Nov 27, 2007 - Factors which determine when pipelines are profitable and efficient are; the ..... Boyle's law was the earliest form of an equation of state. Robert ...
CHAPTER ONE INTRODUCTION 1.1
BACKGROUND OF STUDY The natural gas hydrates technology includes-production,
transportation and the process of re-gasifying natural gas hydrates. Natural gas hydrate production works with the principle of using standard reactor, pumps, separators and heat exchangers in process operation. The origin of this research dated back to the works of Gudmundsson and Borrehaug (1996) who discovered that hydrate production process could be technically feasible. This led to Gudmundsson suggestion that a process by which natural gas with the addition of water are made to have contact in a stirred tank reactors at typically 65 bar and 100c. Natural gas hydrate contains up to 182 SCF of gas per volume of hydrate. Gudmundsson, Parlaktuna and Khokhar (1992, 1994) concluded that produced hydrates can be transported and stored at
1
atmospheric pressure and temperatures below 00c, typically at-150c, without significant loss of gas. The storage capacity and the stability of hydrates at atmospheric pressure, make NGH technology, a safe and economical method for handling natural gas. Two concepts, the dry hydrate concept and the hydrate slurry concept were being development at NTNU. In the dry hydrate concept natural gas hydrates are produced in stirred tank reactors and with reactors water. After being produced, the hydrates are separated from the water, and simultaneously frozen and depressurized to low temperature and atmospheric pressure. Hydrate crystals are then available for transport by bulk carriers to the market. In the hydrate slurry concept, the gas first produced in a water continuous phase and then in an oil continuous phase. The final product is a hydrate-oil slurry with free water, in pipelines to shore or transportable in shuttle tankers (Gudmundsson et al, 1998). In both concepts, gas hydrates are produced by contacting water with natural gas in a chain of stirred tank reactors.
2
The hydrate formation process has some special features. A hydrate reactor is a three-phase slurry reactor as the production water is slightly denser than the hydrate. The kinetics of hydrate formation is poorly understood, and mass and heat transfer effects are believed to be important for the overall rate of hydrate formation. Most of the reported studies of hydrate formation have been carried out in small scale batch or semi-batch reactors and not in continuous stirred tank reactor (CSTRs) as needed in a commercial process. Models based on reactor experiments are seen to be system specific. By studying the hydrate formation process in this type of reactor, established knowledge can provide valuable
information in
understanding the process. Hydrate formation is an exothermic process by producing in a water continuous system such as a stirred tank reactor, the water phase as a heat sink for the heat of hydrate formation, so that the temperature in the reactor can be controlled. The upper temperature in hydrate
3
production is restricted by be equilibrium temperature for the existence of hydrates. The
production of hydrates and its transportation, which is the
primary focus of this thesis , seeks to obtain a deeper knowledge of the processes that occurs in hydrate formation using theoretical modeling, such understanding is vital for an effective operation of a hydrate production design, commercial transportation initially for the scale-up from laboratory to pilot plant size. 1.2
STATEMENT OF PROBLEM As there tends to be a growing demand for gas consumption in
Nigeria and in the world; there may be a requirement for improved synthetic hydrate formation processes which can provide new and economic storage, transportation, safe and processing capabilities. The major problem here is when these gases are transported as hydrate the delivery time is of great concern as such even in the humid region, Ice and Promoters are added to boost delivery time
4
without untimely re-gasification while on transit. Hence, these project look at the problems of (a) Heat loss as a major factor of hydrate transportation (b) Suitable carrier materials and design (c) The economic of carrier material design, with the aim to bringing solution especially at warm temperate region as ours. Other motivating problems includes such as, -
Economic perspective: Building new pipelines and/or railway systems for transportation of natural gases are expensive and labour intensive. In addition compressed natural gas [(CNG) at 20.67 TO 24.82 Mpa (3000 to 3600 psia)] and liquefied natural gas [(LNG) requiring cryogenic temperatures at less than 112k (-1610c), also requires large capital investment.
-
Safety perspective: Elaborate safety systemsassured. For instance, on 27th November, 2007, twenty eight persons were killed and unspecified numbers wounded as a result of pipeline
5
explosion in Saudi Arabia; officials ruled out terrorism. (Saudi pipeline fire). -
Pipeline failures: Most of the pipeline used in transportation of natural gas, most often met with leakages as a result of corrosion (due to humid environment) and technical defect during designs. This is a great concern to the transportation of natural gas.
