Sea Water Softening with the Lime-Magnesium Carbonate (LMC) Process J. Darius Mavis* Burns and Roe Construction Corporation, Fountain Valley, California 92708
Andrew Checkovich Burns and Roe, Incorporated, Oradeii, New Jersey 07646
A pilot plant to remove 70% of the calcium from sea water using the lime-magnesium carbonate (LMC) process was tested and improved. The calcium-deficient LMC product water was used as feed to a distillation desalting plant which was operated scale-free at 335'F and above. LMC pretreatment is estimated to cost 5.1&/1000 gal of treated water for a plant sized to supply a 50 million gal/day desalting plant. The process may find use with desalting processes using feed water more saline than the oceans.
Sea Water Softening with the LMC Process The total cost for commercially desalting sea water is the sum of operating and capital costs. For large distillation desalting plants, more than half of the operating cost is for energy used to heat the brine. As energy costs increase, desalting plants must become increasingly more efficient if product water costs are to remain competitive with other sources of fresh water. Higher thermal efficiency can be achieved by raising the maximum brine temperature of the distillation plant, but there is a practical limit a t 250-265OF due to the formation of calcium sulfate scale on the heat transfer surfaces. The lime-magnesium carbonate process (LMC) was devised to remove the maximum temperature restriction by precipitating calcium from sea water as calcium carbonate (W. R. Grace & Co., 1966). The economic advantage of this process does not hinge on the sale of byproducts as has been the case with many other proposed schemes for the removal of scale-forming materials. All materials produced in the process are recycled for use in some other part of the process, and only nominal chemical makeup is required. Process Description Two feed stocks are required to sustain the LMC process: limestone and fuel oil. All other materials required in the process are derived, directly or indirectly, from these two. The process can be represented by the four basic operations shown in Figure 1. Softening. The softening step is described by two reactions in which calcium precipitates as calcium carbonate and bicarbonate is converted to carbonate. Ca?' + MgC0,.3H20 CaCOl + Mg*' + 3H20 (I) --+
Ca" + 2HC0,- f C a O 2CaC01 + H 2 0 (10 These reactions are carried out in a sludge contact reactor. The clarified water from this reactor is the final LMC product, depleted of 70% of its original calcium content, and is used as distillation plant feed water. Calcining. Calcium carbonate sludge from the softener is supplemented with purchased limestone and both are calcined. Offgassed carbon dioxide is used in the manufacture of magnesium carbonate trihydrate from magnesium hydroxide, and lime is used to neutralize bicarbonate and to produce magnesium hydroxide from sea water. Magnesium Hydroxide Production. Magnesium hydroxide is precipitated from a sea water slip stream with lime. The magnesium hydroxide floc is separated and used in the carbonation process, and the calcium-rich liquid phase is discharged as waste. +
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Magnesium Hydroxide Carbonation. Magnesium hydmxide and carbon dioxide from the kiln are combined in a sparged reactor to form magnesium carbonate trihydrate. This product is added to the main sea water stream to precipitate calcium carbonate.
