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The main factors considered in this calculation are the hydrocarbon dew point, the natural gas physical properties, and conformity to pipeline specifi...
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Uncertainty Analysis Applied to Thermodynamic Models and Fuel Properties − Natural Gas Dew Points and Gasoline Reid Vapor Pressures Samaneh Hajipour,† Marco A. Satyro,*,‡ and Michael W. Foley† †

Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 ‡ Virtual Materials Group Inc., 1829 Ranchlands Boulevard NW, Calgary, Alberta, Canada T3G 2A7 ABSTRACT: A simple, consistent, and self-contained error propagation algorithm was developed using the uncertainty information related to pure component physical properties, binary interaction parameters, and thermodynamic model parameters combined with Monte Carlo simulation along with the Latin Hypercube Sampling (LHS) method. This algorithm is generally applicable to simulate the error propagation in process flow sheets of arbitrary complexity as long as the thermodynamic model parameters encode uncertainty information. In this work, two significant problems related to hydrocarbon processing are studied under the light of uncertainty analysis. First, the injection of a valuable liquid hydrocarbon into an existing natural gas pipeline for transportation was studied in order to find the optimum injection rate of liquid n-butane that can be safely added to the flowing gas without undesired condensation. The main factors considered in this calculation are the hydrocarbon dew point, the natural gas physical properties, and conformity to pipeline specifications. Second, uncertainties on Reid vapor pressure (RVP) calculations were taken into account for the calculation of optimal rate of liquid n-butane blending into gasoline. Gasoline blending is an important operation in refineries where gasoline must be produced with enough volatility for the proper operation of engines in cold climates.

1. INTRODUCTION In the first part of these series,1 a comprehensive database for pure hydrocarbons containing basic physical properties associated with uncertainties, two correlations for estimation of pseudocomponents properties with required uncertainty information, and the variance-covariance matrix for the reparameterized Peng−Robinson equation of state parameters were developed. In the second part of this series of papers,2 a database for 87 binary mixtures containing the experimental values of vapor−liquid equilibrium (VLE) data and their uncertainties was developed and the reparameterized Peng− Robinson equation of state1 was extended to mixtures using simple quadratic mixing rules, adequate for the modeling of light hydrocarbon systems. Binary interaction parameters and their uncertainties were determined by data regression using a thermodynamically consistent and carefully evaluated VLE database taking into consideration the uncertainties of measured binary data. With the available uncertainty information for pure components, model parameters, and binary interaction parameters, it is possible to perform a rigorous uncertainty analysis for process design through a self-contained and consistent computational procedure. In this work, a simple error propagation algorithm integrated with an internally consistent thermodynamic data set and correlations along with associated uncertainties was developed and coupled with the VMGSim process simulation software3 to evaluate the effect of thermodynamic uncertainties on the final simulation results. 1.1. Liquid Hydrocarbon Injection into an Existing Natural Gas Pipeline. In this process, liquid hydrocarbon is injected into the existing pipeline with flowing stream of natural © 2013 American Chemical Society

gas in order to increase the pipeline capacity and/or heating value of the gas. This method also allows for the transportation of high value hydrocarbon liquids through a gas pipeline. The amount of liquid hydrocarbon that can be safely injected into the pipeline is controlled by the hydrocarbon dew point of the gas and the existing pipeline specifications. Commonly used cryogenic fluids such as liquefied natural gas (LNG) and liquefied petroleum gas (LPG) or a mixture of hydrocarbons such as propane and butane with air are added into the gas pipeline in the supply and distribution of natural gas to increase the capacity during peak demand periods.4,5 A liquid hydrocarbon, normally butane, sometimes is added to the fuel gas to increase the heating value of the gas used to provide heat in refining processes.6 Stark et al.4 presented an innovative method and apparatus for adding LNG or mixture of hydrocarbon/air to the fuel gas pipelines using a Venturi jet. Different possibilities of injection were explored wherein the kinetic energy of the flowing gas is used to aspire the cryogenic liquid into the high pressure gas stream and the heat of the flowing gas stream is used to supply the latent heat of vaporization. The amount of liquid to be injected is determined by the temperature of the pipeline or the gravity of the natural gas. They claimed that using the Venturi system reduces the capital investment and required power by eliminating the need for pumps or compressors to increase the pressure of the injected liquid and vaporizers to supply the heat required for liquid vaporization. Received: October 1, 2013 Revised: December 23, 2013 Published: December 24, 2013 1569