1.3 AIMS AND OBJECTIVE OF STUDY
To develop a theoretical model for the transportation of stranded gases as hydrate effectively.
From the model, produce program design for gas hydrate formed to delivery, guiding against heat losses.
To look at the economic viability of the said heat loss material design enhancing natural gas storage and transportation.
6
1.4 1.
METHODOLOGY To develop analytically, a theoretical model using heat and mass transfer coefficient and generate data for hydrate formation.
2.
From the model, produce a program that will be able to calculate hydrate transportation duration for a specified number of days.
3. 1.5
Software to be used is Hygas, of Visual Basic. NET application. LIMITATION OF THE METHODOLOGY.
It is limited to theoretical modeling with assumptions, without laboratory and bench work data.
7
CHAPTER TWO 2.0
LITERATURE REVIEW 2.1 DEFINING STRANDED GAS ‘Stranded gas’ could be defined as any hydrocarbon-based gas
which is not economical to be produced or transported to the market. Examples are associated gas, flared/vented gas, and gases that are been re-injected into the reservoir to comply to regulatory bodies of hydrocarbon exploration other than its use for maintaining reservoirpressure. Factors which determine when pipelines are profitable and efficient are; the resource volume, transportation route, the regulatory environment involved, the size of the market and the growth in demand. Other times, excess reserves might also be regarded as stranded, if they require a paltry delivery rate in the bid to oversupply the products to the local markets. Moreso, a negative economics could be as a result of technical issues and complexities or very high expenses which might be associated with the recovery or gathering of the gas.
8
2.2
REVIEW OF GAS HYDRATE FORMATION The review of hydrate formation studies with emphasis on the
rate of gas hydrate formation in water continuous system under stirred conditions is ever important for the purpose of this work. With some exceptions, the review is limited to systems where the hydrate forming components are hydrocarbons. The macroscopic studies of hydrate formation, which are most relevant to the present work, are stressed. 2.3
HYDRATE STRUCTURES Gas hydrates are crystalline solids. They are more properly
called clathrat hydrates to distinguish them for stoichiometric hydrates found in inorganic chemistry. The crystalline structure is composed of polyhedra of hydrogen-bonded water molecules. The polyhedral from cages that contains at most one guest molecule each. The cages are stabilized by van der waals forces between the water molecules and the enclatherated guest molecule. In extraordinary
9
situations, two guest molecules may enter the same cage (Solan, 1998). Only a few kind of cages may form depending on the size of the guest molecule. These cages arrange into different hydrate structures known as structure I (sI), structure II (sII) and structure H (sH) (Sloan, 1998). Methane gas forms sI hydrates, while natural gas usually forms II hydrate. 2.4
HYDRATE EQUILIBRIUM CONDITIONS At hydrate equilibrium conditions, solid hydrate may exist in
equilibrium with liquid water or ice, gas and some additive. Such temperature and pressure conditions are defined by the hydrate equilibrium curve for a given gas and water composition. Hydrates can only form at temperatures lower than the equilibrium temperature and simultaneously at pressures higher than the equilibrium pressure. The distance from the equilibrium conditions is the driving force for hydrate formation hence; the hydrate
10
equilibrium curve represents the pressure and temperature condition where the hydrates dissociate. Hydrate equilibrium curve can be predicted from statistical thermodynamics using the van der waals and Platteeuw model with some modifications, which is thoroughly explained by Sloan (1998). Also other simpler methods based on hand calculation and phase diagrams exist (Sloan, 1998). Available computer programs based on statistical thermodynamic models, such as CSMhyd (1998) or PVTsim (2001) can predict hydrate equilibrium conditions. The hydrate equilibrium curves temperature at I atm is-730c for methane gas and -360c for natural gas mixture (PVTsim, 2001). At these temperatures, hydrates are thermodynamically stable. On the other hand, Gudmundsson, Parlakuna and Khokhar (1992) observed that the natural gas concentration, in hydrate samples remained almost unchanged when stored at atmospheric pressure and at-180c for periods up to 10 days. This behaviour was explained by the formation of an ice layer around the hydrate crystals. The self-
11
preserving effect is utilized in the NGH process for atmospheric and adiabatic storage of natural gas hydrates. 2.5
MECHANISMS OF GAS HYDRATE FORMATION The thermodynamic behavior of hydrate system forms the basis
for understanding the mechanisms of hydrate formation. The timedependent phenomenon of hydrate formation kinetics is described by applying crystallization theories including nucleation, growth, agglomeration and breakage. So far, the most studied phenomena of gas hydrate formation are the nucleation and growth processes. Nucleation is a microscopic stochastic phenomenon where gas-water clusters (nuclei) grow and disperse until some nuclei have grown to a critical size. Nucleation may occur spontaneously (homogeneous nucleation), or it may be induced around impurities (heterogeneous nucleation). In contrast to primary nucleation, where nucleation commences without crystals present, secondary nucleation occurs in the vicinity of already growing crystals in the system.
