410
Ind. Eng. Chem. Fundam. 1980, 79, 410-414
It would be valuable to have an analytic means of calculating the Jacobian matrix for the case of nonideal liquid solutions, and work on this is continuing. Notation General. A lower case symbol is a scalar. An upper case symbol is a column vector. An upper case symbol with a superscript is a column vector formed by removing the column indicated by the superscript from the matrix with the same name. Boldface upper case symbols are matrices. If a symbol is defined as a vector, then when boldface, it represents a diagonal matrix with the same elements as the vector. A matrix with a superscript is always a diagonal matrix. Nomenclature E = error vector; e, is the error in the sum of compositions in stage i e = the error in the sum of composition in stage n in the constant relative volatility case f = moles of component j in the feed per mole of feed = identity matrix J = Jacobian correction matrix, j , is (de,/dt,) or (de,/dx,) K = equilibrium ratio matrix; k , is the equilibrium ratio for component j in stage i rn = number of components n = number of equilibrium stages rD = distillate fraction, moles of distillate per mole of feed RJ = vector; rl = fJ/rD; r, = 0, i # 1 Rl = matrix; rll = fJ/rD; rlJ.= 0, ij # 1 S = an n X n identity matrix with r D / ( l - rD) in the nth row and first column T = temperature vector; t , is temperature in stage i
f
W = matrix ( n X n ) with a 1 in elements to the right of the main diagonal X = liquid composition matrix; xi, is composition of component j in liquid in stage i x = iteration vector of bottoms compositionsfor nonideal case; contains (rn - 1) elements; xi is equal to xn, Y = vapor composition matrix; yii is composition of component j in stage i 2 = matrix (n X rn) which is all zeroes except for row n; zni is fj/(1 - r d a,* = relative volatility of component j with respect to component 1, assumed constant ai,= relative volatility of component j with respect to component 1 in stage i v = subscript used to indicate iteration number u = Euclidean norm of E t j = size of the increment in composition of component j for numerical calculation of derivatives L i t e r a t u r e Cited Bennett, C. O., Brasket, E. J., Tierney, J. W., AIChEJ., 6, 67 (1960). Fenske, M., Ind. Eng. Chem., 24, 482 (1932). Filippov, G.. Shevyreva, L., Chem. Technol. Fuels Oi/(USSR),10, 460 (1974). Holland, C. D., "Multicomponent Distillation", Chapter 9, McClaw Hill, New Ywk, 1963. Holland, C. D., Pendon, G. P., Hydrocarbon Process., 54(7), 148 (1974). Lyster, W.N., Sullhran, S . L., Jr., Mcbnough, J. A,, Holknd, C. D., Pet. Refiner, 39(8), 121 (1960). Prausnitz, J. M., Eckert, C. A., Orye, R. V., O'Connell. J. P., "Computer Calculations for Multicomponent Vapor-Liquid Equilibria", pp 216-224, Prentice-Hall, Englewood Cliffs, N.J., 1967. Underwood, A., Trans. Inst. Chem. Eng., 10, 112 (1932). White, R., Pet. Process., 8 , 1336 (1953).
Received for review September 18, 1979 Accepted July 7 , 1980
Experimental Thermodynamic Parameters for the Prediction of Natural Gas Hydrate Dissociation Conditions P. B. Dharmawardhana, W. R. Parrlsh,' and E. D. Sloan' Chemical & Petroleum Refining Engineering Department, Colorado School of Mines, Golden, Colorado 8040 1
Two basic parameters are needed for accurate prediction of natural gas hydrate dissociation conditions. These two parameters, the chemical potential difference and the enthalpy difference between the empty hydrate lattice and ice at 273.15K and 0 kPa, were measured experimentally using cyclopropane hydrates. The accuracy of these two parameters was demonstrated by comparing the predicted values for the dissociation pressure of simulated natural gas mixtures of methane, ethane, and propane with the experimental data. A favorable comparison with other current prediction methods was made along the three phase (vapor-hydrate-aqueous phase) boundary for t h e above simulated mixtures.
Introduction Traditionally natural gas hydrates have been of interest in the gas industry mainly because they can plug transmission lines. However, there is additional interest in hydrates because there may be vast amounts of natural gas reserves in the form of hydrates in the arctic area and in the deep sea formations as shown by Katz (1971,1972). An accurate method of predicting the thermodynamic properties of hydrates is necessary for preventing hydrate Phillips Petroleum Co., Bartlesville, OK 74003.
formation in pipelines and for determining the feasibility of producing the gas hydrate reserves. Natural gas hydrates are nonstoichiometric compounds which form in either of two structures. Each structure is composed of cavities of different sizes wherein the gas molecule is encaged. The structure that forms will depend on the composition of the gas and the temperature of the system. Natural gas mixtures that contain propane and/or isobutane normally form Structure I1 whereas mixtures that do not have these components normally form in Structure I; Holder (1976) has indicated exceptions. Therefore, any accurate method for describing hydrate
0196-4313/80/1019-0410$01,00/00 1980 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
411
Table I. Comparison of Values for A p o . o P - a and Aho.ofl-O:between the Empty Hydrate and Ice 273.15 K and 0 kPa A p o , o p - o r J/mol ,
investigator
basis
van der Waals and Platteeuw (1959) Barrer and Ruzicka (1962) Sortland and Robinson (1964) Child (1964) Parrish and Prausnitz (1972) Holder (1976) Holder and Corbin (1979) present work (1979)
expt. expt. expt. estimate estimate estimate estimate expt.