The Pilot Plant
A prototype LMC plant was operated a t the Interior Department's San Diego Test Facility, administered by the Office of Saline Water. The plant was designed to produce 950,000 lb/hr of sea water containing about 30% of the calcium normally present in the sea water feed. This plant did not contain equipment for magnesium hydroxide production and calcining since design data for these operations are widely available. The San Diego plant consisted only of the softening and magnesium carbonate trihydrate production steps as shown in Figure 2, which includes flow sheet and mass balance. Carbon dioxide, calcium oxide, and magnesium hydroxide were obtained commercially to supply the pilot plant. In order to study calcium carbonate precipitation, separate reaction vessels were used for carrying out calcium carbonate precipitation, bicarbonate neutralization, and solids separation instead of the recommended sludge contact reactor. Softening Tests. At the outset, laboratory studies were run to identify factors influencing the precipitation rate. Three groups of tests were run to determine (I) whether the softening reaction rate was limited by the dissolution of magnesium carbonate trihydrate or the precipitation of calcium carbonate; (11) whether the softening reaction rate was enhanced in the presence of finely divided calcium carbonate; and (111) whether, within the practical process limits of 8.2-9.5, pH would influence the softening reaction rate (this pH range is bounded by natural sea water, p H 8.2, and the magnesium hydroxide precipitation threshold a t ambient temperature, p H 9.5). Case I. (a). Two 1-1. quantities of 0.01 M calcium chloride solution, each containing 15 g of calcium carbonate powder, were treated with stoichiometric quantities of sodium carbonate solution and magnesium carbonate trihydrate slurry, respectively, and the rate of calcium ion disappearance was followed. Both reactors were stirred a t room temperature. The calcium concentration fell rapidly for 10 min in both reactors, then stabilized. There was no detectable difference in the calcium disappearance rates in the two reactors. (b). Two 1-1. quantities of raw sea water, each containing 15 g of calcium carbonate powder, were treated as above, with stoichoimetric amounts of sodium carbonate and magnesium carbonate trihydrate, respectively. Again
Seawater
calcium concentration out of the softening reactor with the concentration in the final product stream (overall plant mean residence time 620 min) showed that 65-7570 of the calcium was precipitated in the softener and the remaining 25-35% precipitated in the clarifier. Magnesium Hydroxide Carbonation
Figure 1. Schematic drawing of the LMC process. The LMC distillation desalting pretreatment process utilizes limestone and fuel oil feedstocks to remove calcium from sea water to prevent calcium sulfate scale at high temperature.
the calcium precipitation rates for sodium carbonate- and for magnesium carbonate trihydrate-treated sea water were the same, although the precipitation rate from sea water was lower than from calcium chloride solution. Case I test results shown in Figure 3 indicate the reaction rate was not limited by magnesium carbonate trihydrate dissolution, and that for sea water some reaction other than calcium carbonate precipitation is rate limiting. Case 11. One liter of raw sea water was treated with magnesium carbonate trihydrate, but the 15 g of calcium carbonate was omitted. As before, the calcium concentration decrease with time was followed, and the data compared with those of case Ib using magnesium carbonate. A comparison of curves Ib and I1 shows that without an initial calcium carbonate seed crystal charge, the initial precipitation rate is slow, but when finely divided calcium carbonate is present, the calcium concentration decreases rapidly. Case 111. During case I and I1 calcium precipitation runs, the p H gradually decreased from 9.2 toward 8.5-8.7 as the calcium concentration fell. At p H 8.5, the carbonate-bicarbonate equilibrium shifts toward bicarbonate ion, so an additional test was made .to determine whether calcium carbonate would precipitate faster a t higher p H levels where the equilibrium favors the carbonate ion. Case Ib with magnesium carbonate trihydrate was repeated, only the p H was maintained a t about 9.4 with sodium hydroxide. Comparing curves Ib and I11 in Figure 3 shows that when the p H was kept high, the calcium concentration dropped faster than when p H was allowed to fall. Softener Residence Time. Stirred batch reactor data from case I11 were used to estimate the required continuous-flow softening reactor residence time. The batch reactor residence time, t, is simply the elapsed time beginning the instant the reaction begins and running until the desired extent of reaction is achieved. Time in a continuous flow stirred tank reactor arises from the quantity (V/c = t), where V is the reactor volume in cubic feet, and is the flow rate in cubic feet/minute. The two terms ( t and t) arise under different circumstances and cannot be used interchangeably; however, a relationship does exist and can be demonstrated analytically (in the mathematical sense) for certain kinetic models. Conventional kinetic models failed to adequately describe the calcium carbonate precipitation data, so a graphical method (Hattiangadi, 1971) was used to determine the continuous flow reactor residence time requirement from the batch data. The continuous flow reactor residence time for 80% of the extent of reaction after 220 min in the batch reactor was 1hr. Direct comparison of residence time estimated from batch data with the observed softener performance was not possible, but the plant softener residence time of 55-60 min was found to be adequate. A comparison of the
