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Developing two analytical solutions for the diffusion equation and two simple mathematical models for prediction of the gas diffusivity
Journal of Natural Gas Science and Engineering 21 (2014) 417e424
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Developing two analytical solutions for the diffusion equation and two simple mathematical models for prediction of the gas diffusivity S.A. Shafiee Najafi*, M. Jamialahmadi, B. Moslemi Petroleum University of Technology, Ahwaz Faculty of Petroleum, Ahwaz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 June 2014 Received in revised form 5 September 2014 Accepted 8 September 2014 Available online
The accurate prediction of the diffusion coefficient of gases in liquid hydrocarbons is of paramount importance in a variety of applications. Two general approaches are frequently employed for calculation of this property namely, empirical correlations and theory-based mathematical models. Due to lack of sound experimental data at high pressure conditions, application of empirical correlations has become limited; hence, most researchers have employed theory-based mathematical models at these conditions to provide more accurate estimates. The Primary aim of this study is to develop two analytical solutions for the diffusion equation In order to predict the diffusion coefficient of gases in liquids. Diffusion coefficients of methane in dodecane and also in a typical Iranian crude oil are estimated using the developed analytical solution. Two mathematical models have subsequently been developed through transferring these analytical solutions into a dimensionless form, making them easier to solve. The novelties and advantages behind the current work is that the solution of this model is not a function of empirical constants, additionally, it has no parameters that need to be adjusted. One of the fascinating features of this model is that it is quite simple, straight forward, and easy to implement.
Keywords: Analytical solution Diffusion coefficient Mathematical model Mass transfer
© 2014 Elsevier B.V. All rights reserved.
1. Introduction Molecular diffusion is concerned with the movement of individual molecules through a substance by virtue of their thermal energy (Treybal, 1980). Mechanism of molecular diffusion occurs due to close contact of gas and liquid phases which are not thermodynamically at equilibrium (Jamialahmadi et al., 2006). This mechanism ultimately leads to a completely uniform concentration of substance throughout a solution which may initially have not been uniform (Treybal, 1980). The mass transfer by molecular diffusion is very important in various fields of science and engineering, including chemical engineering, petroleum engineering and Biotechnology (Jamialahmadi et al., 2006). Diffusivity of gaseliquid systems is the most important factor to determine the transfer rate of species from one phase to another. As a common application, it is necessary to predict the rate of mass transfer between gas and oil due to a diffusion process for planning
* Corresponding author. E-mail addresses: [email protected], Amin.najafi[email protected] (S.A. Shafiee Najafi). http://dx.doi.org/10.1016/j.jngse.2014.09.006 1875-5100/© 2014 Elsevier B.V. All rights reserved.
and evaluation of gas injection projects. Molecular diffusion coefficient at reservoir condition is the most important parameter required for determining the rate of mass transfer between gas and oil phases (Jamialahmadi et al., 2006; Dorao, 2012). Diffusion coefficient, also called Diffusivity, is an important parameter indicative of the diffusion mobility. Diffusion coefficient is not only encountered in Fick's law, but also in numerous other equations of physics and chemistry (Sidiq and Amin, 2009). For accuracy prediction of Molecular diffusion coefficient requires two important decisions: (i) to define a reliable experimental method; and (ii) to choose accurate models for data interpretation. Experimental methods of calculating the molecular diffusion coefficient are divided into two categories, namely direct and indirect methods (Sheikha et al., 2005; Etminan et al., 2010). First method requires the composition of the liquid phase to be known. To provide an accurate model, fluid composition and its physical and thermodynamic properties have to be updated periodically, mainly owing to continuous changes in the fluid composition (Jamialahmadi et al., 2008). Conclusively, the direct methods are usually time consuming, expensive and quite sensitive to the accuracy of the experiment (Policarpo and Ribeiro, 2011). Moreover,
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the recorded laboratory data applicable to direct methods are highly scarce and rather scattered particularly at high pressure conditions. Considering the aforementioned limitations, direct methods are less suited to predict the diffusion coefficient of hydrocarbon gases in hydrocarbon liquids within wide pressure ranges (Jamialahmadi et al., 2006). In contrast, the need for analyzing the liquid phase composition is obviated using the indirect experimental data. So the Indirect experimental methods are simpler than the direct methods while having sufficiently accurate results (Policarpo and Ribeiro, 2011). The indirect experimental methods can be classified into several types, the most important of which are pressure decay and volume-time methods. In the pressure decay (PD) method, the gas phase is injected into the liquid phase at an isothermal condition, during which the changes in gas phase pressure are recorded versus elapsed time. However, alterations in volume of the liquid phase are generally neglected and its quantity is assumed constant in this method. Finally, the diffusion coefficient is determined using the recorded pressure decay data in conjunction with an appropriate model (Renner, 1988; Zhang et al., 2000; Sheikha et al., 2006). In the volume-time (VT) method, the gas phase is injected into the liquid phase at constant temperature and pressure, during which the volume of the liquid phase is being recorded versus elapsed time. The diffusion coefficient is then calculated using the recorded data together with an appropriate model (Jamialahmadi et al., 2006). This model takes into account the oil phase swelling so as to provide more accurate results. To perform the PD method, it is possible to use any PVT cell but, for the VT method, the cell must possess visual windows and a cathetometer to permit volume readings. Accordingly, the PD method is preferred because of its simplicity in experimental measurements. Although the assumption of negligible oil phase swelling in the PD method has a rather slight effect on the measured diffusion coefficient at low pressures, some deviations and errors are expected to happen at moderate and high pressures (Zhang et al., 2000; Sheikha et al., 2006).Furthermore, According to Etminan et al. (2010), the mathematical models are more complicated for the PD method compared to that of the VT method due to the pressure decline occurring mostly at the interface (Etminan et al., 2010). In this paper, two analytical solutions to the diffusion of hydrocarbon gas molecules into crude oil and dodecane will be presented, one for the semi-infinite boundary condition and the other for the finite boundary condition. The analytical solution for the finite boundary condition is evaluated using the experimental data presented in the work of Jamialahmadi et al., (2006), based on the VT method within a wide pressure range. The data is resulted from the laboratory tests done by means of an accurate, high pressure diffusion cell with a finite domain moving boundary behavior. Finally, the diffusion coefficients of methane-dodecane and methane-crude oil are determined. Furthermore, two mathematical models are developed for the semi-infinite boundary condition and the finite boundary condition through transferring these analytical solutions into a dimensionless form. The diffusion coefficient obtained from these mathematical models are then compared with those determined by the analytical solution developed for the finite boundary condition.
2 vC vC vC vC v C v2 C v2 C ¼ D$ þ RA ux $ þ uy $ þ uz $ þ þ þ vx vy vz vt vx2 vy2 vz2
For a one-dimensional diffusion cell in the absence of chemical reaction and natural convection, the equation of continuity is simplified into a form represented by Equation (2). The schematic of the diffusion process is shown in Fig. 1.
vCA v2 C A ¼D vt vx2
(2)
The diffusion process at high pressures can also be described by Fick's second law provided that the diffusion coefficient, D, is improved by a thermodynamic factor for non-ideal mixtures as follows (Riazi, 1996):
vln 4i D ¼ Da 1 þ vln xi
(3)
Where 4i is the Fugacity coefficient and xi is the mole fraction of i in the liquid phase. Ignoring the above factor for the experimental data used in this paper does not introduce a significant error into our calculations (Jamialahmadi et al., 2006). The diffusion coefficient could be determined after solving the Equation (2) according to initial and boundary conditions imposed on the problem. One initial condition and two boundary conditions are needed to solve this equation. It is assumed that the solute concentration in the liquid phase is negligible at initial condition, thus the initial condition is defined as:
CA ¼ 0
For t ¼ 0 and 0 x Z
(4)
According to the film theory of Whitman (1923) (Policarpo and Ribeiro, 2011), the gas and liquid phases at the interface (i.e.x ¼ Z) shown in Fig. 1 are at thermodynamic equilibrium. Thus, the interfacial solute concentration remains constant as long as the pressure and temperature of the diffusion cell are kept unaltered. The first boundary condition is then defined as:
CA ¼ CAi
For x ¼ Z and t > 0
2. Mathematical modeling The mathematical model, which is used to predict the gas diffusivity in this study, is obtained from the equation of continuity of solute components. The general form of continuity equation is given in Equation (1): (Jamialahmadi et al., 2006).
