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An accurate explicit form of the Hankinson-Thomas-Phillips correlation for prediction ofthe natural gas compressibility factor
Hooman Fatoorehchi, Hossein Abolghasemi,Randolph Rach
PII: S0920-4105(14)00064-3DOI: http://dx.doi.org/10.1016/j.petrol.2014.03.004Reference: PETROL2618
To appear in: Journal of Petroleum Science and Engineering
Received date: 6 December 2013Accepted date: 12 March 2014
Cite this article as: Hooman Fatoorehchi, Hossein Abolghasemi, RandolphRach, An accurate explicit form of the Hankinson-Thomas-Phillips correlationfor prediction of the natural gas compressibility factor, Journal of PetroleumScience and Engineering, http://dx.doi.org/10.1016/j.petrol.2014.03.004
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An accurate explicit form of the Hankinson-Thomas-Phillips correlation for prediction of the natural gas compressibility factor
Hooman Fatoorehchi1, Hossein Abolghasemi1,2,*, Randolph Rach3
1- Center for Separation Processes Modeling and Nano-Computations, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran.
2- Oil and Gas Center of Excellence, University of Tehran, Tehran, Iran.
3- 316 South Maple Street, Hartford, MI 49057-1225, USA.
Email addresses: [email protected]; [email protected] (H.F.), [email protected]; [email protected] (H.A.), [email protected] (R.R.)
*Corresponding Author. Tel: +9821-66954048, fax: +9821-66498984.
Abstract- The use of the Hankinson-Thomas-Phillips correlation for prediction of the
natural gas compressibility factor is a common practice in natural gas engineering
calculations. However, this equation suffers a serious deficiency from a computational
viewpoint; in that it is not explicit with respect to the z-factor and hence is subject to
time-consuming trial and error procedures. In this paper, we propose an explicit series
expansion equivalent to the Hankinson-Thomas-Phillips equation by the aid of a
powerful mathematical technique known as the Adomian decomposition method.
Furthermore, we have enhanced our formula by a applying nonlinear convergence
accelerator algorithm, namely the Shanks transform. The proposed equation is simple,
easy to use, and is shown to be extremely accurate in reproducing the experimental PVT
data of natural gases. Moreover, in contrast to the previous numerical algorithms such as
the Newton-Raphson algorithm, the explicit nature of our formula obviates the need for
any initial guess as an input for calculation of the z-factor. Such independence permits
our formula to always quickly converge to the correct z-factor.
Keywords: Hankinson-Thomas-Phillips correlation; Adomian decomposition method;
Adomian polynomials; Shanks transformation; natural gas compressibility factor.
1. Introduction
The mathematical expression relating pressure, temperature and volume for a gas with
molecules of infinitesimal size and devoid of intermolecular forces is known as the ideal
or perfect gas law. Real gases follow the ideal gas law near atmospheric pressures within
an acceptable degree of accuracy, provided that the temperature is sufficiently higher
than their condensation temperatures. In order to generalize the ideal gas law to real
gases, a variable correction factor known as the compressibility factor, or the z-factor, has
been adopted. For a gas mixture, the compressibility factor is a function of temperature,
pressure and the composition of the mixture. In regard to practical natural gas production
and transportation, which entails pressures much higher than the atmospheric pressure, up
to several thousand psi at wellheads and typically 200 to 1200 psi in transmission
pipelines, the use of the ideal gas law in natural gas engineering practice is severely
limited. Consequently, the compressibility factor is a key parameter required for many
natural gas engineering calculations. A number of these calculations include gas
metering, gas compression, design of processing units, design of pipeline systems, etc.
The gas compressibility factor can be determined thorough experimental PVT data in
laboratories, equations of state, or empirical correlations. The experimental measurement
of the natural gas compressibility factor is the most accurate among all the methods;
however, it is very costly and time-consuming. Therefore, the use of the latter two
approaches has become increasingly preferred (Elsharkawy, 2004; Mokhatab and Poe,
2012). Some of the most common and popular empirical, and semi-empirical,
correlations for estimation of the natural gas z-factor include the Papay equation, the
Hall-Yarborough equation, Brill and Beggs’ z-factor correlation, the Dranchuk-Purvis-
Robinson correlation, and the Shell Oil Company correlation (Ahmed, 1989; Al-Anazi et
al., 2011; Guo et al., 2007). Other recent methods for calculation of the natural gas
compressibility factor are due to Ghedan et al. (1993), Nasrifar and Bolland (2006),
Kamyab et al. (2010), Heidaryan et al. (2010), and Shokir et al. (2012).
Hankinson, Thomas and Phillips presented a correlation for the compressibility factor of
natural gases as a function of pseudo-reduced properties, which is theoretically based on
the Benedict-Webb-Rubin (BWR) equation of state (Hankinson et al., 1969; Thomas et
al., 1970). The BWR equation of state is, in turn, an efficient modification of the Beattie-
Bridgeman equation of state featuring eight experimentally determined parameters
(Benedict et al., 1951). In terms of the compressibility factor, the HTP equation is
� �2
64 2 3 12 2 2 3 3
5 2 21 5 7 8 8
6 6 2 2 2 2
1 1
1 exp 0.
pr prpr pr
pr pr pr
pr pr pr
pr pr pr
p pAA T A A T Az T z T z T
A A A p A p A pz T z T z T
� �� � � �� � � � � �� � � � � �
� � �� � � �
� � � �� � � � � �
(1)
To further improve the accuracy of their equation, Hankinson, Thomas and Phillips
proposed two sets of values for the constants 1A to 8A . The first set corresponds to prp
from 0.4 to 5.0 and the other one is for prp from 5 to 15; see Table 1. Common practice
for solving Eq. (1) for the compressibility factor involves the Newton-Raphson algorithm
in an iterative manner (Ahmed, 1989). The HTP correlation has gained a good reputation
due to its sufficient degree of accuracy and applicability for high temperatures (Ahmed,
1989; Elsharkawy and Elkamel, 2001; Wu et al., 2011). It is understood that the HTP
correlation is invalid for pseudo-reduced temperatures less than 1.1 (Ahmed, 1989).
