Experimental Determination of Gas Hydrate Equilibrium below the Ice Point Benjamin J. Falabella and Marcel Vanpee* Chemical Engineering Department University of Massachusetts. Amherst. Massachusetts
07 002
The equilibrium curves of the methane and ethane gas hydrates were experimentally determined at pressures below 1 atm with corresponding temperatures as low a s 148'K. The hydrates were formed by direct reaction of the gases with ice crystals under agitation. The pressure range observed was 67.6-4.0 cm Hg for methane and 73.6-6.2 cm Hg for ethane. Extrapolation of the data to 1 atm yielded equilibrium temperatures of 193.2 and 240.8"K for methane and ethane, respectively. These results agree with the extrapolations of the lowest pressure data previously available but disagree with the values of 244 and 257°K currently presented in the literature.
Gas hydrates, which are among the best known examples of a class of nonstoichiometric inclusion complexes called clathrates, are ice-like phases formed from gases and water under suitable conditions of temperature and pressure. Hydrate stability has been traditionally treated as an example of three-phase equilibrium between the gas, the gas hydrate, and liquid or solid water. The two hydrate equilibrium values most often tabulated in texts and data collections are the equilibrium pressure a t 0°C and the equilibrium temperature a t 1 atm. The first value has generally been determined by direct experiment. However, the equilibrium temperature at 1 atm is below the ice point for most gases and, until recently, no technique was available for its direct experimental determination. The published temperature values are rough estimates obtained from extrapolations of the available equilibrium data above the ice point. The purpose of the present research was to obtain accurate equilibrium values for the hydrates of methane and ethane which are the main components of natural gas. As will be shown. the values obtained greatly differ from the previously available extrapolations. These results indicate that measurements on other hydrates are warranted. Previous Work
Villard (1888) reported the first equilibrium measurements on the methane and ethane hydrates a t temperatures of 0°C and higher. At O"C, pressures of 27 atm for methane and 6 a t m for ethane were observed. The technique used was to determine the decomposition pressure by visually observing the disappearance of the solid hydrate phase in the liquid water-gas system as the pressure was reduced or the temperature increased. This method will not work below the ice point, because the hydrate decomposition is not accompanied by a visible phase change in this region. Considerable effort has been directed toward extending the experimental hydrate equilibrium curves to higher pressures. For example, Marshall (1964) reported equilibrium measurements on the methane hydrate a t pressures up to 3900 atm. Van der Waals and Platteeuw (1959) developed an application of statistical mechanics for the thermodynamic treatment of gas hydrates. Low Pressure. The first tabulation of equilibrium temperatures a t atmospheric pressure appears to be that of de Forcrand (1902b). Because experimental data below the ice point were not available, estimated values were reported which ignored the changes in AH a t the ice point, the magnitudes of which were a t the time unknown. Deaton and Frost (1946) reported the first equilibrium data on the methane and ethane hydrates below the ice 228
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point. The technique used was first to form the hydrate a t 0°C and then cool the pressure vessel to the desired temperature. Gas was then released in small increments while the pressure was monitored for the plateau which indicated hydrate decomposition. The minimum pressures obtained were 17.7 and 3.1 atm for the methane and ethane hydrates, respectively. The data allowed accurate calculation of the heats of formation below 0°C and permitted reasonable extrapolation of the equilibrium curves to atmospheric pressure. Barrer and Edge (1967) reported the first experimental determinations of the equilibrium of the argon, krypton, and xenon hydrates a t pressures of 1 atm and lower. These hydrates were directly formed from gas and ice. Equilibrium conditions were studied a t temperatures as low as 902°K by pressure-volume observations. Their method was used in the present study. Experimental Apparatus
In the experimental plan, ice crystals under bombardment by metal balls in a low-temperature reaction vessel were contacted with the hydrate forming gas. The agitation was found to be necessary by Barrer and Ruzicka (1962) if reasonable reaction rates were to be obtained. The gas was supplied by a volumetric system with which the total amount of gas absorbed as well as its pressure could be monitored. The apparatus, which closely parallels that described by Barrer and Edge (1967), was thus composed of three main units: the reaction vessel, the low-temperature thermostat (cryostat), and the volumetric system. A schematic diagram of the apparatus is shown in Figure 1. Reaction Vessel. The hydrate formation took place in a small glass bulb which was about half full of YB-in. diameter chromed-steel balls. It was designed for deep immersion into the cryostat and a minimum dead volume. Agitation was accomplished by a mechanical shaker which grasped the vessel near its top and rocked it back and forth a t about 5 cycles/sec. A small but very important detail was the connection of the reaction vessel to the volumetric system. The line had to be flexible and light in weight so that the vessel could be shaken without danger of breaking the glass. In addition, it was required to have a very low leak rate even under high vacuum. These requirements could not be met by any of the elastomeric tubings tested, so a double-walled line was adopted. By controling the pressure in the annular space and thus reducing the pressure difference ( a common vacuum sealing technique), leakage was reduced to a negligible rate. Cryostat. The cryostat, shown in Figure 2, was built
thus increase the sensitivity of the system to gas absorption due to hydrate formation. Evacuation of the system, when required, was accomplished with a small mechanical-oil-diffusion pump combination. The gases, methane and ethane, were obtained in high purity (99.97% minimum). They were introduced into the system via a valve arrangement that permitted the entire apparatus up to the tank valve to be evacuated, thus avoiding any contamination.