(1)
Fig. 1. One-dimensional diffusion process in a test diffusion cell.
(5)
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To express the second boundary condition, the diffusion processes are divided into two categories, namely semi-infinite boundary systems and finite boundary systems. 2.1. The semi-infinite boundary model The main assumption behind the semi-infinite boundary model is that solute does not reach the bottom of the cell during the diffusion process. When the gas reaches the bottom of the cell, the second case, i.e. the finite boundary condition dominates. Therefore, the second boundary condition for the semi-infinite model is defined as:
CA ¼ 0
For x ¼ 0 and t 0
The process of deriving a mathematical model for the finite boundary condition is similar to the semi-infinite boundary condition, the details of which have been provided in Appendix B. The final form of the mathematical model for the finite boundary condition may be expressed as:
CAav ðtÞ ¼
rffiffiffiffi inf rffiffiffiffi pffiffiffiffi CAi D t X t n2 Z2 2 ð4ð1Þn þ e Dt p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 4nð1Þ D Dt
(9)
(6)
In this study, the analytical solution of Equation (2) for the semiinfinite model is developed using the Laplace transform. The details of which have been provided in Appendix A. The final form of the mathematical model for the semi-infinite boundary condition may be expressed as:
rffiffiffiffi rffiffiffiffi pffiffiffiffi inf C t X t n2 Z2 D CAav ðtÞ ¼ Ai ð4ð1Þn þ e 4Dt 2 p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 2nð1Þ D 2 Dt
419
(7)
2.3. Dimensionless form of the mathematical model for the semiinfinite boundary condition Equation (7) can be rewritten as follows:
rffiffiffiffiffiffiffiffiffi inf rffiffiffiffiffiffiffiffiffi X CAav ðtÞ Dt Dt n2 Z2 n ¼2 4ð1Þ þ e 4Dt CAi pZ 2 n¼1 pZ 2 nZ þ 2nð1Þðnþ1Þ erfc pffiffiffiffiffiffiffiffi 4Dt
(10)
Defining the dimensionless parameters as follows: Equation (7) is valid as long as the solute does not approach the bottom of the diffusion cell. In the other word, Equation (7) is applicable when the contact time between gas and liquid phases is short or the diffusion cell is long enough (Jamialahmadi et al., 2006). At early times of the diffusion process, especially at high pressure condition, experimental data are affected by convective mixing phenomenon. Since Equation (7) has been solved assuming that no convection occurs, the application of the semi-infinite solution is limited at early times. The convective mixing phenomenon decreases with time, becoming relatively insignificant at middle and late time data. However during these times, for a short length of diffusion cell, the solute reaches the bottom of the cell in the course of diffusion and the semi-infinite boundary model is no longer dominant. Therefore, this model is not really practical, at least for experimental data obtained by a short length diffusion cell, and provided results are not accurate enough to calculate the diffusion coefficient of gases in oil. This analytical solution may be applicable only for experimental data obtained by a long length diffusion cell in which the solute does not reach the bottom of the diffusion cell even at middle and late times of the diffusion process, not considering the effect of convective mixing. Nevertheless, conducting these experiments by a long length diffusion cell appears to be highly costly, difficult and time consuming. Taking into accounts the aforementioned statements and also the limited length of the diffusion cell employed in the experiments of Jamialahmadi et al. (2006), the finite boundary model is believed to provide better estimations of the diffusion coefficient. 2.2. The finite boundary model In the finite boundary model, gas approaches the bottom of the cell during the diffusion process. Moreover, the cell is sealed so that no mass transfer is expected to occur across the ends of the cell. Therefore, the second boundary condition for the finite boundary model is defined as (Jamialahmadi et al., 2006):
vCA ¼0 vx
For x ¼ 0 and t 0
(8)
X¼
Z2 Dt
(11)
Y¼
Cav CAi
(12)
And inserting these dimensionless parameters into Equation (10), yields:
Y ¼2
rffiffiffiffiffiffiffi inf rffiffiffiffiffiffiffi X 1 1 n2 X2 nX þ e 4 þ 2nð1Þðnþ1Þ erfc ð4ð1Þn pX n¼1 pX 2 (13)
2.4. Dimensionless form of the mathematical model for the finite boundary condition The process of deriving a dimensionless form of the mathematical model for the finite boundary condition is identical to the semi-infinite boundary condition. The final form for the finite boundary condition could be expressed as:
! rffiffiffiffiffiffiffi inf rffiffiffiffiffiffiffi X 1 1 n2 X 2 n ðnþ1Þ þ e Y ¼2 þ 4nð1Þ erfc nX 4ð1Þ pX n¼1 pX (14) Implementing these dimensionless forms, the graph of Y versus X could be plotted for the semi-infinite boundary condition and for the finite boundary condition as well. Utilizing the results obtained from the plotted graphs, simple relationships could be obtained between Y and X for both conditions. 3. Methodology The parameters, needed to predict the diffusion coefficient from the mathematical model for the finite boundary condition, are obtained using the experimental data presented by Jamialahmadi
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et al. (2006). These experimental measurements have been carried out based on the VT method within a wide pressure range employing an accurate high pressure diffusion cell with the finite domain moving boundary behavior. The parameters of the mathematical model are divided into two categories, namely direct and indirect methods. Direct parameters include those which are obtained directly from experimental data as the data illustrated in Fig. 2. According to this figure, the variations of the height of the liquid phase are recorded as a function of time for several pressures, ranging between 3.5 and 35 MPa (or 500 to 5000 psi), for methane-dodecane system. Indirect parameters include the parameters which are obtained by modifying the experimental data, as represented in Fig. 3, showing the result of solubility of methane and molar volume of the solutions versus pressure for methane-dodecane system, and in Fig. 4, showing the average methane concentration in the liquid phase as a function of time at several operating pressures. Similar experimental data have also been used for the methane-crude oil system. According to film theory, equilibrium exists at the interface which in effect causes the interface concentration to be equal to solubility (or equilibrium concentration) where this parameter could be determined analytically (through an exact method) from thermodynamics. To this end, CMG software was tuned according to operational conditions and using the available information, the quantity of CAi was computed at different conditions. Since it is not possible to solve the mathematical model for the finite boundary condition, a trial and error procedure could be carried out in order to estimate the diffusion coefficient. A computer code with high iteration numbers was then developed in MATLAB environment to achieve satisfactory results. In this computer code, experimental data of the liquid phase height (Z) and its respective time (t) are entered into the model. The most accurate value of the diffusion coefficient D, is defined so that a good agreement could be obtained between average concentration data, CAav and experimental measurements. 4. Results and discussion In this study the results obtained from the analytical solution for the finite boundary condition are presented. Then the mathematical models obtained from the dimensionless forms of analytical solutions are expressed. A comparison is also drawn between the
Fig. 2. Height of the liquid phase as a function of time and pressure for the methanedodecane system (Jamialahmadi et al., 2006).
