1-'
Ai
where Aj = L/KjG,and all of the other terms have already been previously defined for complex, multicomponent, systems. The numbers 1 and 2 have the customary meaning of tower bottom and tower top, respectively. Values of HtoGj and HtoLj can be obtained from the following equations. G HtOGj
F o ~ jand equations.
FOL,
=
FOGja
may be estimated from the following
512
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Since for each component present in the systems the height of packing is given by eq 11and 12, if we calculate the height of packing for the “key” component, we can then use this value to back calculate the composition of each of the other components present in the system. The only problem in doing this is what values to use for G, L, and K, in these calculations, or expressed differently, what values should be used for the absorption factors A, and the stripping factors l/Aj. Two approaches were attempted in answering this question as described in the following paragraph. 1. Simple Approach. In the first and simplest approach taken, the values of G and L used were those at the inlet to the tower and the values of K, were obtained at a temperature corresponding to the inlet liquid. This approach, which we shall call the “simple approach,” has the advantage of simplicity in that the calculation can be started with the data immediately at hand, without necessitating preliminary calculations for estimating the conditions of the outlet streams from the tower. The use of the inlet gas value for G is conservative in that for absorption problems it is higher than the outlet gas and thus its use leads to a slightly higher estimation of the required packing. In addition, its use together with the inlet liquid quantity and temperature conservatively approximate the conditions at the top of the tower for which the bulk of the packing is normally required. As described in the next section, this simple approach gave results which were reasonably close to those calculated rigorously except in the case of very high recovery of the “key” component where the conservativeness of using the inlet gas value for G led to a calculated packing height that was much too high. 2. Edmister-Type Approach. The simple approach described above proved to work well except when the “key” component recovery was very high. It was felt that the results for problems involving a high degree of recovery could be improved if a better method of selecting the average values of L , G, and Kj could be found. To help find a solution to this problem, reference was made to the work of Edmister (1943), who developed a simplified calculation method for tray-type absorbers and strippers. It was hoped that some analogy could be drawn between his work on tray-type towers and the work described here on packed towers. A careful study of his work revealed that no rigorn2s analogy could be derived, but it was found that his definition of “effective” absorption and stripping factors could be analogously used as an approach to obtain results in the case of high recovery. When this is done, the values of G, L , and K, are separately evaluated a t the top and at the bottom of the packing, and these values are then used in the equations for “effective” absorption and stripping factors, AE and SE,given below. A E = [AXp(A1 + 1) 0.25]0’5- 0.5 (19)
+
SE =
[SI(SNp + 1) + 0.251O.’ - 0.5
(20)
A I = absorption factor at tray 1 at top of tower (by analogy, at the top of the packing); ANp = absorption factor at tray N P at bottom of tower (by analogy, a t the bottom of the packing); SI = stripping factor at tray 1 at top of tower (by analogy, at the top of the packing); and S N p= stripping factor at tray N P at bottom of tower (by analogy, at the bottom of the packing). A, = absorption factor a t tray 1 at top of tower (by analogy, at the top of the packing); A m = absorption factor at tray N P at bottom of tower (by analogy, at the bottom of the packing); S1 = stripping factor at tray 1 at top of tower (by analogy, at the top of
Table 111. Comparison of Calculated Height of Packing Using Rigorous and Short-Cut Methods calcd packing height, ft c 3
recovery, case
%
1
10
2 3 4
50 60 90
rigorous method 0.5074 0.5537 0.6361 0.4230
short-cut short-cut (simple) (Edmister) 0.471 0.540 0.665 0.750
0.539 0.480 0.555 0.446
Table IV. Comparison of Calculated Outlet Compositions Using Rigorous and Short-Cut Methods C3 re- comcov- POcase ery, % nent
1
10
c, c 2
2
50
C8
C,
c 2
3
60
C8 C,
c2 C8
outlet stream
mol%, rigorous method
mol%, short-cut (simple)
liquid liquid gas liquid gas gas liquid gas gas
19.993 4.544 0.005 20.406 5.067 0.020 21.561 4.843 0.021
15.3 4.35 0.01 13.5 4.94 0.036 14.6 4.57 0.0383
the packing); and S N p = stripping factor at tray N P at bottom of tower (by analogy, at the bottom of the packing). Four test problems, each representing a complex, multicomponent, adiabatic system consisting of four components, were used to make the comparison between the short-cut method and the rigorous computer solution. The operating conditions used in each of the problems were similar to those of McDaniel(1970) from a standpoint of temperature, pressure, and major components present in the system. The recovery of the “key” component, which was taken as propane, was varied from 10% up to 90%. Also varied was the quantity and the composition of the inlet streams. Table I11 gives a comparison, for each of the four test problems, between the height of packing as calculated rigorously by computer and the height of packing as calculated by the short-cut method. For the short-cut method, two values are presented, the value using a simple approach and the value using the Edmister-type approach. As can be readily seen, except for case 4 in which there is a 90% propane recovery, the simple approach gives results which compare favorably with the computer solution. In fact, for cases l through 3, there is no improvement obtained by using the Edmister-type approach and, indeed, the extra effort cannot be justified in terms of greater accuracy. For case 4, however, the conservativeness of the simple approach gives a value for the calculated packing height that is much too high and thus the simple approach is not adequate. Because of this, there is justification for using the Edmister-type approach which yields a calculated height of packing agreeing favorably with the rigorous solution. Table IV gives a comparison between the outlet composition as calculated rigorously by computer and by the short-cut method utilizing the simple approach. Results are given for cases 1 through 3. It can be seen that the best results are obtained for the component which is most similar to the “key” component. In this case, ethane is most similar to the “key” component propane. Case 4 is not reported since the use of the simple approach is not effective for this high-recovery case. In fact, for case 4, even the Edmister-type approach, although good for estimating the height of packing, gave poor results for estimating composition. No reliable short-cut method was