A high on-stream factor and overall process reliability were essential prerequisites for technical feasibility. Magnesium carbonate trihydrate production presented the greatest single obstacle. Early pilot plant work showed that carbon dioxide was absorbed into magnesium hydroxide slurry much more effectively by sparging the gas through the slurry than by passing the slurry over a packed column. Moreover, when carbonating magnesium hydroxide in a packed vessel, frequent shutdowns were necessary to remove tenaceous magnesium carbonate trihydrate deposits from the packing, product pump, and piping. The following areas were found to need refinement: a carbon dioxide mass transfer coefficient was needed for scale up; adherent scale deposits limited continuous on-stream time to 2 weeks or less and had to be prevented to increase on-stream time to a t least 1 year; for economic reasons, a t least 37.5 mol % of the carbon dioxide and 80 mol % of the magnesium hydroxide fed to the carbonator had to be converted to magnesium carbonate trihydrate. M a s s Transfer Coefficient. The carbon dioxide absorption rate had originally been thought to be gas diffusionlimited (W. R. Grace & Co., 1967). Under this assumption, mass transfer coefficients (&a) were computed for a large body of operating data obtained over a wide range of operating conditions. The calculated coefficients ranged over an order of magnitude, with the value falling rapidly a t magnesium hydroxide feed slurry concentrations less This indicated that gas diffusion was than about 1.2 wt 70. not the only factor influencing carbon dioxide absorption and that the magnesium hydroxide concentration was also a factor. When &a values were divided by the log mean magnesium hydroxide concentration in the carbonator, a relatively constant coefficient was obtained over a wide range of carbon dioxide partial pressures and magnesium hydroxide concentrations. The value was 11.3 lb-mol C 0 2 / hr/ft3-atm CO2-wt.fr. Mg(OH)2 for carbon dioxide feed partial pressures from 0.10 to 0.35 atm and magnesium hydroxide feed slurry concentrations of from 1.0 to 6.5 wt %. The standard deviation in the coefficient was 2.6 (units). There are at least three factors which are believed to contribute to the relatively large standard deviation. (1) Computation of the log-mean magnesium hydroxide concentration involved numerical computation with values which could be determined only with limited certainty. (2) Though the log-mean magnesium hydroxide concentration was used in computing the mass transfer coefficient, it would be more reasonable to interpret the mass transfer coefficient in terms of the magnesium hydroxide surface area. While we did not direct our work toward this approach, Lawrence (1970) has shown that freshly precipitated magnesium hydroxide reacts faster than the commercially obtained material used a t the San Diego Test Facility. (3) The only agitation provided for the carbonation reactor was that imparted by the injected gas, whose rate ranged from 275 to 480 cfm. Better agitation may have increased the gas absorption rate. G a s Dispersion. Early sparged carbonator test runs were made by injecting the gas through a 2-in. pipe near the base of vessel. Mass transfer coefficients were deterInd. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
205
Soft en i ng Reactor
Carbonator
Lime
1.Slurry
.7
A
PROCESS
FLOWS
( X
to-’
I
Iblhr)
Figure 2. Revised flowsheet of the LMC test plant. The mass balance shown here is the basis for scale up of the softener (calcium precipitation) and MgC03.3H20 production segments of larger plants. It is typical of the performance of the test plant at San Diego.
CaCO,)
ILL‘ 30
60 Tlme
90
120
IS0
(minJ
Figure 3. Calcium carbonate precipitation rates from aqueous calcium chloride solution and from sea water. Case Ia. the rate of Ca2+ disappearance from 0.01 M CaC12 solution containing 1.5% CaCO&) seed crystals is the same whether Na2C03.HzO or MgC03.3H20 is the precipitating agent. Case Ib. the disappearance of Ca2+ from sea water containing 1.5% CaCOa(s) seed crystals is slower than from aqueous CaClz solution (Ia), but the rate is not affected by the choice of precipitating agent Na2C03.H20 us. M g C 0 3 . 3 H ~ 0 .Case 11. the omission of CaCOs(s) seed crystals from case Ib results in a slow CaZ- disappearance rate. Case 111. the Ca2- disappearance rate from sea water increases when the p H is maintained at 9.4 for the duration of the reaction.