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On the other hand, Arenson5 claimed that the liquefied cryogenic fluid must be preheated, vaporized and superheated prior to combining with the flowing gas transported through the pipeline, and therefore he provided a new method to vaporize and combine the cryogenic gas to the pipeline gas. The possibility of condensation and controlling the amount of liquid hydrocarbon added to the gas were not discussed by the author. While Chin and co-workers6 invented a device to control the amount of butane injected to fuel gas which works based on monitoring the butane saturation point of the gas. Regardless of the method used for the injection, the main objective of this work is to investigate the effect of variation in the liquid hydrocarbon injection rate on the calculated dew point. As previously discussed,1,2 the uncertainty of input parameters affects the calculation of dew point in natural gas, and this effect would be considerable in the presence of a trace amount of heavy components. Therefore, the uncertainty analysis using the error propagation algorithm is performed in order to estimate the optimum amount of liquid hydrocarbon that can be safely injected into the existing gas pipeline without liquid formation. 1.2. Gasoline Blending. Gasoline is a complex mixture of many different hydrocarbons, and its composition varies depending on the crude oil source, refining process, and additives. There are standard specifications established by American Society for Testing of Materials (ASTM) for gasoline dealing with the performance requirements such as volatility.7 In order to meet the standard specifications for a product, different gasoline cuts are blended together with additives and lighter hydrocarbons. The volatility of the gasoline blend is one of the most important properties affecting the performance of engines and their ability to function properly independent of the weather. Volatility is directly related to the Reid vapor pressure (RVP) which is the vapor pressure of the gasoline blend at 310.93 K (100 °F) measured using a specific apparatus and vapor fraction. The maximum allowable limit of this property varies with seasonal temperature changes and geographical location. In the summer or high altitude regions, it is important to have a lower RVP gasoline to reduce evaporation losses and prevent vapor lock. On the other hand, in the winter and lowaltitude areas a higher RVP gasoline must be produced to improve engine starting characteristics.8,9 As such, the RVP ranges from 49.64 kPa (7.2 psia) in the summer to 93.08 kPa (13.5 psia) in the winter.8 Since n-butane is relatively inexpensive, has a lower sales value than gasoline on a volume basis, and has a high RVP equal to 358 kPa (52 psia), it is often used in refineries as a blending component to produce gasolines within proper specification. In this section, we illustrate the use of the error propagation algorithm to evaluate the amount of n-butane required to be added to a gasoline to produce a desired RVP. Since the RVP of the gasoline depends on the quantity and RVP of each component in the blend, the uncertainty in vapor pressure of each individual component affects the quality of the overall RVP calculation. As previously shown,1 uncertainties of input parameters affect the quality of the calculated vapor pressure of pure components. Therefore, physical property uncertainties influence the quality of the overall RVP calculations and must be taken into account for reliable calculation of the amounts added during blending.

2. DEVELOPMENT OF THE ERROR PROPAGATION ALGORITHM When calculating any quantity of interest through mathematical relations, the uncertainties associated with the independent variables are propagated into the final quantity. Evaluation of the uncertainty in the final result using the principles of error propagation based on Taylor linearization is an exceptionally tedious procedure1 and difficult to generalize from a process simulation point of view. Therefore, the error propagation equation is rarely applied for evaluation of uncertainties in complex calculations. Hajipour et al.2 demonstrated that a specially constructed version of the Monte Carlo method can be used for the error propagation calculations for flow sheets of any complexity. This method is simple, adaptable, and reasonably fast for complex computations. The basic requirements for development a selfcontained and consistent error propagation algorithm for physical property calculations is described in detail elsewhere,1,2 and a very brief summary of its major points follows. The sequence of the overall error propagation evaluation is shown in Figure 1. For pure components, critical temperature, critical pressure, and acentric factor data and their associated uncertainties are taken from the pure component database developed in the first part of these series. The redeveloped Riazi-Daubert or Lee-

Figure 1. Sequence of overall error propagation evaluation process. 1570

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Kesler models1 and their variance-covariance matrices of model parameters are used for estimation of critical properties and acentric factors of any undefined oil fractions or plus fractions and their uncertainties. The reparameterized Peng−Robinson equation of state and associated variance-covariance matrix1 along with the van der Waals quadratic mixing rules is used for thermodynamic calculations, and binary interaction parameters and their uncertainties are taken from the database evaluated for 87 binary mixtures and applied for mixture calculations.2 In order to use the Monte Carlo technique for error propagation it is assumed that all input uncertainty values are characterized by normal (Gaussian) distributions with means equal to the listed values in the database and with standard deviations equal to half of their uncertainty values based on the 95% confidence interval (CI). We should stress that the sound application of our proposed error propagation method rests solidly on the applicability of a thermodynamic method for the sound modeling of the behavior of pure components and mixtures. In other words, the thermodynamic models should be devoid of consistent bias when estimating relevant pure component properties such as vapor pressures and mixture properties such as saturation pressures or vapor fractions. Thermodynamic models must thereby be constructed according to the best practices related to data regression and should be verified before use for extensive error propagation calculations. The LHS method10 is applied to generate a sample set containing all input parameters subject to uncertainty with a sample size equal to 100. The sample size was determined from studies previously done1,2 and used in this work without modification. The quantities of interest and their associated uncertainties are calculated through multiple simulations using the sample sets. The distribution of the calculated quantities shows the effect of the input uncertainties. The mean value of the calculated quantities is reported as the true value with the uncertainty equal to two times of the standard deviation. This approach is similar to Whiting et al. method11 in studying the uncertainty of process performance. The main difference being the use of a comprehensive database of thermodynamic parameters with associated uncertainties in this work.