12
The time from the first gas-liquid contact to the first detection of a hydrate phase is called the induction time. In macroscopic studies, the induction time has been used as a measure of the nucleation period (Skovborg et al., 1993, Monfort and Nzihou, 1993, Yousif, 1994, Natarajan, Bishnoi and Kalogerakis, 1994). These studies indicate that the induction time increases dramatically when the driving force for hydrate formation approaches zero. In 1991, Sloan and Fleyfel presented a molecular mechanism for gas hydrate nucleation from ice. The mechanism is based on observations of methane and krypton hydrate formation from ice, where induction times were clearly observed, and ethane hydrate formation, where no induction time was observe. Sloan and Fleyfel suggested that on an ice surface, free water molecules rearrange around a quest molecule to form an unstable 512 cage. Methane and krypton molecules, which enter this cage, oscillate between the 512cages of sI and sII before the cages reach a stable nuclei size. Ethane does not experience the induction time because ethane
13
stabilizes the large cages of sI, and hence, ethane does not induce oscillation between the small cages. Yousif (1994) argued that hydrate formation always starts by stabilizing the large cages of sI and sII regardless of the size of the guest molecule. Christiansen and Sloan (1994) extended the mechanism of Sloan and Fleyfel (1991) to explain hydrate formation in liquid water. They suggested for stages: molecular species, labile clusters, metastable agglomerates and stable nuclei. As gas dissolves in water, hydrogenbonded water molecules gather around a polar guest molecule to form labile clusters. Depending on the guest size, the number of water molecules in each cluster, defined by as coordination number, varies. Because the guest molecules inside the clusters are attracted to each other, the labile clusters agglomerate. However, two different coordination numbers of each hydrate structure are required to form agglomerates, or else the nucleation is inhibited until the clusters transform to fulfill the coordination number requirement. The agglomerates are in quasi equilibrium with each other and other
14
labile clusters until they exceed the critical size for stable nuclei. Stable growth begins. In contrast to Christiansen and Sloan (1994), who assumed hydrate nucleation in the liquid water phase, Rodger (1990) developed a model based on nucleation at a surface. Rodger suggested that the hydrate forming molecules are adsorbed on a water or ice surface and condensed water molecules orient around the guest molecules to form cages. Some cages may rearrange to form liquid water and free guest molecules, others may grow to eventually form stable nuclei. Kvamme (1994) critically reviewed the mechanisms of Christiansen and Sloan (1994) and Rodger (1990). He stated that the main difference is that Christiansen and Sloan relate nucleation to the liquid phase or at least the liquid side of the gas-liquid interface, while Rodger relates it to the gas side of the gas-liquid side of the gas-liquid interface. Kvamme noted that the mechanism of Christiansen and Sloan contained no element of randomness or probability. Considering the mechanism of Rodger, non-hydrate
15
formers may adsorb on the surface and cause inhomogeneous hydrate forming properties of the surface. This introduces an element of probability that can explain observed variety of induction times in parallel experiments. Kvamme (1991); proposed a new hydrate formation theory the Kvamme proposed an hypothesis which state that, initial hydrate formation occurs at the gas-liquid interface and towards the gas end (Rodger, 1990). This theory was then modeled with comparism to several experimental data. An extended diffusive interface theory models the growth rates of the metastable clusters. Comparison of modeled growth rates towards the side and towards the liquid side of the gas-liquid interface indicates that the gas side growth rates are two orders of magnitude higher than the liquid side growth rates (Kvamme, 2002). Mechanisms to describe growth have received less attention. Sloan (1998) considered growth to be a combination of mass transfer of components to the growing surface and growth at the hydrate
16
surface. He suggested a hypothesis adopted from classical crystallization including adsorption of the cluster at the hydrate surface, diffusion over the surface and integration of the cluster into a
Stable
Metastable
Labile
Concentration
kink.