mined for these runs. Subsequently, efforts were made to distribute the gas more evenly by installing piping transversely across the vessel with small orifices drilled on the upper surfaces. Three sets of interchangeable transverse arms were fabricated, with hole sizes of Y4, lis and H6 in., respectively. The total number of holes in 206
Ind. Eng. Chem., Process Des:Dev., Vol. 14, No. 3, 1975
each set was chosen so that the total cross sectional area of all the holes was equal to the area of the 2-in. pipe. A comparison of the mass transfer coefficients obtained with the various spargers failed to show any significant improvement in carbon dioxide absorption with decreasing hole sizes. We did find, however, that the smaller the hole size, the faster the sparger became plugged with magnesium carbonate trihydrate deposits. The problem could be avoided by eliminating the spargers and injecting the gas through the 2-in. pipe a t a rate above 275 cfm. Under these conditions, the gas injection line stayed free of serious scaling for over a year of normal operation. Scale Deposits. Magnesium carbonate trihydrate scaling in the carbonator discharge pump and piping limited the maximum on-stream time to less than 2 weeks. The deposition of scale did not result from low product slurry velocities, since the pump rotating member became coated almost as fast as the piping. By comparison, the carbonation vessel itself did not form deposits nearly as rapidly as the piping because the vessel surface area-to-volume ratio is smaller for the carbonator than for the pipes. The magnesium carbonate trihydrate slurry pipe scale problem was eliminated by removing the carbonator discharge system entirely and installing an overflow trough sloping from the carbonator to the softening vessel. Though the trough gradually became scaled, it could be mechanically cleaned while the process remained on line. This improvement was sufficient for long-term continuous operation, and the technical feasibility of the LMC process could then be demonstrated. COz and MG(0H)Z Utilization. With the establishment of the mass transfer coefficient, operating conditions could be selected to give the desired carbon dioxide and
Table I. Magnesium Hydroxide and Carbon Dioxide Utilization Efficiencies Observed in the Production of Magnesium Carbonate Trihydrate during Typical Runs
co2
Mg(OH),
Run no. 16 20 32 34 38 41 51 60 64 188 193
Mg (OH)? feed, lb/hr 384 398 462 355 373 42 5 472 447 450 162 145
hlg(OH),
feed, wt % 4.05 4.35 5.64 3.63 3.75 4.20 3.87 4.24 4.27
1.90 1.77
utilization, % of feed
CO2 feed, lb/hr
feed, vol % ’
utilization, 90 of feed
86 76 60 89 70 80 80 82 74 76 64
662 499 632 706 496 733 709 632 481 385 41 6
20 16 19.5 20 16 21 20.5 20 16 19 21.5
39 46 32 32 44 35 43 43 48 25 18
magnesium hydroxide utilization efficiencies. Data in Table I show that magnesium hydroxide and carbon dioxide utilization efficiencies (conversion into magnesium carbonate trihydrate) exceeded the respective 80 and 37.5% minimum requirements with proper choice of operating conditions. More detailed design and performance information is available in an OSW report (Mavis, 1971).
LMC Product Properties T o establish a theoretical LMC product water composition with which to compare actual plant product, a relationship was derived which gives the totai alkalinity in terms of the calcium ion concentration and the hydrogen ion concentration. By computing the minimum theoretical alkalinity under given conditions, a convenient yardstick is established which indicates the amount of acid necessary to dispel carbon dioxide and prevent calcium carbonate and magnesium scale formation on heat exchange surfaces. (Acidification of sea water feed as a means of alkaline scale prevention has become a common pretreatment process for distillation desalting plants in the past 10-15 years.) The apparent dissociation constants for carbonate, bicarbonate, borate. and water which were used in the derivation were calculated from empirical relationships by Sverdrup, et al. (1942), and the apparent solubility product constant for aragonite was taken from Riley and Skirrow (1965). Equilibrium was assumed. The total alkalinity expressed as equivalents per liter is given by the sum of the contributions by carbonate, bicarbonate, borate, and hydroxide, ignoring any contribution by organic matter or other trace materials.