Figure 2. Schematic pressure−temperature envelope for natural gas and thermodynamic positions of the pipeline with temperatures of higher (T1) and lower (T2) than dew point at pressure of P.

the condensation occurs at the temperatures (T2) lower than the dew point. This is illustrated through the injection of liquid n-butane into an existing natural gas pipeline designed to transport a maximum natural gas flow rate of 25.485 MMSCMD (900 MMSCFD) at 288.71 K (60 °F) from a source (A) to a delivery (B) location, 130 km away, with a delivery pressure of 5515.8 kPa (800 psia). The composition of the gas and the pipeline specifications are given in Table 1 and Table 2, respectively. Table 1. Natural Gas Composition

3. CASE STUDY PROBLEMS 3.1. Injection of Liquid n-Butane into an Existing Natural Gas Pipeline. Although it is desirable to maximize the amount of liquid hydrocarbons in the natural gas in order to maximize the hydrocarbon transport capacity and heating value of the gas, the injection of liquid hydrocarbons is limited by the dew point of the components and the specifications determined for the existing pipeline. With too much liquid hydrocarbon injected into the pipeline, the gas will be oversaturated, and some of the mixture will condense. One of the essential factors in gas pipeline design is avoiding the condensation of liquid hydrocarbons. Increased pressure drop, capacity reduction, and in-line equipment problems such as compressor damage are the main issues caused by hydrocarbon liquid dropout. In order to prevent hydrocarbon condensation the operating temperature of the pipeline must be kept above the hydrocarbon dew point as commonly shown on the pressure−temperature envelope. Figure 2 represents a typical sketch of the pressure−temperature envelope of a natural gas that shows the effect of the operating temperature of the pipeline on hydrocarbon condensation and indicates that

component

mole fraction

oxygen nitrogen carbon dioxide methane ethane propane i-butane n-butane i-pentane n-pentane n-hexane

0.0002 0.0149 0.0070 0.9500 0.0250 0.0020 0.0003 0.0003 0.0001 0.0001 0.0001

The existing natural gas pipeline was simulated using the VMGSim process simulator3 in order to calculate the number of compressor stations, the required mechanical work to be supplied by the compressors, the compressor discharge Table 2. Existing Natural Gas Pipeline Specifications Used in This Work specs nominal pipe size (NPS) pipe wall thickness pipe wall roughness length pressure temperature gas heating value hydrocarbon dew point a

1571

30 in 0.5 in 1.8 × 10−5 m (0.0007 in) 130 km MAOPa 7584.2 kPa (1100 psia) max. 323.15 K min. 36 MJ/m3 max. 41 MJ/m3 max. 263.15 K at 5515.8 kPa

Maximum allowable operating pressure. dx.doi.org/10.1021/ef4019838 | Energy Fuels 2014, 28, 1569−1578

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Figure 3. Schematic view of the existing natural gas pipeline used in this work.

Since the hydrocarbon dew point calculation is strongly dependent on the composition of natural gas, especially its heaviest components, the dew point of the gas will change due to the injection of n-butane. The maximum allowable hydrocarbon dew point of 263.15 K was considered when designing the existing pipeline. By increasing the amount of nbutane in the gas stream, the hydrocarbon dew point increases. Therefore, the injection rate of liquid n-butane is limited by this design parameter such that the dew point does not exceed the specified value. In this study, the maximum allowable amount of n-butane was calculated using VMGSim under the assumptions that the liquid is added into the pipeline after the first compression station at point A and the pressure of the pipeline does not change during the mixing process due to the minute amounts of liquid n-butane injected into the natural gas. The flow rate of the final gas mixture was kept constant at the maximum gas flow rate of 25.485 MMSCMD throughout the study. Without taking into account the uncertainties, the maximum standard flow rate of n-butane is 137.52 m3/h (116,550 ft3/day) which is added to 24.711 MMSCMD (872.26 MMSCFD) of natural gas. The ratio of the added liquid n-butane to the transported natural gas is equivalent to 0.0134% in standard volume (in other words, without taking into account the vaporization of n-butane that will happen at the actual pipeline conditions). As shown by Hajipour et al.,1,2 the dew point calculation is also dependent on the uncertainty of the input parameters used by the thermodynamic model. Since results from a simple dew point calculation as normally done are not complete, under- or overestimation of liquid injection have to be applied. 3.2. Gasoline Blending. Blending n-butane into gasoline not only increases the RVP of the gasoline but also increases the capacity of the gasoline supplies and reduces the gasoline price. The amount of n-butane that can be added to the blend is limited based on the gasoline product specifications. Addition of n-butane increases the RVP and the tendency of gasoline to vaporize at high temperatures and high-altitude areas. The presence of vapor in the fuel line and the combination of the vapor and liquid feeding the fuel pump interrupts the normal car engine operation and presents a safety hazard. Similarly, gasoline with too low RVP does not provide enough volatility to start the engine in cold weather. In order to minimize gasoline evaporation the amount of n-butane in gasoline blending process must be carefully controlled. This process is illustrated by a particular example of blending n-butane into 331.23 m3/h (50,000 bbl/day) low RVP gasoline at standard conditions with the chemical composition given in