Temperature Figure 2.1:
Shows a generalized solubility supersolubility diagram after Mullin (1993).
The continuous line is the solubility curve and the broken line is the supersolubility curve.
17
2.6
RATE OF GAS HYDRATE FORMATION/ LIMITATION OF EXISTING THEORIES. The rate at which hydrates are formed have been measured by
many researchers after nucleation, (which is the hydrate growth stage) .The rate of formation of hydrates is been expressed in terms of the rate of gas consumption. In the works of Knox et al, (1961); who investigated the rate of formation of propane hydrate for development of a desalination process. In that research, hydrates were produced continuously by the recycling of the water phase and the excess gas was vented. It was observed that the yield from the reactor was affected by the sub-cooling and the liquid residence. Moreover, using the principle of designation of sea water, Pangbom and Barduhn (1970) researched on the rate of formation of methyl bromide hydrate in a continuous stirred tank reactor (CSTR) of 3.8 litre. The reactor was baffled and had a downward pumping impeller. It was found that an increase in the subcooling process yielded a higher rate of hydrate formation. Increasing the impeller power input (calculated from the stirring rate) caused an increased 18
rate, however, at a certain power input no further increase in rate occurred. This was interpreted as strong evidence that above a certain power input, heat and mass transfer are no longer limiting factors on the formation rate. They did not expect that the liquid residence time influences the formation rate, however, they found that the decreases as the slurry thickens, especially at the lowest stirring rates. Experiments carried out with liquid methyl bromide resulted in higher formation rates than experiments carried out with gaseous methyl bromide. One of the very important works done on this subject was that of Vysniauskas and Bishnoi (1983) who started a research to measure the rate of hydrate formation in a semi-batch reactor .the results from this work gave rise to the theory that the rate of hydrate formation depends on the gas-liquid pressure, interfacial area, temperature and subcooling. The effect of water memory was also investigated, and the result showed that the gas consumption rate was not affected after nucleation. The conclusion of this research was that the gas
19
liquid interface is place where hydrate formation would most likely occur since in the methane concentration is very low liquid bulk. Further Experiments conducted with ethane as reported by (Vysniauskas and Bishnoi, 1985) was a confirmation to this effects. Englezos et al. (1987b) measured the formation rate from mixtures of methane and ethane and found that the gaseous mixture composition has a significant effect on the formation rate, mainly because the composition alters the magnitude of the driving force. It should be noted, however, that no true comparison of formation rates for different gas mixtures is possible because the experimental conditions (driving forces or pressures) will never be equal. In 1994, Happel, Hnatow and Meyer studied the rate at which both methane and nitrogen hydrate are formed in order to develop a process for the separation of nitrogen from methane. Experiments were carried out on a 1 litre continuous stirred tank reactor (CSTR) as the gas enters the CSTR an upward counter current is generated to a recycled water stream (In 1994, Happel, Hnatow and Meyer). The
20
research showed that measured methane formation rates obtained was higher than that which was reported by Bishnoi and Coworkers (Vysniasuskas and Bishnoi, 1983, 1985, Englezos et al, 1987b). Contrary to Bishnoi and Coworkers, et al used a baffled tank and employed higher stirring rates. Bourgmayer, Sugier and Behar (1989) investigated the rate of formation of methane and ethane mixtures using a semi-batch reactor but this time in the presence of a condensate and found that hydrates was formed at two interfaces , which include the gas-hydrocarbon interface and secondly at the gas-water interface. This confirmed the works done earlier by Vysniauskas and Bishnoi (1983), they identified the interfacial area, subcooling and pressure as important. In addition, they found that the formation rate increases five times when condensate is present. However, Skovborg (1993) found that a hydrate formation rate was not significantly affected by a hydrocarbon liquid phase rather suggested that the transport of gas
21
in the water phase via a liquid water would dominate the rate of formation of hydrate From experiments with a flow loop with liquid hydrocarbon, water and dissolved gas, Gaillard, Monfort and Peytavy (1999) concluded that the rate-determining step was the major factor that the transportation of dissolved gas in the hydrocarbon phase to the water phase depends considering the whole process of hydrate formation. The formation rate increases as the liquid flow rate increases because of turbulence. Ostergaard et al (2000) measured, in a rocking cell, the hydrate formation rate from gas dissolved in a North Sea black oil to investigate the possibilities of separating gas from oil by producing hydrates. They found that the memory effect of water and the hydrodynamic conditions greatly influence the formation rate. Also, they pointed out that better design of the reactor will significantly improve the rate, and thus, reduce the required volume of the reactor. In the same equipment, Tohidi et al. (1996) studied the rate of
22
sH hydrate formation from pressure drop measurements at constant temperature.