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978 513
found for estimating the composition for a high-recovery case, and it would seem that in such a case, the composition must be determined rigorously without the use of short-cut methods. From Tables I11 and IV we can see that the short-cut method using the simple approach can be used to obtain reasonably close agreement to a rigorously calculated solution of packing height and outlet composition for problems which do not involve a high recovery. For a high-recovery problem, the packing height can be reliably estimated using the Edmister-type approach t o the short-cut method, but no simple method was found to calculate the outlet composition. Nomenclature a = interfacial surface per unit volume of packing, area/ volume A = absorption factor, L/K,G, dimensionless D = diffusivity, area/time F = mass-transfer coefficient, mol/time (interfacial area) G = gas mass velocity, mol/time (area) h = heat-transfer coefficient, heat/time (area) degree h' = heat-transfer coefficient corrected for mass transfer, heat/time (area) degree H = enthalpy, heat/mol HJ,= partial enthalpy of component J at temperature Ti and concentration x J L ,heat/mol Ht = height of a transfer unit, length K = vapor-liquid equilibrium constant, dimensionless L = liquid mass velocity, mol/time (area) N = flux of mass transfer, mol/time (interfacial area) Nt = number of transfer units, dimensionless P = partial pressure Q = flux of heat transfer, heat/time (interfacial area) R = defined by eq 10 S = stripping factor, K , G / L , dimensionless Sc = Schmidt number,'p/pD, dimensionless T = temperature, deg x = liquid concentration, mole fraction y = gas concentration, mole fraction YJ = gas concentration, mol of J/mol of insoluble carrier gas 2 = packing height, length
A = increment p =
viscosity, mass/length (time)
K
= total pressure
p
= density, mass/length3
Subscripts and Superscripts A = component A, a solute E = effective value G = gas i = interface j = a typical component in a system consisting of a total of n components, ranging from 1 through n; jk is the "key" component J = a typical component in a system consisting of a total of J components, ranging from A through J L = liquid n = component n, last in a system of n components next = condition at next level of packing N P = bottom tray of tray power 0 = overall S = component S,the solvent 1 = at bottom of packed tower; also tray 1 at top of tray tower in eq 19 and 20 2 = at top of packed tower L i t e r a t u r e Cited Bassyoni, A. A., McDaniel, R., Holland, C. D., Chem. Eng. Sci., 25,437 (1970). Dwyer, 0. E.. Ph.D. Dissertation, Yale University, New Haven, Conn., 1941. Edmister, W. C., Ind. Eng. Chem., 35 (8),837 (1943). Feintuch, H. M., Ph.D. Thesis, New York University, New York, N.Y., 1973. Holland, C. D., McMahon, K. S.,Chem. Eng. Sci., 25, 431 (1970). McDaniei, R. Ph.D. Dissertation, Texas A & M University, College Station, Texas,
1970. Oldershaw, C. F., Simenson, L., Brown, T., Radcliffe, F., Chem. Eng. Prog., 43 (7),371 (1947). Othmer, D. F., Gilmont, R., Ind. Eng. Chem., 36 (9),858 (1944). Othmer, D. F., Scheibel, E. G., Trans. A.I.Ch.E., 37, 211 (1941). Raal, J. D., Khurana, M. K., Can. J . Chem. Eng., 51, 162 (1973). Rubac, R. E., McDaniel. R.. Holland, C. D., A .I.Ch.E. J., 15 (4), 568 (1969). Sherwood, T. K., Pgford. R. L., "Absorption and Extraction", 2nd ed,McGraw-Hill, New York, N.Y., 1952. Simon, M. J., Govinda Rau, M. A., Ind. Eng. Chem., 40 (l),93 (1948). Treybal, R. E., "Mass-Transfer Operations", 2nd ed, McGraw-Hill, New York, N.Y., 1968. Treybal, R. E., Ind. Eng. Chem., 61 (7),36 (1969).
Received for review December 7, 1977 Accepted April 18, 1978
Selective Hydrocracking of n-Paraffins in Jet Fuels Nal
Y. Chen' and William E. Garwood
Mobil Research and Development Corporation, Princeton and Paulsboro Laboratories, Princeton, New Jersey 08540 and Paulsboro, New Jersey 08066
Lowering the freeze point of paraffinic jet fuel by the selective conversion of n-paraffins is shown to be feasible over metal loaded molecular sieve zeolites, Ca-A and H-erionite, under hydrocracking conditions. Comparable experiments over large-pore Zeolite X show that in the absence of shape selectivity, the freeze point of the jet fuel is raised instead of lowered on account of the preferential conversion of the more reactive branched molecules. The reaction over Ni/H-erionite is shown to be sensitive to the partial pressure of hydrogen and sulfur. The presence of sulfur, and/or the absence of high hydrogen partial pressure (- 1000 psig) inhibits the conversion of n-paraffins. The decrease in activity is explained by the accumulation of intracrystalline inhibitor concentration which increases with hydrocarbon partial pressure, but decreases with increasing hydrogen partial pressure. Sulfur had the additional effect of promoting the conversion of non-normals. Similar activation by sulfur was reported by Dudzik on Zeolite K-A .
Introduction Normal paraffins are present in all petroleum fractions. Their concentrations depend on the crude source and 0019-7882/78/1117-0513$01.00/0
intervening processing history. Despite the relative inertness of n-paraffins, Weisz and Frilette (1960) demonstrated the selective conversion of n-paraffins over a
0 1978 American Chemical Society