Fig. 3. Solubility of the methane in dodecane and molar volume of solution as a function of pressure and temperature (Jamialahmadi et al., 2006).
diffusion coefficients determined from the mathematical models with those obtained from the analytical solution for the finite boundary condition. 4.1. Prediction of the diffusion coefficient From the mathematical model developed for the finite boundary condition, the diffusion coefficients of methane-dodecane and methane-crude oil were measured from the experimental data provided in the work of Jamialahmadi et al. (2006). The results of these calculations are shown in Fig. 5 as a function of pressure, respectively. These figures also illustrate a comparison between the diffusion coefficients calculated by this model and those obtained by Jamialahmadi et al. (2006). An excellent agreement exists between the obtained results and those reported by Jamialahmadi et al. (2006) within the whole pressure range confirming the accuracy and reliability of the mathematical model developed in this study. The model presented by Jamialahmadi et al. (2006) includes the parametersCAi CAav yA Z0 , D and a. The parameter a is unknown in the model as well as the diffusion coefficient, D. Determination of the parametera is requisitely necessary in order to calculate D. a is
Fig. 4. Concentration of methane in dodecane as a function of time and pressure for T ¼ 65 C (Jamialahmadi et al., 2006).
S.A. Shafiee Najafi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 417e424
421
Fig. 6. Comparison between the dimensionless form of analytical solution for semiinfinite boundary condition and model predictions.
Section 2, since analytical solution for the semi-infinite boundary condition was solved with no convection assumption, the application of the semi-infinite solution is limited at early times and the finite boundary model would provide better estimations of the diffusion coefficient at these times. 4.3. Results for dimensionless form of analytical solution for the finite boundary condition Fig. 7 is also obtained from Equation (14) i.e. dimensionless form of analytical solution for the finite boundary condition. According to this figure, the maximum value of Y for the finite boundary condition is 1. Using the data of Y versus, the best match resulted in a squared correlation coefficient (R2) of R2 ¼ 0.9989. Equation of the linear regression line is defined as: Fig. 5. (a): Diffusion coefficient of methane-dodecane as function of pressure. (b): Diffusion coefficient of methane-crude oil as function of pressure.
the adjustable parameter that could be determined via multiple regression analysis. While a single value of 0.56 was obtained for a in the work of Jamialahmadi et al. (2006), our calculations show that this parameter is not constant and slight changes are rather expected particularly at high pressures. 4.2. Results for dimensionless form of analytical solution for the semi-infinite boundary condition
Y ¼ 1:0801X 0:492
(17)
Now, substituting X and Y from Equations (11) and (12) into Equation (17) yields:
Cav ¼ 1:0801CAi
Dt Z2
0:492 (18)
The Equation (18) can be used instead of analytical solution for the finite boundary condition to predict the diffusion coefficients of gases in liquid hydrocarbon.
Fig. 6 is obtained from Equation (13) i.e. dimensionless form of analytical solution for the semi-infinite boundary condition. According to this Figure, the maximum value of Y for the semi-infinite boundary condition is 0.5. The best match resulted in squared correlation coefficient (R2) of R2 ¼ 0.9843 using the data of Y versus X. As shown in Equation (15):
Y ¼ 0:9614X 0:47
(15)
Now, substituting Y and X from Equations (11) and (12) into Equation (15) yields:
Cav ¼ 0:9614CAi
Dt Z2
0:47 (16)
Therefore, the Equation (16) can be used instead of analytical solution for the semi-infinite boundary condition. As discussed in
Fig. 7. Comparison between dimensionless form of analytical solution for finite boundary condition and model predictions.
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Table 1 Comparison of measured diffusion coefficient of methane in dodecane with values calculated according to Equation (18). P (MPa)
31.028 28.959 26.201 23.443 20.685 17.237 13.790 10.343 6.895 3.448
D(m2/s) 108
Error percent (%)
Analytical solution
Modeling
1.547 1.485 1.408 1.336 1.265 1.190 1.130 1.085 1.064 1.051
1.580 1.517 1.433 1.358 1.282 1.202 1.136 1.095 1.076 1.050
2.145 2.141 1.761 1.641 1.336 1.000 0.580 0.896 1.170 0.119
Table 2 Comparison of measured diffusion coefficient of methane in crude oil with values calculated according to Equation (18). P (MPa)
23.443 20.685 17.237 13.790 10.343 6.895 3.448
D(m2/s) 108
Error percent (%)
Analytical solution
Modeling
1.330 1.246 1.145 1.048 0.955 0.870 0.788
1.360 1.261 1.156 1.053 0.955 0.865 0.778
1.463 1.124 0.998 0.451 0.055 0.609 1.320
Tables 1 and 2 compare the diffusion coefficient values from the Equation (18) with the results obtained by analytical solution for the finite boundary condition. Apparently, a good agreement exists between the diffusion coefficients determined from Equation (18) with the results obtained from this analytical solution within the whole pressure range, further corroborating the accuracy and reliability of the mathematical model developed in this study. 5. Conclusions In this study, two analytical solutions were developed for prediction of the diffusion coefficient of hydrocarbon gasses in liquid hydrocarbons. By using the analytical solution for the finite boundary condition and experimental data provided in the work of Jamialahmadi et al. (2006), accurate quantities of the diffusion coefficients of methane in dodecane and also in a typical Iranian crude oil were determined within a wide pressure range. Furthermore, two mathematical models were developed for the semi-infinite boundary condition and the finite boundary condition through transferring these analytical solutions to dimensionless form. A comparison was drawn between the diffusion coefficients determined from the mathematical models and those obtained from the analytical solution for the finite boundary condition. The diffusion coefficients obtained from the mathematical models are in well agreement with the outputs of the analytical solution developed for the finite boundary condition within the whole pressure range, further verifying the accuracy, validity, and reliability of the developed mathematical models. These mathematical models could be employed to predict the diffusion coefficients of gases in liquid hydrocarbon with a solution procedure which is much simpler than the analytical solutions. The advantages behind the current work is that the diffusion coefficient has directly been obtained from the model presented in this work. Additionally, the solution of this model is not a function of empirical constants, thereby it has no parameters that need to be adjusted.