temperature, and the intercooler duty. The number and location of the compressor stations required to transport the natural gas were calculated by neglecting temperature and elevation differences along the pipeline. The reparameterized Peng−Robinson equation of state1 was used for the process calculations, and the pure component critical properties, acentric factors, and binary interaction parameters are taken from the databases developed by Hajipour et al.1,2 For the design of the pipeline, it was assumed that the first compressor station at point A has a discharge pressure of 7584.2 kPa (MAOP) and the intercooler outlet temperature is 288.71 K. The intermediate compressor station is located halfway between A and B (65 km) with a suction and discharge pressure of 5515.8 and 7584.2 kPa, respectively. Based on the calculation done by VMGSim, the gas can be transported to B using two compressor stations, and gas pressure at the delivery point would be greater than 5515.8 kPa which can be adjusted by installing a pressure regulator at B. The schematic view of the gas pipeline is shown in Figure 3. The basic information for the in-line equipment associated with the uncertainties propagated from the uncertainties in all input variables including pure components properties, binary interaction parameters, and thermodynamic model parameters are summarized in Table 3. The uncertainty analysis was Table 3. Existing in-Line Equipment Performance Data equipment

specifications

compressor adiabatic efficiency (%) pressure ratio inlet temperature (K) adiabatic work (kW)

80 1.375 288.71 10,825.1 ± 6.2

inlet temperature (K) outlet temperature (K) pressure drop (kPa) duty (kW)

323.15a 288.71 68.95 19,803.4 ± 9.2

intercooler

a

Maximum allowable temperature of the pipeline.

performed using the error propagation algorithm previously described, and assumed that the inlet and outlet temperatures and pressure drop of the intercooler, the adiabatic efficiency, and pressure ratio of the compressor are specified and they have no errors associated with them. We note that if uncertainty information related to these parameters was available they could easily be used in the Monte Carlo simulations. 1572

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interaction parameters and their uncertainties is straightforward as previously discussed.2 In this example, the uncertainty analysis was conducted by taking into account only the uncertainties on critical properties and acentric factors of each of the individual components using the APR thermodynamic model, and the uncertainties related to thermodynamic models and binary interaction parameters were not considered. Without taking into consideration the uncertainties of input parameters, the RVP of the gasoline with the specified composition is 70.72 kPa (10.26 psia), and the flow rate of n-butane required to provide a RVP equal to 93.08 kPa is 23.73 m3/h (3582.44 bbl/day) at standard conditions. The volume of the blended n-butane is equivalent to 7.17% of the initial volume of the gasoline. Similarly to the dew point calculation, the RVP also depends on the composition. Consequently, the uncertainty analysis was carried out in order to estimate the under or over estimation of the blended n-butane rate.

Table 4 in order to increase RVP from 70.72 kPa (10.26 psia) to 93.08 kPa (13.5 psia). Table 4. Low RVP Gasoline Blend Chemical Composition

a

compound

weight fractiona

standard volume fraction

i-butane n-butane i-pentane n-pentane n-hexane n-heptane 2,2,4-trimethylpentane n-octane 2-methyldecane 2-methyl-2-butene 2,3-dimethyl-1-butene benzene toluene m-xylene o-xylene p-xylene 1,2,4-trimethylbenzene i-butylbenzene

0.030 0.030 0.050 0.050 0.050 0.050 0.050 0.140 0.050 0.050 0.050 0.020 0.150 0.034 0.033 0.033 0.080 0.050

0.0393 0.0379 0.0589 0.0585 0.0555 0.0535 0.0528 0.1463 0.0499 0.0553 0.0541 0.0167 0.1267 0.0289 0.0275 0.0281 0.0671 0.0430

4. UNCERTAINTY ANALYSIS RESULTS AND DISCUSSION In this section, the uncertainty in dew point and Reid vapor pressure calculations was quantified using the error propagation algorithm developed in this work. The uncertainty analysis results make it possible to evaluate the under- or overestimation for the rate of n-butane that can be safely added to the natural gas stream and gasoline blend to reach the desired specification. For the first case study problem, depending on the amount of liquid hydrocarbon injected into the pipeline the composition of the gas will change and hence the dew point of gas will vary. The uncertainty analysis was performed to estimate the safe flow rate of injected n-butane regarding the uncertainty of dew point of gas at 5515.8 kPa. In the previous section, the maximum standard flow rate of 137.52 m3/h was estimated for n-butane by setting the dew point of final gas mixture to 263.15 K without considering the uncertainties of input parameters. In order to find the uncertainty of dew point with the maximum injection and compare the design parameters of the process after injection with the existing pipeline specifications, the uncertainty analysis was performed using the Monte Carlo method with the LHS scheme as previously discussed. Figure 4 shows the pressure− temperature envelopes and associated uncertainties for the natural gas before and after the n-butane injection. Points A and

Data are from ref 12.