Each of the terms in eq 1 can be expanded to express the alkalinity in terms of the hydrogen ion concentration, with the calcium ion concentration assumed to be constant a t 0.003 mol/l. The hydroxide ion concentration is given by
COS
~t~~
=
[c~’+][co,’-]
(3)
where [Caz+]and [C032-] are the molar quantities of calcium and carbonate ion and K’sp is the apparent solubility product constant. The molecular weight of carbonate is twice the equivalent weight, so = 2
[co,’-j
(4)
The contribution of bicarbonate ion concentration to alkalinity is calculated from the second association constant of carbonic acid [H+][CO
K’? =
’-3
[HC03;7-
where [HCOB-] and K’z are the molar bicarbonate ion concentration and apparent second dissociation constant for carbonic acid, respectively. Equations 3 and 5 can be combined and rearranged to give the bicarbonate ion concentration in terms of K’,p and the calcium and hydrogen ion concentrations.
The contribution of borate ion to alkalinity can be computed from the expression for the first dissociation constant for boric acid.
[HzB03-] and [H3B03] are the molar quantities of monovalent borate, and boric acid, and K’H is the apparent first dissociation constant. At the p H of LMC product water, the total boron concentration, designated [B], is the sum of boric acid and borate ion concentrations.
[g]=
[H3B03] + [H,BO,-]
(8)
Substituting eq 8 into eq 7 and rearranging gives where [OH- ] and [H ’1 are the molar quantities of hydroxide and hydrogen ion. and K’, is the apparent dissociation constant for sea water. The contribution of carbonate ion to the total alkalinity is calculated from the apparent solubility product constant for aragonite, since a t equilibrium L M C product water is saturated.
By substituting eq 2 , 4, 6, and 9 into eq 1, the equilibrium alkalinity for LMC product water can be calculated from combinations of p H and calcium expressed as milligrams per liter of calcium carbonate. Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
207
Table I1
400
LMC
t
80
Component (lb/106 lb) Sea wateP
~~
a 5
9 5
90
100
P H
Figure 4. LMC product water alkalinity-calculated us. observed. The LMC product water calcium content was held at 0.003 M , the pH was varied by adjusting the Ca(0H)Z dosage, and the product alkalinity was determined ( w ) . At pH 5 8.5 the alkalinity increases faster than the theoretical curve (solid line). This is due to the decreasing rate of reaction at low pH values.
Chloride Sodium Magnesium Sulfate Calcium Potassium Bicarbonate Carbonate Boron pH (unitless)
18,980 10,561 1,272 2,649 400 380 136 7 4.6 8.2
= 50,000
{&
K’B
i[H’]
+
[rs]
LMC Product Water Sea water softened by the LMC process undergoes a 70% reduction in calcium content, a magnesium augmentation of 14-18%, and a reduction in total alkalinity of 12-15%. Both the reduced calcium content and the formation of soluble magnesium sulfate complexes a t high temperatures (Marshall and Slusher, 1968; Templeton and Rodgers, 1967) make possible distillation desalting a t maximum brine temperatures in excess of 335°F. This was demonstrated in tests on a high-temperature distillation plant operated a t the San Diego Test Facility using LMC pretreatment. In addition, the lower total alkalinity permits the use of less sulfuric acid for decarbonation, keeping the sulfate in concentration a t a lower level than with conventional acid pretreatment. The composition of LMC product water is compared with natural sea water in Table 11. LMC-treated sea water has several advantages over natural sea water. (1) An extended flashdown due a higher maximum brine temperature results in a higher thermal efficiency. (2) A 12-15% reduction in total alkalinity results in a corresponding decrease in sulfuric acid consumption. (3) Silt and other particulate matter normally present in sea water are removed during LMC processing. 208
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
X
l o 6 lb/hr
Cents/1o3 gal
+
KrB
where A = total alkalinity equivalent to mg/l. of CaC03; K‘, = 1.194 X mo12/1.2; K r s p = 1.1 X 10-6 mo12/ 1.2; K’z = 9.583 X 10-lO mol/l.; K‘g = 1.869 X 10-9 mol/ 1.; [B] = 4.255 X mol/l.; [Ca2+] = 0.003 mol/l. (for LMC product water); and [H+]= 10-PH mol/l. Using eq 10 LMC product containing 120 mg of calcium/l. (0.003 mol/l.) would exhibit a minimum alkalinity a t pH 9.62 of 63 mg-equiv of calcium carbonate/. In practice the LMC process must be operated below p H 9.5 to prevent magnesium hydroxide precipitation. The LMC product water design pH was 9.2, which a t 120 mg of calcium/l. would have an equilibrium alkalinity of 65 mg-equiv of calcium carbonate. In practice LMC product water with 120 mg of calcium/l. and pH 9.2 typically has an alkalinity around 100 mg-equiv of calcium carbonate. Alkalinity measurements made on plant product water are compared in Figure 4 with the theoretical curve from eq 10.