Acknowledgment The first author would like to thank Peyman Kamranfar and Mehran Moradi, graduate students at Petroleum University of Technology, for their assistance. Nomenclature CA CAav CA ðx; tÞ BðsÞ D Da p RA S t u x xi Z
Mass concentration of solute (kg/m3) Average Solute concentration (kg/m3) Solute concentration at position x at time t (kg/m3) Laplace transform of Solute concentration Diffusion coefficient (m2/sec) Activity-corrected diffusion coefficient (m2/s) Pressure (Pa) Rate of reaction (kg/m3s) Laplace transform variable, dimensionless Time (sec) Velocity (m/sec) Coordinate direction (m) The mole fraction of i in the liquid phase Position of interface (m)
Greek symbols 4 Fugacity coefficient y Molar volume (m3) Subscript A x i t 0
and superscript Salute Coordinate Initial At time, t Standard condition
Appendix A. The process of deriving a mathematical model for the semi-infinite boundary condition Applying the Laplace transform to the PDE given in Equation (2) with respect to time and then substituting the initial condition, this equation is converted to a simpler linear ODE as shown in Equation (A-1):
v2 CA x; S 1 CA ðx; SÞ ¼ 0 D vx2
(A-1)
General solution of the Equation (A-1) is obtained as:
pffiffiS pffiffiS CA ðx; SÞ ¼ A S e Dx þ B S e Dx
(A-2)
The boundary conditions should be transformed into the Laplace form to be used in Equation (A-2) as the following:
CA ðx; SÞ ¼ 0
(A-3)
Since the pressure and temperature are constant during the gas diffusion process, the interfacial concentration, CAi , does not change with time. The Laplace transform of CAi is then calculated as:
CA ðZ; SÞ ¼
CAi S
(A-4)
Substituting Equation (A-3) into Equation (A-2), gives:
AðSÞ ¼ BðSÞ
(A-5)
And by substituting Equations (A-4) and (A-5) into Equation (A-
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2), the coefficients of AðsÞ and BðsÞ are calculated according to the following expressions:
pffiffiS CAi e DZ AðSÞ ¼ pffiffi S 2 Z D 1 S e B S ¼
(A-6)
pffiffiS CAi e DZ
(A-7)
pffiffi S 2 Z D 1 S e
CA ðx; SÞ ¼ S
pffiffi S 2
Z D
e
vCA 0; S ¼ 0 vx
Since the pressure and temperature are constant during the gas diffusion process, the interfacial concentration, CAi , does not change with time. The Laplace transform of CAi is then calculated as:
CA ðZ; SÞ ¼
Z 0
CAav ðZ; SÞ ¼
CA Z
pffiffiS pffiffiS e Dx e Dx
(A-8)
1
(A-9)
0
Substituting Equation (A-8) into Equation (A-9) and performing the calculation gives:
pffiffis 1 pffiffiffiffi 0 Z D 1 CAi D @ e pffiffis A CAav ðZ; SÞ ¼ 3 S2 e DZ þ 1
(A-10)
3
ZS2
S
inf X
en
pffiffiS
Z
D
! ð1Þn
2
Z
D
S
S Z D
2
e
(B-6)
þ1
Now, substituting of AðSÞ and BðSÞ from Equations (B-5) and (B6) into Equation (B-1) yields:
pffiffiS pffiffiS pffiffiS CAi e DZ e Dx þ e Dx CA ðx; SÞ ¼ pffiffi S 2 Z D þ1 S e
(A-11)
And finally by calculating the inverse Laplace transforms (Chen et al., 2001) of Equation (A-11), the mathematical model for the semi-infinite boundary condition is obtained as:
(B-7)
Z
Z
0
CAav ðZ; SÞ ¼
CA x; S dx Z
(B-8)
Z
dx 0
Substituting Equation (B-7) into Equation (B-8) and performing the calculation gives:
(A-12)
Appendix B. The process of deriving a mathematical model for the finite boundary condition The general solution of Laplace transform for the PDE given in Equation (2) with respect to time could be written as:
pffiffiS pffiffiS CA ðx; SÞ ¼ A S e Dx þ B S e Dx
pffiffiS
pffiffi
n¼1
rffiffiffiffi rffiffiffiffi pffiffiffiffi inf C t X t n2 Z2 D 2 CAav ðtÞ ¼ Ai ð4ð1Þn þ e 4Dt p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 2nð1Þ D 2 Dt
Z
D
e
CAi e
(B-5)
þ1
pffiffi S
Since Equation (2) is used for a one-dimensional diffusion cell, the average gas concentration can be obtained by the following integration:
The series form of the Equation (A-10) is calculated as:
1þ2
(B-4)
pffiffiS CAi e DZ
B S ¼
x; S dx
Z
pffiffiffiffi CAi D
(B-3)
And by substitution Equations (B-3) and (B-4) into Equation (B1), the coefficients of AðsÞ and BðsÞ are calculated according to the following expressions:
A S ¼
dx
CAav ðZ; SÞ ¼
CAi S
AðSÞ ¼ BðSÞ
Since Equation (2) is used for a one-dimensional diffusion cell, the average gas concentration can be obtained by the following integration:
Z
(B-2)
Substitution Equation (B-2) into Equation (B-1), gives:
Now, substituting AðsÞ and BðsÞ from Equations (A-6) and (A-7) into Equation (A-2) yields:
pffiffiS CAi e DZ
423
(B-1)
The boundary conditions should be transformed into the Laplace form to be used in Equation (B-1) the following:
1 pffiffi C B 2 DS Z C pffiffiffiffi B B e 1 C C CAi D B C B CAav ðZ; SÞ ¼ 3 C B 2 ZS B pffiffiS C B 2 DZ C A @ e þ1 0
(B-9)
The series form of Equation (B-9) is calculated as:
CAav ðZ; SÞ ¼
pffiffiffiffi CAi D 3
ZS2
1þ2
inf X
e
2n
pffiffiS
Z D
! ð1Þ
n
(B-10)
n¼1
And finally by calculating the Inverse Laplace Transforms (Chen
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et al., 2001) of Equation (B-10), the mathematical model for the finite boundary condition is obtained as:
CAav ðtÞ ¼
rffiffiffiffi rffiffiffiffi pffiffiffiffi inf CAi D t X t n2 Z2 2 ð4ð1Þn þ e Dt p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 4nð1Þ D Dt
(B-11)
References Chen, Y., Petras, I., Vinagre, B., 2001. A List of Laplace and Inverse Laplace Transforms Related to Fractional Order Calculus. Dorao, C.A., 2012. Dispersion phenomena in gas liquid systems. J. Nat. Gas Sci. Eng. 5, 25e30. Etminan, S.R., et al., 2010. Constant-pressure technique for gas diffusivity and solubility measurements in heavy oil and bitumen. Energy & Fuels 24 (1), 533e549.
Jamialahmadi, M., Emadi, M., Müller-Steinhagen, H., 2006. Diffusion coefficients of methane in liquid hydrocarbons at high pressure and temperature. J. Pet. Sci. Eng. 53 (1), 47e60. Jamialahmadi, M., Emadi, M., Mueller-Steinhagen, H., 2008. Prediction of diffusion coefficients of hydrocarbon gases in hydrocarbon liquids at high pressure and temperature. Oil Gas-Eur. Mag. 34 (2), 69e77. Policarpo, N., Ribeiro, P., 2011. Experimental measurement of gas-liquid diffusivity. Braz. J. Pet. Gas 5 (3), 171e188. Renner, T., 1988. Measurement and correlation of diffusion coefficients for CO2 and rich-gas applications. SPE Reserv. Eng. 3 (2), 517e523. Riazi, M.R., 1996. A new method for experimental measurement of diffusion coefficients in reservoir fluids. J. Pet. Sci. Eng. 14 (3), 235e250. Sheikha, H., Pooladi-Darvish, M., Mehrotra, A., 2005. Graphical methods for obtaining the diffusion coefficient of gases in bitumen. In: Canadian International Petroleum Conference. Sheikha, H., Mehrotra, A.K., Pooladi-Darvish, M., 2006. An inverse solution methodology for estimating the diffusion coefficient of gases in Athabasca bitumen from pressure-decay data. J. Pet. Sci. Eng. 53 (3), 189e202. Sidiq, H., Amin, R., 2009. Mathematical model for calculating the dispersion coefficient of super critical CO2 from the results of laboratory experiments on enhanced gas recovery. J. Nat. Gas Sci. Eng. 1 (6), 177e182. Treybal, R.E., 1980. Mass-transfer Operations, vol. 3. McGraw-Hill, New York. Zhang, Y., Hyndman, C., Maini, B., 2000. Measurement of gas diffusivity in heavy oils. J. Pet. Sci. Eng. 25 (1), 37e47.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Developing two analytical solutions for the diffusion equation and two simple mathematical models for prediction of the gas diffusivity S.A. Shafiee Najafi*, M. Jamialahmadi, B. Moslemi Petroleum University of Technology, Ahwaz Faculty of Petroleum, Ahwaz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 June 2014 Received in revised form 5 September 2014 Accepted 8 September 2014 Available online
The accurate prediction of the diffusion coefficient of gases in liquid hydrocarbons is of paramount importance in a variety of applications. Two general approaches are frequently employed for calculation of this property namely, empirical correlations and theory-based mathematical models. Due to lack of sound experimental data at high pressure conditions, application of empirical correlations has become limited; hence, most researchers have employed theory-based mathematical models at these conditions to provide more accurate estimates. The Primary aim of this study is to develop two analytical solutions for the diffusion equation In order to predict the diffusion coefficient of gases in liquids. Diffusion coefficients of methane in dodecane and also in a typical Iranian crude oil are estimated using the developed analytical solution. Two mathematical models have subsequently been developed through transferring these analytical solutions into a dimensionless form, making them easier to solve. The novelties and advantages behind the current work is that the solution of this model is not a function of empirical constants, additionally, it has no parameters that need to be adjusted. One of the fascinating features of this model is that it is quite simple, straight forward, and easy to implement.