Although a mixture of pure compounds does not truly represent physical and chemical characteristics of gasoline, this simplifying assumption is reasonably accurate, and a surrogate for gasoline is represented as a mixture of pure compounds. The chemical composition proposed by Kreamer and Stetzenbach12 as a reference surrogate low RVP gasoline for environmental research studies is used in this work. After selecting the gasoline stock, the quantity of n-butane required to give the desired RVP was calculated using the VMGSim process simulation software.3 For the calculation, the input properties of pure components (critical properties and acentric factors) were taken from the database previously developed.1 The critical properties of xylene(s) and 2methyldecane which are not available in the database were taken from ThermoData Engine (TDE),13 and the acentric factors associated with uncertainties were calculated using a similar approach employed to develop the database.1 The properties of these components are listed in Table 5. Table 5. Properties of Pure Components Tc (K)a

compound m-xylene o-xylene p-xylene 2-methyldecane a

616.85 630.43 616.19 624.10

± ± ± ±

0.56 0.61 0.15 9.70

ω

Pc (kPa)a 3540 3745 3528 1811

± ± ± ±

13 25 16 23

0.328 0.312 0.323 0.545

± ± ± ±

0.002 0.003 0.002 0.006

Data are from ref 13.

Since the database of binary interaction parameters2 was developed based on the components present in natural gas, the uncertainty information of binary interaction parameters for some of the components considered in this case study is not available. For those binaries, the calculation was done using the default binary interaction parameters provided by the VMGSim for the Advanced Peng−Robinson (APR)14 thermodynamic model. The expansion of the database to include the missing binary

Figure 4. Pressure−temperature envelopes for natural gas before and after the liquid n-butane injection. 1573

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specified condition. Figure 5(b) also shows that half of the Monte Carlo (MC) results are located above the maximum allowable dew point line. Two green dashed-dotted lines show the lower and upper limits of calculated dew points at 5515.8 kPa which are represented the uncertainty of the dew point on the 95% confidence interval (CI). The uncertainty analysis shows that there is a possibility to violate the existing pipeline specification due to uncertainties in the physical properties as calculated by the thermodynamic model, and the amount of injected n-butane should be reduced. In order to find the maximum rate of n-butane such that the upper limit of the calculated dew point at 5515.8 kPa is below 263.15 K, the uncertainty analysis was performed for different standard volume proportions of the liquid n-butane introduced to the flowing gas with the composition assumed in this study. Figure 6(a) shows the results of uncertainty analysis. The intersection of the curve with the maximum allowable dew point line (T = 263.15 K) determines the maximum safe liquid n-butane/gas standard volume ratio which is observed more clearly in the zoomed-in version of the plot, Figure 6(b). Therefore the maximum safe standard volume ratio of the injected liquid n-butane to the natural gas would be around 0.0132% which is equivalent to 135.45 m3/h of liquid n-butane at standard conditions. The dew point temperature with the new injection rate was calculated using the error propagation algorithm to ensure that the upper uncertainty of temperature is still below 263.15 K. So, the reduction of n-butane injection rate from 137.52 Sm3/h to 135.45 Sm3/h leads to a reduction of target dew point from 263.15 k to 262.71 K. Figure 7 shows the oscillation of data points around 262.71 K with the uncertainty of 0.34 K indicated with two green dashed-dotted lines. This type of guided determination of safety factors for design is one of the greatest benefits of uncertainty analysis as applied to process design.

B indicate the thermodynamic positions of the source and delivery locations. As shown in Figure 4, the dew point curve is strongly affected by changing the gas composition, and increasing the amount of n-butane in the gas leads to a significant increase in cricondentherm. This effect was previously discussed by Hajipour and Satyro.1 The results of uncertainty analysis for the natural gas before and after the injection are summarized in Table 6. As indicated Table 6. Results of the Phase Envelopes Uncertainty Analysis property 3

n-butane injection rate (Sm /h) cricondentherm temperature (K) cricondentherm pressure (kPa) cricondenbar temperature (K) cricondenbar pressure (kPa) dew temperature at 5515.8 Kpa (K)

before injection

after injection

− 218.18 2529.8 199.65 5257.3 −

137.52 263.35 4978.0 239.10 9110.5 263.15

± ± ± ±

0.77 41.3 2.60 24.6

± ± ± ± ±

0.34 12.5 0.20 39.4 0.34

in this table, there is no dew point reported for the natural gas before injection at 5515.8 kPa, since that pressure is greater than the cricondenbar and there is no possibility for condensation of the gas at that pressure by reduction of temperature. The calculated dew point of the gas after injecting of 137.52 Sm3/h liquid n-butane is 263.15 ± 0.34 K at 5515.8 kPa. For clarity purposes, the zoomed-in version of the pressure− temperature envelope of gas after injection and the distribution of dew points calculated using the Monte Carlo for 100 sample sets are indicated in Figure 5(a) and 5(b), respectively. As seen in Figure 5(a), the maximum dew point is located in the retrograde condensation region for the curves with the hydrocarbon dew points being greater than 263.15 K at 5515.8 kPa, and therefore some gas will condense at the

Figure 5. (a) The zoomed-in version of Figure 4 for pressure−temperature envelope of gas after the injection of 137.52 m3/h and (b) Monte Carlo simulation results for the dew point calculation at 5515.8 kPa. 1574

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Figure 6. (a) Calculated dew point and associated uncertainty at 5515.8 kPa against the injected liquid/gas standard volume ratio and (b) zoomed-in version of (a) in the vicinity of maximum dew point.