1456 120 26.4 36.8 4.6 9.2
Table 111. Total Operating Cost Summary in Terms of 1972 Dollars 6.8
A
product water
Capital charges Electricity Oil Makeup limestone Labor Maintenance Total
3.664 1.022 1.180 0.540 1.679 0.202 8.287
% 45 12
14 7 20 3 100
34 x
lo6 lb/hr
Cents/lo3 gal 2.139 0.752 1.180 0.540 0.382 0.104 5.097
% 42 15 23 11 7 2 100
(4) Brines unsuitable for desalting plant feed may become usable with LMC pretreatment. (5) Mineral byproduct recovery may be feasible with the higher desalting plant waste brine concentration ratios possible with LMC pretreatment (est. max. C.F. = 5.5 (g T < 90°F for LMC treated sea water).
Economics Operating and capital cost estimates were made for LMC production plants with capacities of 6,800,000 lb/hr and 34,000,000 lb/hr of treated sea water with 70% calcium removal. These plants would support a 10 million gal/ day (MGD) and a 50 MGD desalting unit a t a brine concentration factor of 2.1. Table I11 summarizes the operating costs for both plants. The costs are given in cents per thousand gallons of treated sea water. Detailed cost breakdown and design data are given in a report to OSW by Mavis, et al. (1972). Conclusions The utility of the LMC process as a pretreatment process for high temperature distillation desalting plants was demonstrated. Testing a t the San Diego Saline Water Test Facility confirmed that an evaporator could be operated a t a maximum brine temperature of a t least 335°F and that an LMC plant could be designed to have a high on-stream factor. The question of economic feasibility must be assessed in comparison with other pretreatment processes for each potential plant siting. It appears that the process would be well suited for use with concentrated sea water brines similar to those found in some areas of the Middle East or inland salt lakes. Acknowledgment Research and development was conducted by Burns and Roe,New Jersey, under contract to the U S . Department of the Interior, Office of Saline Water.
Literature Cited Grace, W . R. 8 Co., "Development of Precipitation Processes for Removal of Scale Formers from Sea Water," OSW R&D Rem. No. 192% May 1966. Grace, W. R. 8 Co. for OSW Research Division. Rept. NO. RES 67-57, 1567. Hattiangadi, U. S., Chem. Eng., 78, 104 (1971) Lawrence, R., Aerojet General Corporation Rept. to OSW. Aerojet Rept. No. 1512-F. Aug 1970. Marshall, W. L., Slusher, R., J . Chem. Eng. Data, 13,83 (1968). Mavis, J. D., "Operation of the Lime Magnesium Carbonate Plant,'' OSW Thermal Processes Division Interim Topical Rept. No. 7, Dec 1971.