sH
hydrates
were
made
from
methane,
methlcyclohexane and water. In a discussion work of Bishnoi and Coworkers (Vysniauskas and Bishnoi 1983, 1985, Englezos et al, 1987b), Happel, Hnatow and Meyer (1994) stated that “applying such information (batch reactor results) to the design of a continuous hydrate former could prove difficult”. For an order of magnitude comparison of different reactor systems, reported rates of hydrate formation are presented in table 2.1. The selected rates of formation are presented in terms of gas consumption rate at similar temperature and pressure conditions. The units of the measured gas consumption rates were inconsistent, hence, some of the rates were recalculated. For the comparison it should be noted that the effect of a stirring rate of 400 RPM highly depends on the size of the reactor and the impeller, the placing of the impeller and the reactor geometry.
23
Impeller power consumption is a better parameter for comparison, but not enough data were available for the calculation. The gas consumption rates in table 2.1 differ with about two orders of magnitude, the consumption rate reported by Happel, Hnatow and Meyer (1994) is one to two orders of magnitude higher than the rates measured in the semi-batch or the batch reactors (except from the result of Takaoki et al (2002). Unfortunately, Happel et al did not report the exact stirring rate, but the selected consumption rate was the lowest of the ones reported, and is therefore believed to represent a stirring rate substantially lower than 2250 RPM. It should be noted that the temperature in the experiment of Happel et al is higher than in the experiments of others. The geometry of the CSTR of Happel et al had some important features compared to the batch reactors. Gas was bubbled through the liquid countercurrent to the liquid phase, giving a larger gasliquid interfacial area than in the batch reactors where gas was injected into a gas-liquid interfacial area than in the batch reactors
24
where gas was injected into a gas pocket at the top of the reactors. In addition, the CSTR was equipped with baffles and a turbine impeller. Happel et al. found that the consumption rate was limited by the increased heat of formation. Thus, for experiments with methanenitrogen mixtures, Happel et al, modified the geometry by installing a sparger and disc-and-doughnut baffles and obtained 25-fold improvement of the reactor throughput. The work of Happel et al, indicates that the geometry has a large effect on the gas consumption rate, probably because of increased gas-liquid interfacial area. Table 2.1 methane gas consumption rates reported in the literature. Consumption rates are given at normal temperature and pressure (1 atm, 00c) per volume of water. Reference
Reactor
Vysniauskas and Bishnoi Semi(1983) batch Englezos et al. (1987a) Semibatch Happel, Hnatow& Meyer CSTR (1994) Parlaktuna and Batch Gudmundsson (1996)a Herri, Gruy&cournil Batch (1996b)
Liq. Vol Pres. (dm3) (bar)
Temp. (oc)
0.3
55
1.2
Stirring rate (RPM) 400
0.3
58.2
3
400
1
54
6
0.1
71
2
Up 2250 400
1
30
1
400
Gas cons. Rate (dm3 (NTP) min1dm-3) 0.23 0.26 (initial rate) 0.076
to 1.3 0.17 0.78 (initial rate) 0.020
25
Arai et al (2000) Takaoki et al (2002)
a.
Semibatch Semibatch
0.6
70
7
700
0.21
5.5b
53
3
914c
1.8
Published rates are higher than the rates originally reported by Parlakuna and Gudmundsson (1995). The data from this report is used for the comparison.
b.
The reactor volume was 10 litre, but it was filed with about 5.5kg of water.
c.