St. I
phase behavior must include properties for both structures. Currently, the gas industry uses the work of Parrish and Prausnitz (1972) to predict hydrate phase behavior. This method uses the statistical thermodynamic model for clathrates developed by van der Waals and Platteeuw (1959). This model requires properties of hydrate lattice along with the difference in chemical potential Apo,op-a enthalpy Aho,Op-a, and volume between the empty hydrate lattice and coexisting phases, water, or ice, a t the ice point. These three parameters Apo,oB-a,Ah0,08-a, and A U ~ , ~aFt "0 "C and 0 kPa pressure are common parameters for all hydrates in a given structure and may be considered as standard bases for prediction of the hydrate formation conditions. The necessary geometric properties and volume difference for both structures were measured by von Stackelberg and Mdler (1954) using X-ray crystallography. This paper presents new values, for Apo,gB-"and Ah0,O"" for both Structures I and I1 based upon experiments with cyclopropane hydrates.
Previous Work The empty hydrate lattice does not exist in nature but it attains stability by encagement of gas molecules in the cavities. Therefore, the empty hydrate lattice properties must be determined indirectly by obtaining the filled hydrate composition or by numerical estimation. Table I lists the various reported values for Apo,08-aand Aho,08-a,along with results of the present work. The chemical potential difference a t a given temperature and pressure can be computed from the hydrate composition using van der Waals and Platteeuw's (1959) model as given below Apo-" =
-RTXu, In (1 - cy,) m
(1)
mi
where u, is the number of cavities of the type "m" per water molecule of a unit cell of the hydrate lattice and ymj is the fraction of cavities of the type "m" filled by the gas molecules of the type j per unit cell. van der Waals and Platteeuw assumed: (1) that there is no multiple occupancy of the cavities by the gas molecules; (2) the gas molecules do not distort the hydrate lattice; (3) no interaction between encaged molecules and hydrate lattice; and (4) that classical statistics is valid. The value of Apo,o@-" that was used for Structure I hydrate by van der Waals and Platteeuw (1959) was based on bromine hydrate which was latter shown by Allen and Jeffrey (1963) to form neither Structure I nor 11. Sortland and Robinson (1964) estimated Apo,o@-a for hydrate I1 by differentiating sulfur hexafluoride P-T dissociation data to obtain heats of formation and then hydrate composition. The values of Apo,o@-areported by Barrer and Ruzicka (1975) are approximately one-half of the other values reported in the literature. This is probably due to the occlusion of mother liquor in the hydrate and/or meta stability of the hydrate. Childs (1964) estimated and Aho,oO-afor both structures by comparing entropies of formation of hydrates
ALho,oU-or,Jim01
St. I
St. I1
699
-
1255 1264 1155 127611294 1297
837 368-536 883 795 -
-
937
St. I1 0
0
-
-
795 808
1151 381 941/2213 1389
0
-
1025
-
PRESSURE INDICATOR TEMPERATURE ULTRASONIC INDICATOR CONDUCTIVITY GENERAmR
1
+PRESSURE
L O W TEMPERATURE BATH
SENSOR
L C Y LGA' i N ER
Figure 1. Hydrate experimental apparatus.
with those of other clathrates. Parrish and Prausnitz (1972), Holder (1976), and Holder and Corbin (1979) numerically optimized A~~O,~B-" for the two structures to obtain better fits of the statistical mechanics model to existing hydrate data. Values for Aho,O@-a and values for Apo,o"-@ based on experimental data for Structure I have not been reported. The diversity of values for Apo,08-" and Aho,ofl-a indicates the problems of occlusion of mother liquor and accurate measurements of the gas to water ratio. Accwate determinations of these parameters can lead to better predictions of dissociation conditions of hydrates. In the present work the amount of gas enclathrated was determined by volumetric measurements of the gas consumed upon hydrate formation. The amount of water consumed was determined by specific conductivity increase in a KCl solution from which hydrates were formed; salt is not enclathrated. The ratio of gas to water in the cavity was used, together with eq 1 to determine App-"(T,p) a t the experimental conditions. An ultrasonic agitator was used to inhibit metastability and t o prevent occlusion of the mother liquor. Apparatus and Procedures The unique property of cyclopropane t o form either Structure I or I1 hydrates as a function of temperature (Hafemann and Miller, 1969) enables a single gas to be used to obtain data for both hydrate structures. The cyclopropane was obtained from the Matheson Gas Co. with a purity of 99.86% cyclopropane and stated impurities of propylene (C1230 ppm), allene (1