Keywords: Analytical solution Diffusion coefficient Mathematical model Mass transfer
© 2014 Elsevier B.V. All rights reserved.
1. Introduction Molecular diffusion is concerned with the movement of individual molecules through a substance by virtue of their thermal energy (Treybal, 1980). Mechanism of molecular diffusion occurs due to close contact of gas and liquid phases which are not thermodynamically at equilibrium (Jamialahmadi et al., 2006). This mechanism ultimately leads to a completely uniform concentration of substance throughout a solution which may initially have not been uniform (Treybal, 1980). The mass transfer by molecular diffusion is very important in various fields of science and engineering, including chemical engineering, petroleum engineering and Biotechnology (Jamialahmadi et al., 2006). Diffusivity of gaseliquid systems is the most important factor to determine the transfer rate of species from one phase to another. As a common application, it is necessary to predict the rate of mass transfer between gas and oil due to a diffusion process for planning
* Corresponding author. E-mail addresses: [email protected], Amin.najafi[email protected] (S.A. Shafiee Najafi). http://dx.doi.org/10.1016/j.jngse.2014.09.006 1875-5100/© 2014 Elsevier B.V. All rights reserved.
and evaluation of gas injection projects. Molecular diffusion coefficient at reservoir condition is the most important parameter required for determining the rate of mass transfer between gas and oil phases (Jamialahmadi et al., 2006; Dorao, 2012). Diffusion coefficient, also called Diffusivity, is an important parameter indicative of the diffusion mobility. Diffusion coefficient is not only encountered in Fick's law, but also in numerous other equations of physics and chemistry (Sidiq and Amin, 2009). For accuracy prediction of Molecular diffusion coefficient requires two important decisions: (i) to define a reliable experimental method; and (ii) to choose accurate models for data interpretation. Experimental methods of calculating the molecular diffusion coefficient are divided into two categories, namely direct and indirect methods (Sheikha et al., 2005; Etminan et al., 2010). First method requires the composition of the liquid phase to be known. To provide an accurate model, fluid composition and its physical and thermodynamic properties have to be updated periodically, mainly owing to continuous changes in the fluid composition (Jamialahmadi et al., 2008). Conclusively, the direct methods are usually time consuming, expensive and quite sensitive to the accuracy of the experiment (Policarpo and Ribeiro, 2011). Moreover,
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the recorded laboratory data applicable to direct methods are highly scarce and rather scattered particularly at high pressure conditions. Considering the aforementioned limitations, direct methods are less suited to predict the diffusion coefficient of hydrocarbon gases in hydrocarbon liquids within wide pressure ranges (Jamialahmadi et al., 2006). In contrast, the need for analyzing the liquid phase composition is obviated using the indirect experimental data. So the Indirect experimental methods are simpler than the direct methods while having sufficiently accurate results (Policarpo and Ribeiro, 2011). The indirect experimental methods can be classified into several types, the most important of which are pressure decay and volume-time methods. In the pressure decay (PD) method, the gas phase is injected into the liquid phase at an isothermal condition, during which the changes in gas phase pressure are recorded versus elapsed time. However, alterations in volume of the liquid phase are generally neglected and its quantity is assumed constant in this method. Finally, the diffusion coefficient is determined using the recorded pressure decay data in conjunction with an appropriate model (Renner, 1988; Zhang et al., 2000; Sheikha et al., 2006). In the volume-time (VT) method, the gas phase is injected into the liquid phase at constant temperature and pressure, during which the volume of the liquid phase is being recorded versus elapsed time. The diffusion coefficient is then calculated using the recorded data together with an appropriate model (Jamialahmadi et al., 2006). This model takes into account the oil phase swelling so as to provide more accurate results. To perform the PD method, it is possible to use any PVT cell but, for the VT method, the cell must possess visual windows and a cathetometer to permit volume readings. Accordingly, the PD method is preferred because of its simplicity in experimental measurements. Although the assumption of negligible oil phase swelling in the PD method has a rather slight effect on the measured diffusion coefficient at low pressures, some deviations and errors are expected to happen at moderate and high pressures (Zhang et al., 2000; Sheikha et al., 2006).Furthermore, According to Etminan et al. (2010), the mathematical models are more complicated for the PD method compared to that of the VT method due to the pressure decline occurring mostly at the interface (Etminan et al., 2010). In this paper, two analytical solutions to the diffusion of hydrocarbon gas molecules into crude oil and dodecane will be presented, one for the semi-infinite boundary condition and the other for the finite boundary condition. The analytical solution for the finite boundary condition is evaluated using the experimental data presented in the work of Jamialahmadi et al., (2006), based on the VT method within a wide pressure range. The data is resulted from the laboratory tests done by means of an accurate, high pressure diffusion cell with a finite domain moving boundary behavior. Finally, the diffusion coefficients of methane-dodecane and methane-crude oil are determined. Furthermore, two mathematical models are developed for the semi-infinite boundary condition and the finite boundary condition through transferring these analytical solutions into a dimensionless form. The diffusion coefficient obtained from these mathematical models are then compared with those determined by the analytical solution developed for the finite boundary condition.
2 vC vC vC vC v C v2 C v2 C ¼ D$ þ RA ux $ þ uy $ þ uz $ þ þ þ vx vy vz vt vx2 vy2 vz2
For a one-dimensional diffusion cell in the absence of chemical reaction and natural convection, the equation of continuity is simplified into a form represented by Equation (2). The schematic of the diffusion process is shown in Fig. 1.
vCA v2 C A ¼D vt vx2
(2)
The diffusion process at high pressures can also be described by Fick's second law provided that the diffusion coefficient, D, is improved by a thermodynamic factor for non-ideal mixtures as follows (Riazi, 1996):
vln 4i D ¼ Da 1 þ vln xi
(3)
Where 4i is the Fugacity coefficient and xi is the mole fraction of i in the liquid phase. Ignoring the above factor for the experimental data used in this paper does not introduce a significant error into our calculations (Jamialahmadi et al., 2006). The diffusion coefficient could be determined after solving the Equation (2) according to initial and boundary conditions imposed on the problem. One initial condition and two boundary conditions are needed to solve this equation. It is assumed that the solute concentration in the liquid phase is negligible at initial condition, thus the initial condition is defined as:
CA ¼ 0
For t ¼ 0 and 0 x Z
(4)
According to the film theory of Whitman (1923) (Policarpo and Ribeiro, 2011), the gas and liquid phases at the interface (i.e.x ¼ Z) shown in Fig. 1 are at thermodynamic equilibrium. Thus, the interfacial solute concentration remains constant as long as the pressure and temperature of the diffusion cell are kept unaltered. The first boundary condition is then defined as:
CA ¼ CAi
For x ¼ Z and t > 0
2. Mathematical modeling The mathematical model, which is used to predict the gas diffusivity in this study, is obtained from the equation of continuity of solute components. The general form of continuity equation is given in Equation (1): (Jamialahmadi et al., 2006).