Mixing Reynolds number reported instead of stirring rate. Stirring rate calculated by assuming an impeller diameter of 9 cm.
Using a batch reactor, Takaoki et al (2002) was able to obtain a high consumption rate. Although no detail of the reactor was given in this work, the report showed that the reactor was particularly constructed for high rate production which had a volume of 10liters and the internal geometry was also designed for achieve efficient production. Moreover with the high stirring rate observe in this equipment , this explained
the
reason
for
its
high
of
rate
consumption. 26
Notwithstanding, a large volume would not necessitate high rate of consumption since the reactor of Happel et al processes an equal volume as that of the batch reactor by Herri, Gruy and Cournil (1996b). Vysniauskas and Bishnoi (1983) and Englezos et al. (1987a) performed experiments in the same reactor, but Vysniauskas and Bishnoi measured the formation rate for 10 minutes while Englezos et al continued for about 85 minutes. The results of Vysniauskas and Bishnoi and Parlaktuna and Gudmundsson (1996) show that in a batch reactor, the initial gas consumption rate is higher than the rate after a longer period of time. The comparison indicates that the consumption rate depends on the geometry of the reactor. Improvements in design affecting the gas-liquid interfacial area may enhance the formation rate by at least one order of magnitude. This means that hydrate formation experiments are highly system dependent and that models developed for one system, may not be applicable for other systems.
27
2.7
GAS HYDRATE FORMATION MODELS Several hydrate formation models have been published on the
basis of experimental results from batch or semi-batch reactors. All the models required fitting of experimental data to obtain empirical constants. The semi-empirical model of Vysniauskas and Bishnoi (1983) represents the first attempt to describe quantitatively the rate of hydrate formation. In their model, the rate of gas consumption is correlated against total gas-liquid interfacial area, temperature, subcooling and pressure. a Y E qvys Aa s exp a exp p RT T
( 2.1)
The constants A, α,β,Υ and the activation energy E a were determined by fitting the measured methane gas consumption rates to the model. The value of Υ is about three, which suggests an overall order of formation of about three with respect to pressure. The gasliquid interfacial area was determined by measuring the contour of the vortex induced by the impeller for different stirring rates at 28
atmospheric pressure. Vysniauskas and Bishnoi reported that the model predicts the rate of hydrate formation within +10% of the measured formation rate. Englezos et al (1987a) presented a growth model with a single adjustable parameter for the formation of methane and ethane hydrates. They included two steps for the growth: the diffusion of dissolved gas molecules from the liquid bulk to the crystal surface, and the integration of the molecules at the surface. The following relation was proposed to determine the growth rate of a single crystal. 1 1 1 dn 1 dt p K *a p f f eq k * k k r d
(2.2)
The driving force is the difference in fugacity of the dissolved gas and the three-phase equilibrium fugacity. To obtain the overall formation rate for all the hydrate crystals, Englezos et al defined the surface area of the crystals using the second moment of the particle size distribution μ2. by integrating over all the crystals of any size, the overall formation rate emerged: 29
r y (t ) 4K * 2( f f eq)
(2.3)
In addition, they combined a two-film theory with above equation, to describe the gas-liquid transport. Also, they added a population balance to calculate the second moment of the particle size distribution. The single adjustable rate constant was fitted to the experimental data for each stirring rate. Based on the observation that the rate constant is independent of the stirring rate above 400 RPM, they concluded that the mass transfer resistance around the hydrate crystals is eliminated and K* is the intrinsic rate constant. They also found out that the consumption rate decreases with time and models limited to methane and ethane gases not composite as in natural gas film. Skovborg and Rasmussen (1994) thoroughly analyzed the model of Englezos et al. (1987a). They used the model to calculate the amount of gas consumed for an extended time scale, and found that the consumption rate increases with time, which is in contrast to the measured consumption rate that decreases with time. Skovborg and
30
Rasmussen suggested that the decrease in measured rate is observed because the consumption rate does not depend on the total surface area, or because the consumption rate is controlled by transport phenomena rather than integration into the crystal structure. By changing the values of the intrinsic rate constant and the gas-liquid mass transfer coefficient, they obtained a better fit between the model and the experimental data. However, the new parameter values imply that the formation is in a region where gas-liquid mass transfer is the rate-determining step in the overall formation process. In a bit have a deeper knowledge of the subject matter, Skovborg and Rasmussen (1994) develop a more easilier model which considered the gas consumption rate alone to depend on the gas from the gas transport phase to that of the liquid bulk phase as expressed in the following equation dn K L Ag 1C wo x int xb dt