Figure 1. Diagram of the experimental apparatus: A, cryostat; B, shaker; C, vacuum manifold; D, flexible tube; E, water bulb; F, gas supply; G, reaction vessel; H, gas buret; I, McLeod gauge; J, manometer.
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Figure 2. Details of the cryostat: A, aluminum foil; B, guard; C, reaction vessel; D, platinum resistance thermometer; E, stirrer; F, heater; G, thermoregulator.
along the lines of the design described by Barrer and Stuart (1959); the main differences were the substitution of metal components for glass and the use of a resistance thermoregulator instead of a bimetallic spiral. A 2-1. stainless'steel bucket was fitted with an aluminum cover sealed with a spring-backed Teflon gasket. The openings in the cover required for the regulating equipment were provided with threaded Teflon sleeves. A larger open hole was provided for the reaction vessel which had to be shaken. The bucket was filled with a low-temperature circulation fluid and the contents were stirred by a glass propeller. Temperature control at the required low value (down to 148°K) was achieved by a thermal balance arrangement. Heat was removed by placing the bucket in thermal contact with liquid nitrogen (LN2) uia an aluminum-foil cylinder. A heater, which added sufficient heat to maintain the desired temperature, was controlled by a thermoregulator employing a platinum resistance-sensing element. The actual temperature was measured by a calibrated platinum resistance thermometer read by a precision bridge. Temperature regulation of icO.05"C or better could be maintained. Volumetric System. This part of the apparatus was all glass. It was basically a mercury-displacement gas buret connected in parallel with a constant-volume mercury manometer. The mercury levels in both were measured with a precision cathetometer. All of the connections between the components were made with 2-mm i.d. capillary tubing in order to 'minimize the dead volume and
Experimental Procedure The experimental method was basically the same as that described by Barrer and Edge (1967). Minor changes were required due to the before-mentioned differences in the apparatus. First, the entire system was pumped down to about one 1 p of mercury in pressure. Then, with the volumetric system isolated from the reaction vessel and water bulb section of the apparatus, a weighed quantity of water, in the range of 0.1 g, was vacuum-distilled from the water bulb into the reaction vessel. This was accomplished by first degassing the water in the bulb by several freeze-thaw cycles using LN2 as the refrigerant. Next, the actual distillation was carried out by having the tip of the reaction vessel immersed in LN2 while the frozen water in the bulb was warmed. After thus charging the reaction vessel with water in the form of fine ice crystals, it was isolated by means of a stopcock and immersed in the cryostat which had been set a t the desired temperature. The evacuated volumetric system was then charged with either methane or ethane. Equilibrium. Having prepared the system in the preceding manner, the shaker was set in motion and the stopcock between the volumetric system and the reaction vessel was opened. The ice crystals were shaken in the presence of the hydrate forming gas under thermodynamicly favorable conditions until an absorption of gas representing a 5-10% conversion of the ice to hydrate was observed. The pressure was then reduced in steps (by lowering the mercury level in the burette) at constant temperature until the pressure was observed to increase, thus indicating hydrate decomposition. Adjustment of the pressure by decreasing increments was then made in order to alternate between formation and decomposition, until a pressure was reached which remained constant when the system volume was held constant. This was taken to be the equilibrium pressure of the hydrate. Experimental Error. A dynamic approach to equilibrium was necessary for a number of reasons. First. near equilibrium the rate of reaction is not only very slow, but if no hydrate is present, the induction period for formation becomes indefinitely long. Secondly, as indicated by the absorption isotherm calculations and experiments of Barrer and Edge (1967), the hydrate structure may exist with varying fractional occupancy of the cavities when exposed to gas a t pressures greater than the minimum required to stabilize the crystal lattice. Since the hydrateice-gas interaction is a relatively slow surface reaction, a considerable length of time may be required for the hydrate to reach thermodynamic equilibrium. It was found that the equilibrium could therefore be located more rapidly and accurately by the search technique. The overall accuracy of the equilibrium measurements was, however, still limited by the hydrate kinetics. By observing the hydrate when small pressure displacements were made from the apparent equilibrium and by comparing check experiments, it was concluded that the uncertainty in the equilibrium pressures was about ic0.5 cm Hg . Ind. Eng. Chem., Fundam., Vol. 13, No. 3, 1974
229
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Figure 3. Gas-hydrate-water equilibrium curve for methane.