(1)
Fig. 1. One-dimensional diffusion process in a test diffusion cell.
(5)
S.A. Shafiee Najafi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 417e424
To express the second boundary condition, the diffusion processes are divided into two categories, namely semi-infinite boundary systems and finite boundary systems. 2.1. The semi-infinite boundary model The main assumption behind the semi-infinite boundary model is that solute does not reach the bottom of the cell during the diffusion process. When the gas reaches the bottom of the cell, the second case, i.e. the finite boundary condition dominates. Therefore, the second boundary condition for the semi-infinite model is defined as:
CA ¼ 0
For x ¼ 0 and t 0
The process of deriving a mathematical model for the finite boundary condition is similar to the semi-infinite boundary condition, the details of which have been provided in Appendix B. The final form of the mathematical model for the finite boundary condition may be expressed as:
CAav ðtÞ ¼
rffiffiffiffi inf rffiffiffiffi pffiffiffiffi CAi D t X t n2 Z2 2 ð4ð1Þn þ e Dt p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 4nð1Þ D Dt
(9)
(6)
In this study, the analytical solution of Equation (2) for the semiinfinite model is developed using the Laplace transform. The details of which have been provided in Appendix A. The final form of the mathematical model for the semi-infinite boundary condition may be expressed as:
rffiffiffiffi rffiffiffiffi pffiffiffiffi inf C t X t n2 Z2 D CAav ðtÞ ¼ Ai ð4ð1Þn þ e 4Dt 2 p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 2nð1Þ D 2 Dt
419
(7)
2.3. Dimensionless form of the mathematical model for the semiinfinite boundary condition Equation (7) can be rewritten as follows:
rffiffiffiffiffiffiffiffiffi inf rffiffiffiffiffiffiffiffiffi X CAav ðtÞ Dt Dt n2 Z2 n ¼2 4ð1Þ þ e 4Dt CAi pZ 2 n¼1 pZ 2 nZ þ 2nð1Þðnþ1Þ erfc pffiffiffiffiffiffiffiffi 4Dt
(10)
Defining the dimensionless parameters as follows: Equation (7) is valid as long as the solute does not approach the bottom of the diffusion cell. In the other word, Equation (7) is applicable when the contact time between gas and liquid phases is short or the diffusion cell is long enough (Jamialahmadi et al., 2006). At early times of the diffusion process, especially at high pressure condition, experimental data are affected by convective mixing phenomenon. Since Equation (7) has been solved assuming that no convection occurs, the application of the semi-infinite solution is limited at early times. The convective mixing phenomenon decreases with time, becoming relatively insignificant at middle and late time data. However during these times, for a short length of diffusion cell, the solute reaches the bottom of the cell in the course of diffusion and the semi-infinite boundary model is no longer dominant. Therefore, this model is not really practical, at least for experimental data obtained by a short length diffusion cell, and provided results are not accurate enough to calculate the diffusion coefficient of gases in oil. This analytical solution may be applicable only for experimental data obtained by a long length diffusion cell in which the solute does not reach the bottom of the diffusion cell even at middle and late times of the diffusion process, not considering the effect of convective mixing. Nevertheless, conducting these experiments by a long length diffusion cell appears to be highly costly, difficult and time consuming. Taking into accounts the aforementioned statements and also the limited length of the diffusion cell employed in the experiments of Jamialahmadi et al. (2006), the finite boundary model is believed to provide better estimations of the diffusion coefficient. 2.2. The finite boundary model In the finite boundary model, gas approaches the bottom of the cell during the diffusion process. Moreover, the cell is sealed so that no mass transfer is expected to occur across the ends of the cell. Therefore, the second boundary condition for the finite boundary model is defined as (Jamialahmadi et al., 2006):
vCA ¼0 vx
For x ¼ 0 and t 0
(8)
X¼
Z2 Dt
(11)
Y¼
Cav CAi
(12)
And inserting these dimensionless parameters into Equation (10), yields:
Y ¼2
rffiffiffiffiffiffiffi inf rffiffiffiffiffiffiffi X 1 1 n2 X2 nX þ e 4 þ 2nð1Þðnþ1Þ erfc ð4ð1Þn pX n¼1 pX 2 (13)
2.4. Dimensionless form of the mathematical model for the finite boundary condition The process of deriving a dimensionless form of the mathematical model for the finite boundary condition is identical to the semi-infinite boundary condition. The final form for the finite boundary condition could be expressed as:
! rffiffiffiffiffiffiffi inf rffiffiffiffiffiffiffi X 1 1 n2 X 2 n ðnþ1Þ þ e Y ¼2 þ 4nð1Þ erfc nX 4ð1Þ pX n¼1 pX (14) Implementing these dimensionless forms, the graph of Y versus X could be plotted for the semi-infinite boundary condition and for the finite boundary condition as well. Utilizing the results obtained from the plotted graphs, simple relationships could be obtained between Y and X for both conditions. 3. Methodology The parameters, needed to predict the diffusion coefficient from the mathematical model for the finite boundary condition, are obtained using the experimental data presented by Jamialahmadi
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et al. (2006). These experimental measurements have been carried out based on the VT method within a wide pressure range employing an accurate high pressure diffusion cell with the finite domain moving boundary behavior. The parameters of the mathematical model are divided into two categories, namely direct and indirect methods. Direct parameters include those which are obtained directly from experimental data as the data illustrated in Fig. 2. According to this figure, the variations of the height of the liquid phase are recorded as a function of time for several pressures, ranging between 3.5 and 35 MPa (or 500 to 5000 psi), for methane-dodecane system. Indirect parameters include the parameters which are obtained by modifying the experimental data, as represented in Fig. 3, showing the result of solubility of methane and molar volume of the solutions versus pressure for methane-dodecane system, and in Fig. 4, showing the average methane concentration in the liquid phase as a function of time at several operating pressures. Similar experimental data have also been used for the methane-crude oil system. According to film theory, equilibrium exists at the interface which in effect causes the interface concentration to be equal to solubility (or equilibrium concentration) where this parameter could be determined analytically (through an exact method) from thermodynamics. To this end, CMG software was tuned according to operational conditions and using the available information, the quantity of CAi was computed at different conditions. Since it is not possible to solve the mathematical model for the finite boundary condition, a trial and error procedure could be carried out in order to estimate the diffusion coefficient. A computer code with high iteration numbers was then developed in MATLAB environment to achieve satisfactory results. In this computer code, experimental data of the liquid phase height (Z) and its respective time (t) are entered into the model. The most accurate value of the diffusion coefficient D, is defined so that a good agreement could be obtained between average concentration data, CAav and experimental measurements. 4. Results and discussion In this study the results obtained from the analytical solution for the finite boundary condition are presented. Then the mathematical models obtained from the dimensionless forms of analytical solutions are expressed. A comparison is also drawn between the
Fig. 2. Height of the liquid phase as a function of time and pressure for the methanedodecane system (Jamialahmadi et al., 2006).