(2.4)
From equation 2.4. The rate of consumption is a function of the mass transfer coefficient of the gas-liquid, the gas-liquid interfacial
31
area and also the mole fraction driving force. Using the data of Englezos et al. (1987a) to obtain the combined mass transfer coefficient, they found that the average absolute deviation between measured and calculated consumption rates is 22% for methane and 14% for ethane. The model was extended to more than one hydrateforming component by assuming that the transport of the components are independent, however, this implies that all the components have the same mass transfer coefficient. That the rate of hydrate formation may be mass transport-limited does not imply that measurements of particle size distributions are redundant. Hem et al (1999) pointed out that information about the particle size distribution is not accessible with measurements of the gas consumption rate only. Lekvam and Ruoff (1993) develop a reaction kinetic model for methane hydrate formation using a method referred to as the pseudo-elementary reaction steps (Lekvam and Ruoff 1993.) . The initial reactants were liquid water and gaseous methane and obtained
32
the product as hydrate. Dissolved methane and some hydrate precursor species were used as reaction intermediates. The gas dissolution in water phase, the buildup of the precursor and also the growth of methane hydrate using an autocatalytic process were used as the dynamic elements. The rate constants were determine for each pseudo-elementary reaction with one gas phase for different reaction at various time. The effect of stirring rate on the formation rate is modeled when gas dissolution is selected as the rate-determining step. (Lekvam and Ruoff 1993.) Gaillard, Monfort and Peytavy (1996) modeled nucleation, growth and agglomeration in their hydrate loop (with liquid hydrocarbons) applying crystallization theory and methane gas consumption measurements. They proposed a population balance for the hydrate crystals in the system and expressed the nucleation, growth and agglomeration rates by empirical correlations. Experiment obtained by Gaillard et al (1999) who worked on nucleation and agglomeration growth in their hydrate loop observed no induction time . Gailard et
33
al after series of experiment expressed the rate of heterogeneous nucleation as follows: A m J Pk poexp f f eq u RT
(2.5)
Where u = liquid velocity f-feq = driving force and kpo, A, M and M’ are constants. The equation to determine or estimate the growth rate and adopted empirical correlation is thus. E m d b G K exp f f eq u L RT
(2.6)
Where L= crystal and K,E, m, d and b are constants. In the model employed for the rate of agglomeration, the rate of agglomeration is proportional to the collision probability of two crystals and their sticking probability.
34
2.8
OVERVIEW OF EQUATION OF STATE The equation of state is mostly used in the correlation of the
densities of various liquids to their temperatures and pressures. The ideal gas law is the simplest example, this is almost accurate provide the gas is a weak polar gas and must be at a moderate temperature low pressure.
However, the equation of state has limitation of
prediction in conditions of condensation from gas to liquid and at high temperature and pressure, hence the development of several models of the equation of state. Practically, the equations of state in recent times have several applications, for example, in PVT calculation and petroleum gas/liquid equilibrium calculations.
BOYLE’S LAW (1662) Boyle’s law was the earliest form of an equation of state. Robert Boyle in 1662 carried out series of experiments in which He used a J-shaped glass tube sealed at one end. He added mercury to the tube which
35
trapped a fixed mass of air in the short sealed end. The gas volume was measured as an additional mercury was been added. By this arrangement the gas pressure was measured using the difference between the mercury level in the sealed end and the open end of the same tube. Boyle observed that the volume of the gas varied inversely with the pressure at constant temperature. Thus pV = constant
( 2.7)
CHARLES’S LAW AND GAY-LUSSAC (1787) A French physicist named Jacques Charles also worked on the relationship of gas volume, temperature and pressure. Charles discovered that, nitrogen and hydrogen, oxygen and carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. Joseph Louis and Gay-Lussac published the results of a similar experiment in 1802 .They developed a linear relationship between volume and temperature as presented in equation 2.8.1
36
V1 V 2 T1 T 2
(2.8.1)
DALTON’S LAW OF PARTIAL PRESSURE (1801) This law states that a mixture of gasses is equal to the sum of the pressures of all of the constituent gases alone. This is expressed mathematically as n
Ptotal P1 P 2 ... P n Ptotal pi
(2.8.2)
i 1
THE IDEAL GAS LAW (1834) Emile Clapeyron investigated the works of Boyle and Charles and came up with the first statement known as the ideal gas law after combining the boyle’s and Charles’ law. This relation was given as pVm=(Tc +267) where T in degrees Celsius), and R is the gas constant.