Figure 4. Gas-hydrate-water equilibrium curve for ethane.
Experimental Results Methane. The pressure-temperature equilibrium locus of the methane hydrate was studied a t pressures between 67.6 and 4.0 cm Hg. Each of the five points shown in Figure 3 is the result of several check experiments. They are, however, not average values. Each is the result of a run in which the hydrate was observed for a long period of time over a small pressure range near equilibrium, which was determined by previous measurements. The typical duration of such an experiment was about 12 hr. Some of the high-pressure data of Deaton and Frost (1946) are also presented in Figure 3 for perspective. A short-range extrapolation of the experimental data to atmospheric pressure yielded an equilibrium temperature of 193.2"K. Ethane. In a similar manner, pressure-temperature equilibrium points of the ethane hydrate were obtained at pressures between 73.6 and 6.2 cm Hg. These results, plus some of the high-pressure data of Deaton and Frost (1946), are presented in Figure 4. Extrapolation of the data to 1 atm resulted in a value of 240.8"K for the equilibrium temperature.
ane hydrate the extrapolated value was 200°K compared to an experimental 193.2"K, and in the case of ethane there was no significant difference between the extrapolated value and the experimental value of 240.8"K. The temperature a t which a hydrate has a dissociation pressure of 1 atm has been used as a characteristic parameter since the pioneering work of de Forcrand (1902). He reported that for hydrates where this temperature was below 0°C the values that he tabulated were obtained by extrapolating the data above 0°C to a pressure of 1 atm without using the change in the heat of formation at the ice point. The magnitude of this change was not known at the time. The error in this estimation was quite large. For example, a temperature of 244°K was found for the methane hydrate and 257.2"K for ethane. The reason behind the preceding analysis is that an extensive literature survey has shown that virtually all of the tabulated values for the equilibrium temperature at atmospheric pressure for the methane and ethane hydrates are either traceable to or nearly identical with the original estimates of de Forcrand. Typical examples are those of King (1961) and Stackelburg and Muller (1954). The work of Barrer and Edge (1967) has shown that a similar situation existed with respect to the hydrates of the inert gases; argon, krypton, and xenon. For example, their direct experiment yielded an equilibrium temperature of 149.6"K at 1 atm for the argon hydrate while the commonly tabulated value was about 230°K (King, 1961).
Comparison with Previous Work The lowest pressure experimental work previously reported appears to be that of Deaton and Frost (1946), where equilibrium measurements were made below the ice point. The minimum pressures reported were 17.7 and 3.1 atm for methane and ethane, respectively. Along the pressure-temperature, three-phase equilibrium between gas hydrate, ice or liquid water, and gas, the system is univariant, and hence the Clapeyron equation may be applied. Making the assumptions of the Clapeyron-Clausius equation, most importantly constant heat of formation and ideal gas phase, gas hydrate data may be extrapolated by plotting In P us. 1/T, the slope being proportional to AH. The combined effect of the nonidealities is to impart a curve to the plot of In P us. 1/T. As a result, extrapolation of the equilibrium locus is accurate only over small temperature ranges. Accepting the uncertainties in extrapolating over large ranges, the data of Deaton and Frost were extended to 1 atm for comparison to the findings of the present study. This is graphicly illustrated in Figures 3 and 4. For meth230
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Conclusions The hydrate equilibria of methane and ethane at pressures below 1 atm have been experimentally determined for apparently the first time. The results of this work agree, to within the accuracy of the comparison, with the extrapolations of the lowest pressure data previously available. A literature review has shown that the widely published values for the equilibrium temperatures of these hydrates at 1 atmosphere are estimates that predate any direct experimental work. Nomenclature AH = heat of formation, kcal/mol 'In = natural logarithm
P = absolute pressure, cm Hg at 0°C T = absolute temperature, O K Literature Cited Barrer. R. M . , Edge, A. V. J.. Proc. Roy. SOC.,Ser. A, 300, 1 (1967). Barrer, R. M., Ruzicka. D. J., Trans. Faraday Soc., 58, 2262 (1962). Barrer, R. M., Stuart, W. I., R o c . Roy. Soc., Ser. A, 249, 464 (1959). Deaton, W. M., Frost, E. M., Jr., U. S. Bur. Mines, Monogr., 8 (1946). de Forcrand, R . , Compt. Rend., 134, 835 (1902a). de Forcrand. R., Compt. Rend.. 135, 959 (1902b).