Figure 7. Monte Carlo simulation results for dew point calculation at 5515.8 kPa after the injection of 135.45 Sm3/h n-butane.

Since the aim of this process is to use the existing pipeline to transport the gas after injection, it is also necessary to check the possibility of using the current in-line equipment to transport the gas with different physical properties. In order to do so, the compressor and intercooler performance were investigated using the uncertainty analysis algorithm. Comparing the results shown in Table 7 with those listed in Table 3 indicates that the current equipment can be used to transport the gas without any changes since the adiabatic work of the compressor and duty of the intercooler required to transport the gas are smaller than those of existing equipment. Adding n-butane to the gas increases the specific gravity and ratio of specific heats which are two main parameters in calculation of work done by compressor. Since the pressure ratio does not change, increasing the values of these two parameters decreases the required compressor horsepower. On the other hand, the specific heat of the gas increases after injection and leads to an increase in the duty of the intercooler; however, it is still below the design duty of the existing intercooler, designed to cool the gas from the maximum allowable pipeline temperature to 288.71 K. The compressor outlet temperature was also calculated to ensure that the upper uncertainty of temperature is below 323.15 K and does not violate the pipeline specification. The physical properties of gas including specific gravity and net and gross heating values were also calculated, and the results are summarized in Table 7. The specific gravity of the gas increases from 0.583 to 0.626 due to injection of 135.45 Sm3/h of liquid n-butane. This significant increase in the gas

Table 7. Physical Properties of Gas and the in-Line Equipment Performance Data before and after the Injection before injection

property

natural gas flow rate (MMSCMD) 25.485 n-butane injection rate (Sm3/h) 0 gas flow rate after injection (MMSCMD) 25.485 specific gravity 0.583 net heating value (MJ/Sm3) 33.98 gross heating value (MJ/Sm3) 37.67 dew point at 5515.8 kPa (K) − compressor before injection inlet temperature (K) pressure ratio adiabatic efficiency (%) adiabatic work (kW) outlet temperature (K) intercooler

after injection 24.711 135.45 25.485 0.626 36.35 40.21 262.71 ± 0.34 after injection

288.71 1.375 80 10,825.1 ± 6.2 316.22 ± 0.01 before injection

288.71 1.375 80 10,529.1 ± 8.6 315.10 ± 0.01 after injection

316.22 ± 0.01 288.71 68.95 46.29 ± 0.02 15,858.0 ± 6.4

315.10 ± 0.01 288.71 68.95 50.08 ± 0.03 16,455.5 ± 9.0

inlet temperature (K) outlet temperature (K) pressure drop (kPa) average specific heat (kJ/kmole-K) duty (kW)

specific gravity results from the butane vaporization at pipeline conditions and its actual volume fraction in the final gas, equal to 3.04 actual volume percent corresponding to a volume flow increase of 0.774 MMSCMD. If there is a specified upper limit 1575

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for the density of the gas, it should be included in the determination of the safe amount of injection as well. The heating value of the gas also increases by increasing the amount of n-butane, but it is still less than the maximum heating value specified in Table 2. Since the dew point temperature varies widely depending on the gas composition and pressure of the pipeline and the model used for the calculation, it may vary by changing any one of these parameters. Note that the effect of other compounds present in small quantities such as methanol and ethylene glycol eventually injected as part of a hydrate formation prevention program can be quantified in the same way. For the second case study problem, the uncertainty analysis was performed to estimate the safe amount of blended n-butane into gasoline based on the uncertainty of RVP of the gasoline blend. The uncertainty in the RVP calculation propagated from the uncertainties of critical temperatures, critical pressures, and acentric factors of each of 18 components of gasoline listed in Table 4 was estimated using VMGSim by applying the APR equation of state and the Monte Carlo technique. The simple calculation of RVP shows that 23.73 Sm3/h of nbutane is required to increase the RVP of the gasoline blend from 70.72 to 93.08 kPa. The uncertainty analysis results indicate that by blending this amount of n-butane to the gasoline, there is a possibility to produce the gasoline with RVP greater than the maximum allowable value of 93.08 kPa. The saturation pressure curves as a function of temperature and their associated uncertainties for the gasoline blend before and after n-butane blending are shown in Figure 8 for the