Mavis, J. D., Still, C. O., Checkovich, A,, "Conceptual Design and Cost Estimates for Lime-Magnesium Carbonate (LMC) Plants," OSW Thermal Processes Division Interim Topical Rept. No. 6, Apr 1972. Riley, J. p . , Skirrow, G., Ed., "Chemical Oceanography," pp 131-134, Academic Press, New York, N.Y., 1965. Sverdrup, H. U . , Johnson, M . W., Fleming, R . H,, "The Oceans," pp 198-200, Prentice-Hall, Englewood Cliffs, N.J.. 1942. Templeton, C. C., Rodgers. J. C., J. Chem. Eng. Data, 12, 536-547 (1567).
Receiced for reuieu. M a y 17, 1974 Accepted December 16, 1974
A Generalized Method for Predicting Second Virial Coefficients J. George Hayden and John P. O'Connell* Department of Chemical Engineering, University of Florida, Gainesville Florida 3261 I
Expressions for predicting pure-component and cross second virial coefficients for simple a n d complex systems have been developed from the bound-pair formalism of Stogryn and Hirschfelder. For pure components, t h e generalized correlation requires t h e critical temperature and pressure, Thompson's mean radius of gyration or t h e parachor, dipole moment, and, if appropriate, a parameter to describe c h e m i cal association which depends only in t h e t y p e of group (hydroxyl, amine, ester, carboxylic acid, etc.). Mixing rules have been developed for predicting cross coefficients and solvation effects can b e a c counted for in a similar manner to association. Agreement with experimental data on 39 nonpolar and 102 polar and associating compounds, 119 mixed nonpolar systems, and 73 mixed systems involving polar compounds, is comparable to or better than that of several other correlations including those which require data to obtain parameters. T h e method should b e most accurate for systems of complex molecules where no data are available In order to accurately predict phase equilibria involving the vapor phase at pressures above atmospheric, deviations from the perfect-gas law usually need to be taken into account (Prausnitz, 1969; Nagata and Yasuda, 1974). The vinal equation terminated at the second coefficient is a simple but accurate method for conditions up to a density of about one-half the critical and has been employed in completely developed methods for predicting vapor-liquid equilibria such as Prausnitz et al. (1967). Several analytical methods for predicting values for the second virial coefficient have been developed (Black, 1958; O'Connell and Prausnitz, 1967; Kreglewski, 1969; Nothnagel et al., 1973; Tsonopoulos, 1974), but except for the last, all suffer from the disadvantage of often requiring one or more parameters that must be obtained from data, or the results are too inaccurate to be acceptable. This work develops an accurate method for predicting second virial coefficients using only critical properties and molecular parameters. all of which may usually be estimated from molecular structure to the required accuracy. From extensive comparisons with pure component and cross vinal coefficient data, the present method appears to be more consistently accurate than any other purely predictive method. In addition, for strongly associating substances, the method predicts association effects at higher densities in a realistic fashion (Nothnagel et al., 1973) using a parameter which depends only on the group interaction.
Basic Expressions The virial equation of state relates the compressibility factor to the independent intensive variables of composition, temperature, and pressure or density. Making suitable thermodynamic manipulation of this equation of
state yields the vapor phase fugacity which is used in obtaining K factors and relative volatilities. Since the accuracy of the fugacity and compressibility are about the same for the pressure-explicit and density-explicit equations truncated at the second virial coefficient (Prausnitz, 1969), and systems are usually specified by temperature, pressure, and composition, the most convenient form of the virial equation to be used is PV RT
z=---=l+-
BP RT
where u is the molar volume and, in a mixture of N components N
N
i.1
j;l
2 1~ i y j B i j ( T )
B =
( 2)
Here y is the mole fraction and B,,(T) is the second virial coefficient characterizing pair interactions between an 'i" and a "j" molecule, a function only of temperature. The vapor fugacity is given by fi'
=
where the fugacity coefficient is given by L
j=1
For substances such as carboxylic acids which associate very strongly, the virial equation is not valid. However, the "chemical theory" for nonideality can give good predictions in such cases when an equilibrium constant for association is available (Nothnagel et al., 1973). Values of second virial coefficients can be related to the equilibrium constant in a simple way, so if a correlation yields accuInd. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1 9 7 5
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