Fig. 3. Solubility of the methane in dodecane and molar volume of solution as a function of pressure and temperature (Jamialahmadi et al., 2006).
diffusion coefficients determined from the mathematical models with those obtained from the analytical solution for the finite boundary condition. 4.1. Prediction of the diffusion coefficient From the mathematical model developed for the finite boundary condition, the diffusion coefficients of methane-dodecane and methane-crude oil were measured from the experimental data provided in the work of Jamialahmadi et al. (2006). The results of these calculations are shown in Fig. 5 as a function of pressure, respectively. These figures also illustrate a comparison between the diffusion coefficients calculated by this model and those obtained by Jamialahmadi et al. (2006). An excellent agreement exists between the obtained results and those reported by Jamialahmadi et al. (2006) within the whole pressure range confirming the accuracy and reliability of the mathematical model developed in this study. The model presented by Jamialahmadi et al. (2006) includes the parametersCAi CAav yA Z0 , D and a. The parameter a is unknown in the model as well as the diffusion coefficient, D. Determination of the parametera is requisitely necessary in order to calculate D. a is
Fig. 4. Concentration of methane in dodecane as a function of time and pressure for T ¼ 65 C (Jamialahmadi et al., 2006).
S.A. Shafiee Najafi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 417e424
421
Fig. 6. Comparison between the dimensionless form of analytical solution for semiinfinite boundary condition and model predictions.
Section 2, since analytical solution for the semi-infinite boundary condition was solved with no convection assumption, the application of the semi-infinite solution is limited at early times and the finite boundary model would provide better estimations of the diffusion coefficient at these times. 4.3. Results for dimensionless form of analytical solution for the finite boundary condition Fig. 7 is also obtained from Equation (14) i.e. dimensionless form of analytical solution for the finite boundary condition. According to this figure, the maximum value of Y for the finite boundary condition is 1. Using the data of Y versus, the best match resulted in a squared correlation coefficient (R2) of R2 ¼ 0.9989. Equation of the linear regression line is defined as: Fig. 5. (a): Diffusion coefficient of methane-dodecane as function of pressure. (b): Diffusion coefficient of methane-crude oil as function of pressure.
the adjustable parameter that could be determined via multiple regression analysis. While a single value of 0.56 was obtained for a in the work of Jamialahmadi et al. (2006), our calculations show that this parameter is not constant and slight changes are rather expected particularly at high pressures. 4.2. Results for dimensionless form of analytical solution for the semi-infinite boundary condition
Y ¼ 1:0801X 0:492
(17)
Now, substituting X and Y from Equations (11) and (12) into Equation (17) yields:
Cav ¼ 1:0801CAi
Dt Z2
0:492 (18)
The Equation (18) can be used instead of analytical solution for the finite boundary condition to predict the diffusion coefficients of gases in liquid hydrocarbon.
Fig. 6 is obtained from Equation (13) i.e. dimensionless form of analytical solution for the semi-infinite boundary condition. According to this Figure, the maximum value of Y for the semi-infinite boundary condition is 0.5. The best match resulted in squared correlation coefficient (R2) of R2 ¼ 0.9843 using the data of Y versus X. As shown in Equation (15):
Y ¼ 0:9614X 0:47
(15)
Now, substituting Y and X from Equations (11) and (12) into Equation (15) yields:
Cav ¼ 0:9614CAi
Dt Z2
0:47 (16)
Therefore, the Equation (16) can be used instead of analytical solution for the semi-infinite boundary condition. As discussed in
Fig. 7. Comparison between dimensionless form of analytical solution for finite boundary condition and model predictions.
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S.A. Shafiee Najafi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 417e424
Table 1 Comparison of measured diffusion coefficient of methane in dodecane with values calculated according to Equation (18). P (MPa)
31.028 28.959 26.201 23.443 20.685 17.237 13.790 10.343 6.895 3.448
D(m2/s) 108
Error percent (%)
Analytical solution
Modeling
1.547 1.485 1.408 1.336 1.265 1.190 1.130 1.085 1.064 1.051
1.580 1.517 1.433 1.358 1.282 1.202 1.136 1.095 1.076 1.050
2.145 2.141 1.761 1.641 1.336 1.000 0.580 0.896 1.170 0.119
Table 2 Comparison of measured diffusion coefficient of methane in crude oil with values calculated according to Equation (18). P (MPa)
23.443 20.685 17.237 13.790 10.343 6.895 3.448
D(m2/s) 108
Error percent (%)
Analytical solution
Modeling
1.330 1.246 1.145 1.048 0.955 0.870 0.788
1.360 1.261 1.156 1.053 0.955 0.865 0.778
1.463 1.124 0.998 0.451 0.055 0.609 1.320
Tables 1 and 2 compare the diffusion coefficient values from the Equation (18) with the results obtained by analytical solution for the finite boundary condition. Apparently, a good agreement exists between the diffusion coefficients determined from Equation (18) with the results obtained from this analytical solution within the whole pressure range, further corroborating the accuracy and reliability of the mathematical model developed in this study. 5. Conclusions In this study, two analytical solutions were developed for prediction of the diffusion coefficient of hydrocarbon gasses in liquid hydrocarbons. By using the analytical solution for the finite boundary condition and experimental data provided in the work of Jamialahmadi et al. (2006), accurate quantities of the diffusion coefficients of methane in dodecane and also in a typical Iranian crude oil were determined within a wide pressure range. Furthermore, two mathematical models were developed for the semi-infinite boundary condition and the finite boundary condition through transferring these analytical solutions to dimensionless form. A comparison was drawn between the diffusion coefficients determined from the mathematical models and those obtained from the analytical solution for the finite boundary condition. The diffusion coefficients obtained from the mathematical models are in well agreement with the outputs of the analytical solution developed for the finite boundary condition within the whole pressure range, further verifying the accuracy, validity, and reliability of the developed mathematical models. These mathematical models could be employed to predict the diffusion coefficients of gases in liquid hydrocarbon with a solution procedure which is much simpler than the analytical solutions. The advantages behind the current work is that the diffusion coefficient has directly been obtained from the model presented in this work. Additionally, the solution of this model is not a function of empirical constants, thereby it has no parameters that need to be adjusted.