Table 8. Vapor Pressures and Uncertainties Calculated Using the Monte Carlo Simulation for the Gasoline before and after n-Butane Blending at Different Temperatures vapor pressure (kPa) temperature (K) 270 280 290 300 310 320 330

before blending 18.02 26.46 37.79 52.64 71.73 95.79 125.6

± ± ± ± ± ± ±

0.26 0.36 0.47 0.61 0.77 0.97 1.2

after blending 24.33 35.51 50.39 69.73 94.35 125.1 162.9

± ± ± ± ± ± ±

0.25 0.33 0.44 0.57 0.73 0.9 1.1

Figure 9 shows the result of Monte Carlo simulation as a function of the sample number. The results are distributed around the maximum allowable RVP of 93.08 kPa (red dashed line) with the maximum amount of 93.95 kPa at sample number 72 and the minimum amount of 92.34 kPa at sample number 20. Two green dashed-dotted lines show the lower and upper limits of RVP determined based on the 95% confidence interval (CI). For the 50 points out of 100 points in the sample set, the calculated RVPs are greater than 93.08 kPa which result from the excessive amount of the blended n-butane. In order to find the flow rate of n-butane based on the uncertainty analysis, the RVP of the gasoline blend was calculated at different volume flow rates of n-butane using the Monte Carlo technique. Figure 10 shows the RVP of the gasoline blend against the standard volume ratio of the blended n-butane into 331.23 m3/h of gasoline. The RVP at volume ratio of zero represents the RVP of the initial gasoline with an average value of 70.72 kPa. The red dashed line indicates the maximum allowable RVP of the final blend (93.08 kPa). The numerical values of the RVP and associated uncertainty estimated based on the 95% confidence interval along with the range of variations of Monte Carlo results of RVP are summarized in Table 9. The maximum volume ratio of the blended n-butane can be estimated using the results of the uncertainty analysis such that the calculated RVP is smaller than 93.08 kPa. The green arrow in Figure 10 shows the value of 6.86 volume percent equivalent to 22.71 m3/h of n-butane that can be added to the 331.23 m3/ h gasoline at standard conditions. The results of the uncertainty analysis at this volume ratio were listed in Table 9. As shown in this table, the calculated RVP is 93.06 kPa which is smaller than the maximum allowable RVP. Figure 11 was developed for the clarity purposes to show the distribution of RVP calculated for the gasoline blend with 6.86 volume percent of blended n-butane using the Monte Carlo method. The green solid line indicates the mean value of the calculated vapor pressures that would be the true value of the reported RVP (92.19 kPa) and will be used as a new specification in the gasoline blending process studied in this work, and two green dashed-dotted lines indicate the lower and upper limits of RVP variations with the 95% CI that represent the uncertainty of the calculated RVP (0.72 kPa). The new specification for RVP can be determined rigorously using the results of the uncertainty analysis such that the RVP does not violate the defined specification. We note that this type of analysis allows for the development of realistic safety parameters for blending based on the uncertainty of the calculations instead of simply basing operational guide lines on rules of thumb.

Figure 8. Pressure−temperature envelopes for the gasoline before and after n-butane blending.

temperature range of 290−330 K. The envelopes were constructed using the results of the Monte Carlo simulation for a sample size of 100 generated by the LHS method for 54 uncertain input parameters. The uncertainty analysis shows that the RVP of the initial gasoline is 70.72 ± 0.76 kPa (10.26 ± 0.11 psia), and after blending of 23.73 Sm3/h n-butane, it will increase to 93.08 ± 0.72 kPa (13.5 ± 0.1 psia). The true values of RVP at initial and final conditions are also indicated by dashed lines in this figure. As shown in Figure 8, increasing the n-butane composition causes the bubble pressure and consequently the RVP to increase. The values of calculated vapor pressures and their estimated uncertainties at each temperature are also listed in Table 8, and the results indicate that the uncertainty of vapor pressure increases as the temperature increases in the thermodynamic region studied in this example. 1576

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Figure 9. Monte Carlo simulation results for RVP calculation of the final gasoline blend with 7.17 volume percent of blended n-butane.

5. CONCLUSIONS A consistent and self-contained error propagation using uncertainties in physical properties using thermodynamic model parameters previously developed was used in two typical hydrocarbon processing problems. This approach can provide valuable understanding about a process thanks to the calculation of uncertainties of key operating or product specification parameters resulting from uncertainties on the thermodynamic model and can assist engineers to develop realistic operational conditions for safer equipment operation or reliable product production. Although not done here, uncertainties resulting from process parameters such as temperatures, pressures, flows, and even equipment performance parameters can be easily implemented thanks to the process simulator modular structure and the Monte Carlo algorithm used in this work. One of the major advantages of the proposed uncertainty analysis is its ability to better estimate safety factors for the design of processes or for product specifications. This method was applied in the process of injection of liquid n-butane into an existing gas pipeline and also the process of blending of liquid n-butane to gasoline, to evaluate the error propagation in the dew point calculation of natural gas and the Reid vapor pressure calculation of gasoline. The safe amount of liquid hydrocarbon that can be added to the existing natural gas pipeline without occurring hydrocarbon liquid dropout was found by taking into account the uncertainties of the pure components properties, binary interaction parameters, and thermodynamic model. The

Figure 10. Calculated RVP and associated uncertainty against the blended n-butane/gasoline standard volume ratio.