37
However, further research showed that the number was actually be closer to 273.2, thus, the Celsius scale was re-defined with 00c273.15k. Hence, pVm = R(Tc +273.15)
(2.8.3)
VAN DER WAALS EQUATION OF STATE (1873) J.D. van der Waals in 1873 introduced the first equation of state derived which made significant assumptions such as a finite volume been occupied by the constituent molecules. This discovery evolved a new era in the study of equation of state
MAJOR EQUATIONS OF STATE The major equation of state , states that , for a fixed mass of a substance which is contained in a system, the volume , pressure and temperature are related in the form;
38
ƒ (p, V,T) = 0
(2.8.4)
Definition of Equation f State Variables When dealing with the equations of state , it is important to be consistent with the units , however, the SI units may be preferred. Absolute temperature This refers to use of the Kelvin (K) or Rankine (oR) temperature scales, with zero being absolute zero. Others are; p
= pressure (absolute)
V
= volume
η
= number of moles of a substance
V m
V
= molar volume, the volume of 1 mole of gas or liquid
T
= absolute temperature
R
= ideal gas constant (8.314472 J/(mol.K)
pc
= pressure at the critical point
Vc
= molar volume at the critical point
Tc
= absolute temperature at the critical point 39
CLASSICAL IDEAL GAS LAW The classical ideal gas law given as pV = RT
(2.8.5)
equation 2.8.5 becomes f ( p,V ,T ) pV RT 0
(2.8.6)
Another form of the ideal gas law could written as p=p ( 1)e
(2.8.7)
where; p = density, Υ= Cp/Cv adiabatic index (ratio of specific heats) e = CvT is the internal energy per unit mass (the “specific internal energy”), Cv is the specific heat at constant volume and Cp is the specific heat at constant pressure.
CUBIC EQUATIONS OF STATE
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This is the re-written form of the equation of state in a cubic function given as Vm VAN DER WAALS EQUATION OF STATE (1873) The van der waals model of the equation of state is expressed as a p 2 Vm b RT Vm
(2.8.8)
Where Vm = molar volume. Where a and b are the specific constants is the substance and could be calculated from the pseudo critical properties such as pc, Tc and Vc Where Vc is the molar volume at critical conditions a 3 p cVc2
(2.8.8a) b
Vc 3
(2.8.8b) a
27( RT c) 64 p c
2
(2.8.8c) b
RTc 8 pc
(2.8.8d)
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From Van der waal’ equation above, a is known as the attraction parameter while b the repulsion parameter or the effective molecular volume. Although the equation is superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where there is liquid formation. The van der waals’ equation might be considered as the “improved” form of the ideal gas law due to two independent reasons namely; 1.
Molecules are thought as particles with volume, not material
points. Thus Vm cannot be too little, less than some constant. So we get (Vm-b) instead of Vm. 2.
While ideal gas molecules do not interact, the attraction of
molecules with others within a distance of several molecules radii. There is actually no effect inside the material rather, the surface molecules are attracted into the material from the surface. This is seen as decrease in the pressure on the outer shell (used in the ideal gas law), hence written as (p+ something) instead of p. to evaluate this
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”something” we need to examine the an additional force which acts on an element of gas surface. let the force acting on each surface molecule be –p, and the force acting on the whole is represented in equation 2.8.9 Element is –p2
1 Vm2
(2.8.9)
With the reduced state variables, i.e V1= Vm/Vc, Pr=P/Pc and Tr=T/Tc, the reduced form of the van der waals equation can be formulated: 3 Pr 2 3Vr 1 8Tr Vr
(2.8.9a)
The benefit of this form is that for given Tr and Pr the reduced volume of the liquid and gas can be calculated directly using Cardano’s method for the reduced cubic form: 1 8T 3V 1 Vr3 r Vr2 r 0 Pr Pr 3 3Pr
(2.8.9b)
FrPr