King, M. B.. “Phase Equilibrium in Mixtures,” pp 160-1 73, Pergamon. New York, N. Y., 1961. Marshall, D. R., Saito, S., Kobayashi, R.,A.l.Ch.E. J., 10, 202 (1964). Stackleberg, M. yon, Muller, H. R., Z.Elekfrochem., 58, 25 (1954). van der Waals, J. H.. Platteeuw, J. C., Advan. Chem. Phys., 2, 1 (1969). Viilard, P., Compt. Rend., 106, 1602 (1888).
Received for review October 5 , 1973 Accepted March 5,1974 This research was sponsored by the Department of the Interior, Bureau of Mines, under Grant No. G0111343.
A Stronger Version of the Discrete Minimum Principle Arthur W. Westerberg* and George Stephanopoulos Department of Chemical Engineering, University of Florida, Gainesville, Florida 3267 7
The discrete form of Pontryagin’s Minimum Principle proposed by a number of authors has been shown by .others in the past to be fallacious; only a weak result can be obtained. Due to the mathematical character of the objective function and the stage transformation equations, only a small class of chemical engineering problems have been solved by the strong discrete minimum principle. This paper presents a method to overcome the previous shortcomings of the strong principle. An algorithmic procedure is developed which uses this new version. Numerical examples are provided to clarify the approach and demonstrate its usefulness.
1. Introduction
Pontryagin’s minimum principle (Pontryagin, et al., 1962) is a well-known method to solve a wide class of extremal problems associated with given initial conditions. A discrete analog of the minimum principle, where the differential equations are substituted by difference equations, is not valid in general but only in certain almost trivial cases. Rozonoer (1959) first pointed out this fact. Katz (1962) and Fan and Wang (1964) later on developed a discrete minimum principle which was shown to be fallacious by Horn and Jackson (1965a), by means of simple counterexamples. As was pointed out by Horn and Jackson (1965a,b) and lucidly presented by Denn (1969), the failure of a strong minimum principle lies in the fact that we cannot deduce the nature of the stationary values of the Hamiltonian from a consideration of first-order variations only. Inclusion of the second-order terms does not help to draw a general conclusion about the nature of the stationary points in advance. A weak minimum principle which relates the solution of the problem to a stationary point of the Hamiltonian exists and is valid (Horn, 1961; Jackson, 1964). In the case of control systems described by differential equations, time, by its evolution on a continuum, has a “convexifying” effect (Halkin, 1966) which does not make necessary the addition of some convexity assumptions to the specification of the problem. Thus a strong minimum principle can be applied for these problems, requiring the minimization of the Hamiltonian even in the case that a continuous problem is solved by discretizing it with respect to the time and using a strong discrete minimum principle. For discrete, staged systems described by difference equations, the evolution of the system does not have any “convexifying” effect and, in order to obtain a minimum principle, we must add some convexity assumptions to the problem specification or reformulate the problem in
an equivalent form which possesses inherently the convexity assumptions. This present work belongs to the second class. In the present work we propose to show a strong version of the minimum principle which relates the solution of the problem to a minimum point, rather than a stationary point, of the Hamiltonian. This is attained through the use of the Hestenes’ method of multipliers, a technique used effectively by the authors (Stephanopoulos and Westerberg, 1973a,b) to resolve the dual gaps of the twolevel optimization method. This method turns a stationary point of the Hamiltonian into a minimum point; thus minimum seeking algorithms can be used. In section 2, the Lagrangian formulation of the problem is given and from this the discrete minimum principle and the two-level optimization procedure are developed. In section 3, the main theorems which constitute the basis of the proposed stronger discrete minimum principle are presented. In section 4, an algorithmic procedure is developed, and in section 5, numerical examples are solved. Finally, in section 6, the various algorithms are compared and some relationships between the discrete minimum principle and the two-level optimization procedure are pointed out. 2. Statement of the Problem, a n d Its Lagrangian Formulation As a basis for the description of the minimum principle, the two-level optimization approach, their success and failure, and the development of a strong minimum principle, consider the following sequential unconstrained problem. (Constraints and recycles do not change the following results (Westerberg, 1973), and we want to keep the presentation here as simple as possible.) s
min F
=C41(x,,u,) 1-1
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