Acknowledgment The first author would like to thank Peyman Kamranfar and Mehran Moradi, graduate students at Petroleum University of Technology, for their assistance. Nomenclature CA CAav CA ðx; tÞ BðsÞ D Da p RA S t u x xi Z
Mass concentration of solute (kg/m3) Average Solute concentration (kg/m3) Solute concentration at position x at time t (kg/m3) Laplace transform of Solute concentration Diffusion coefficient (m2/sec) Activity-corrected diffusion coefficient (m2/s) Pressure (Pa) Rate of reaction (kg/m3s) Laplace transform variable, dimensionless Time (sec) Velocity (m/sec) Coordinate direction (m) The mole fraction of i in the liquid phase Position of interface (m)
Greek symbols 4 Fugacity coefficient y Molar volume (m3) Subscript A x i t 0
and superscript Salute Coordinate Initial At time, t Standard condition
Appendix A. The process of deriving a mathematical model for the semi-infinite boundary condition Applying the Laplace transform to the PDE given in Equation (2) with respect to time and then substituting the initial condition, this equation is converted to a simpler linear ODE as shown in Equation (A-1):
v2 CA x; S 1 CA ðx; SÞ ¼ 0 D vx2
(A-1)
General solution of the Equation (A-1) is obtained as:
pffiffiS pffiffiS CA ðx; SÞ ¼ A S e Dx þ B S e Dx
(A-2)
The boundary conditions should be transformed into the Laplace form to be used in Equation (A-2) as the following:
CA ðx; SÞ ¼ 0
(A-3)
Since the pressure and temperature are constant during the gas diffusion process, the interfacial concentration, CAi , does not change with time. The Laplace transform of CAi is then calculated as:
CA ðZ; SÞ ¼
CAi S
(A-4)
Substituting Equation (A-3) into Equation (A-2), gives:
AðSÞ ¼ BðSÞ
(A-5)
And by substituting Equations (A-4) and (A-5) into Equation (A-
S.A. Shafiee Najafi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 417e424
2), the coefficients of AðsÞ and BðsÞ are calculated according to the following expressions:
pffiffiS CAi e DZ AðSÞ ¼ pffiffi S 2 Z D 1 S e B S ¼
(A-6)
pffiffiS CAi e DZ
(A-7)
pffiffi S 2 Z D 1 S e
CA ðx; SÞ ¼ S
pffiffi S 2
Z D
e
vCA 0; S ¼ 0 vx
Since the pressure and temperature are constant during the gas diffusion process, the interfacial concentration, CAi , does not change with time. The Laplace transform of CAi is then calculated as:
CA ðZ; SÞ ¼
Z 0
CAav ðZ; SÞ ¼
CA Z
pffiffiS pffiffiS e Dx e Dx
(A-8)
1
(A-9)
0
Substituting Equation (A-8) into Equation (A-9) and performing the calculation gives:
pffiffis 1 pffiffiffiffi 0 Z D 1 CAi D @ e pffiffis A CAav ðZ; SÞ ¼ 3 S2 e DZ þ 1
(A-10)
3
ZS2
S
inf X
en
pffiffiS
Z
D
! ð1Þn
2
Z
D
S
S Z D
2
e
(B-6)
þ1
Now, substituting of AðSÞ and BðSÞ from Equations (B-5) and (B6) into Equation (B-1) yields:
pffiffiS pffiffiS pffiffiS CAi e DZ e Dx þ e Dx CA ðx; SÞ ¼ pffiffi S 2 Z D þ1 S e
(A-11)
And finally by calculating the inverse Laplace transforms (Chen et al., 2001) of Equation (A-11), the mathematical model for the semi-infinite boundary condition is obtained as:
(B-7)
Z
Z
0
CAav ðZ; SÞ ¼
CA x; S dx Z
(B-8)
Z
dx 0
Substituting Equation (B-7) into Equation (B-8) and performing the calculation gives:
(A-12)
Appendix B. The process of deriving a mathematical model for the finite boundary condition The general solution of Laplace transform for the PDE given in Equation (2) with respect to time could be written as:
pffiffiS pffiffiS CA ðx; SÞ ¼ A S e Dx þ B S e Dx
pffiffiS
pffiffi
n¼1
rffiffiffiffi rffiffiffiffi pffiffiffiffi inf C t X t n2 Z2 D 2 CAav ðtÞ ¼ Ai ð4ð1Þn þ e 4Dt p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 2nð1Þ D 2 Dt
Z
D
e
CAi e
(B-5)
þ1
pffiffi S
Since Equation (2) is used for a one-dimensional diffusion cell, the average gas concentration can be obtained by the following integration:
The series form of the Equation (A-10) is calculated as:
1þ2
(B-4)
pffiffiS CAi e DZ
B S ¼
x; S dx
Z
pffiffiffiffi CAi D
(B-3)
And by substitution Equations (B-3) and (B-4) into Equation (B1), the coefficients of AðsÞ and BðsÞ are calculated according to the following expressions:
A S ¼
dx
CAav ðZ; SÞ ¼
CAi S
AðSÞ ¼ BðSÞ
Since Equation (2) is used for a one-dimensional diffusion cell, the average gas concentration can be obtained by the following integration:
Z
(B-2)
Substitution Equation (B-2) into Equation (B-1), gives:
Now, substituting AðsÞ and BðsÞ from Equations (A-6) and (A-7) into Equation (A-2) yields:
pffiffiS CAi e DZ
423
(B-1)
The boundary conditions should be transformed into the Laplace form to be used in Equation (B-1) the following:
1 pffiffi C B 2 DS Z C pffiffiffiffi B B e 1 C C CAi D B C B CAav ðZ; SÞ ¼ 3 C B 2 ZS B pffiffiS C B 2 DZ C A @ e þ1 0
(B-9)
The series form of Equation (B-9) is calculated as:
CAav ðZ; SÞ ¼
pffiffiffiffi CAi D 3
ZS2
1þ2
inf X
e
2n
pffiffiS
Z D
! ð1Þ
n
(B-10)
n¼1
And finally by calculating the Inverse Laplace Transforms (Chen
424
S.A. Shafiee Najafi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 417e424
et al., 2001) of Equation (B-10), the mathematical model for the finite boundary condition is obtained as:
CAav ðtÞ ¼
rffiffiffiffi rffiffiffiffi pffiffiffiffi inf CAi D t X t n2 Z2 2 ð4ð1Þn þ e Dt p n¼1 p Z ! nZ ðnþ1Þ Z pffiffiffiffi erfc pffiffiffiffiffiffi þ 4nð1Þ D Dt
(B-11)
References Chen, Y., Petras, I., Vinagre, B., 2001. A List of Laplace and Inverse Laplace Transforms Related to Fractional Order Calculus. Dorao, C.A., 2012. Dispersion phenomena in gas liquid systems. J. Nat. Gas Sci. Eng. 5, 25e30. Etminan, S.R., et al., 2010. Constant-pressure technique for gas diffusivity and solubility measurements in heavy oil and bitumen. Energy & Fuels 24 (1), 533e549.
Jamialahmadi, M., Emadi, M., Müller-Steinhagen, H., 2006. Diffusion coefficients of methane in liquid hydrocarbons at high pressure and temperature. J. Pet. Sci. Eng. 53 (1), 47e60. Jamialahmadi, M., Emadi, M., Mueller-Steinhagen, H., 2008. Prediction of diffusion coefficients of hydrocarbon gases in hydrocarbon liquids at high pressure and temperature. Oil Gas-Eur. Mag. 34 (2), 69e77. Policarpo, N., Ribeiro, P., 2011. Experimental measurement of gas-liquid diffusivity. Braz. J. Pet. Gas 5 (3), 171e188. Renner, T., 1988. Measurement and correlation of diffusion coefficients for CO2 and rich-gas applications. SPE Reserv. Eng. 3 (2), 517e523. Riazi, M.R., 1996. A new method for experimental measurement of diffusion coefficients in reservoir fluids. J. Pet. Sci. Eng. 14 (3), 235e250. Sheikha, H., Pooladi-Darvish, M., Mehrotra, A., 2005. Graphical methods for obtaining the diffusion coefficient of gases in bitumen. In: Canadian International Petroleum Conference. Sheikha, H., Mehrotra, A.K., Pooladi-Darvish, M., 2006. An inverse solution methodology for estimating the diffusion coefficient of gases in Athabasca bitumen from pressure-decay data. J. Pet. Sci. Eng. 53 (3), 189e202. Sidiq, H., Amin, R., 2009. Mathematical model for calculating the dispersion coefficient of super critical CO2 from the results of laboratory experiments on enhanced gas recovery. J. Nat. Gas Sci. Eng. 1 (6), 177e182. Treybal, R.E., 1980. Mass-transfer Operations, vol. 3. McGraw-Hill, New York. Zhang, Y., Hyndman, C., Maini, B., 2000. Measurement of gas diffusivity in heavy oils. J. Pet. Sci. Eng. 25 (1), 37e47.