Table 9. Results of Uncertainty Analysis of RVP Calculation Depending on the Volume Ratio of the Blended n-Butane to Gasoline at Standard Conditions volume ratio (%) 0.00 1.00 2.00 3.00 4.00 5.00 6.86 7.17 8.00

RVP (kPa)

range (min.-max.) (kPa)

± ± ± ± ± ± ± ± ±

69.94−71.61 73.30−74.95 76.25−78.22 79.78−81.42 82.90−84.54 85.96−87.58 91.45−93.06 92.34−93.95 94.72−96.32

70.72 74.09 77.36 80.55 83.67 86.72 92.19 93.08 95.47

0.76 0.75 0.75 0.74 0.73 0.73 0.72 0.72 0.71

Figure 11. Monte Carlo simulation results for RVP calculation of the final gasoline blend with 6.86 volume percent of blended n-butane. 1577

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(9) Leffler, W. L. Petroleum Refining in Nontechnical Language, 4th ed.; PennWell: Tulsa, OK, 2008. (10) Iman, R. L.; Shortencarier, M. J. A Fortran-77 Program and User’s Guide for the Generation of Latin Hypercube and Random Samples for Use with Computer Models; Report NUREG/RC-3632, SAND832365; National Technical Service: Springfield, VA, 1984. (11) Whiting, W. B.; Tong, T. M.; Reed, M. E. Ind. Eng. Chem. Res. 1993, 32, 1367−1371. (12) Kreamer, D. K.; Stetzenbach, K. J. Ground Water Monit. Rem. 1990, 10, 135−145. (13) Frenkel, M.; Chirico, R. D.; Diky, V.; Muzny, C. D.; Kazakov, A. F. NIST ThermoData Engine (TDE), NIST Standard Reference Database 103b-Pure Compounds, Binary Mixtures, and Chemical Reactions, version 5.0; Standard Reference Data Program, National Institute of Standards and Technology: Gaithersburg, MD, 2010. http://www.nist.gov (accessed Jan 2, 2014). (14) Robinson, D. B.; Peng, D. Y. The Characterization of Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs, Gas Processors Association research report RR-28, Gas Producers Association (GPA): Tulsa, OK, 1978. https://www.gpaglobal.org (accessed Jan 2, 2014).

uncertainty in specifications defined for design of the existing pipeline and in-line equipment performance were also evaluated using the error propagation algorithm to ensure that the n-butane injection does not cause off-specification pipeline issues. For the gasoline blending case, the safe rate of n-butane that can be added to gasoline without violating the RVP specification of the final blend was calculated using the APR model and default values of the VMGSim software for the binary interaction parameters, and only by taking into consideration the uncertainties of pure components properties. The quality of the estimated uncertainties for mixtures containing heavier hydrocarbons can be bettered through data regression as discussed by Hajipour et al.2 This self-contained and consistent computational procedure can be used by modular process simulation systems such as VMGSim.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +1 403-457-4595. E-mail: marco.satyro@ virtualmaterials.com. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support provided by Shell Canada Ltd. for this work and support given by Virtual Materials Group Inc. for providing access to NIST’s TDE software and a copy of VMGsim.

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NOTATION MMSCFD million standard cubic feet per day MMSCMD million standard cubic meters per day P pressure, kPa Pc critical pressure, kPa Q duty, kW RVP Reid vapor pressure, kPa T temperature, K Tc critical temperature, K W work, kW Greek Letters

ω acentric factor

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REFERENCES

(1) Hajipour, S.; Satyro, M. A. Fluid Phase Equilib. 2011, 307, 78−94. (2) Hajipour, S.; Satyro, M. A.; Foley, M. W. Fluid Phase Equilib. 2013, DOI: http://dx.doi.org/10.1016/j.fluid.2013.12.004. (3) Virtual Materials Group Inc. VMGSim Process simulator, version 6.5; Virtual Materials Group Inc.: Calgary, Alberta, Canada, 2011. http://www.virtualmaterials.com (accessed Jan 2, 2014). (4) Stark, V.; Wastie, A. E.; Sliosberg, J. Fuel Gas Pipeline System. U.S. Patent 3,689,237, Sept. 5, 1972. (5) Arenson, E. M. Method of Vaporizing and Combining a Liquefied Cryogenic Fluid Stream with a Gas Stream. U.S. Patent 3,771,260, Nov. 13, 1973. (6) Chin, T. G.; Klaver, R. F.; Alston, A. Gas Saturation Monitoring System and Method for Controlling Condensable Fuel Added to a Gas. U.S. Patent 4,675,028, Jun. 23, 1987. (7) American Society for Testing and Materials. Annual Book of the ASTM Standards; Vol. 05.01; ASTM: Philadelphia, PA, 2013, http:// www.astm.org (accessed Jan 2, 2014). (8) Gary, J. H.; Handwerk, G. E. Petroleum Refining: Technology and Economics, 4th ed.; Marcel Dekker, Inc.: New York, NY, 2001. 1578

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