Furthermore, according to Elsharkawy et al., the HTP correlation can also be used for
predicting the compressibility factor of natural gas condensates at pseudo-reduced
pressures less than 5 (Elsharkawy et al., 2001).
Our aim in this paper is to develop an explicit equivalent formula derived from the HTP
correlation in the form of a rapidly convergent series by applying a powerful analytical
technique, namely the Adomian decomposition method. As we will show in the sequel,
this explicit variation of the HTP correlation is much easier to use and significantly
reduces the computational effort facilitating comprehensive natural gas transportation or
processing simulations. Besides, as we will demonstrate, the Newton-Raphson iterative
algorithm is not always reliable for solving Eq. (1) for the z-factor.
2. Basic concepts of the Adomian decomposition method
Proposed and developed by Professor George Adomian (1922-1996), the Adomian
decomposition method (ADM) has been successful in accurately solving different types
of nonlinear functional equations, and systems of such equations, including ordinary
differential equations (Adomian, 1998; Wazwaz, 2005), partial differential equations
(Adomian, 1983, 1984, 1991; Wazwaz, 2002a), integral equations (Madbouly et al.,
2001; Wazwaz, 2002b; Ziada, 2013), integro-differential equations (Babolian and Biazar,
2002a; Biazar, 2005; Hashim, 2006), algebraic equations (Adomian and Rach, 1985a,
1985b, 1986), differential-algebraic equations (Hosseini, 2006a, 2006b), differential-
difference equations (Wu et al., 2009; Wang et al., 2011), etc. There is an extensive
mathematical literature on applications of the ADM in the study of problems arising from
the applied sciences and engineering; e.g. see (Adomian, 1990, 1994; Bougoffa et al.,
2011; Geng and Cui, 2011; Fatoorehchi and Abolghasemi, 2011a, 2012a, 2012b, 2013a,
2013b, 2013c, 2014; Kundu and Miyara, 2009; Kundu and Wongwises, 2011; Rach,
2012; Rach and Duan, 2011; Saravanan and Magesh, 2013).
Before proceeding, we present a brief review of the basics of the ADM for the
convenience of the reader.
Consider, without loss of generality, the following functional equation,
� �u N u f� � , (2)
where N is a nonlinear operator on a Banach space E, f is a specified element of E and we
are seeking u E , which satisfies Eq. (2). Assuming that Eq. (2) has a unique solution
for every f E , then the ADM decomposes the solution u as an infinite series
0 iiu u�
��� and the nonlinearity as � � 0 ii
N u A�
��� , where the iA are called the
Adomian polynomials (Adomian, 1994) and are defined as
� �0 10 0
1, , ,!
ik
i i i kik
dA A u u u N ui d
�
��
�
� �
� �� � �
��� . (3)
By selecting the initial solution component as 0u f� , the ADM uses the following
recursion relation to generate components of the solution as
0
1
,, 0.i i
u fu A i�
��� � ��
(4)
The convergence and reliability of the ADM have been ascertained in prior research; e.g.
see (Abbaoui and Cherruault, 1994; Abdelrazec and Pelinovsky, 2011; Babolian and
Biazar, 2002b; Cherruault and Adomian, 1993).
Elsewhere, Fatoorehchi and Abolghasemi (2011b) have developed a new improved
algorithm to rapidly generate the Adomian polynomials of any desired analytic nonlinear
operator. The algorithm primarily relies on string functions and symbolic programming;
see the MATLAB code included in Appendix A.
Other techniques for calculation of the Adomian polynomials are available in the
literature (Biazar et al., 2003; Duan, 2010, 2011; Rach, 1984, 2008).
As the final remark of this section, we want to clarify the difference between the ADM
and a class of seemingly similar methods, namely the weighted residual methods (WRM).
In a WRM, we first approximate the solution of Eq. (2) by a finite set of linearly
independent basis functions � �k x� , such that � �1
nk kk
u c x��
�� . By substituting this trial
solution into Eq. (2), a residual, namely � � 0R u N u f� � � � , is obtained. In order to
find the unknown coefficients kc , we shall force the obtained residual to be zero in the
average sense by setting a weighted integral of the residual equal to zero. In this manner,
a system of n knowns and n unknowns whose solution affords the values for kc , and
hence an approximate solution of Eq. (2), is obtained. Obviously, a different choice for
the weight functions leads to a different variant of the WRM such as the subdomain
method, the collocation method, the least squares method, the Galerkin method, etc. In
view of these explanations, the ADM and a WRM may seem alike in suggesting the
solution in the form of a series of decomposed elements. However, a WRM offers the
solution by minimizing a residual functional while the ADM constructs the solution
components through replacing the nonlinear terms by the Adomian polynomials.
Moreover, the ADM does not impose the solution of a system of algebraic equations, in
contrast to a WRM. The interested reader is recommended to consult the mathematical
literature for more details of various WRMs (Hoffman and Frankel, 2001; Majumdar,
2005).
3. Derivation of an explicit equivalent formula for the HTP correlation
3.1 The standard formula
In order to make Eq. (1) explicit with respect to the z-factor by the ADM, we first need to
convert the equation into its canonical form. Hence, we rewrite Eq. (1) as
� �2
64 2 3 12 2 2 3
5 2 21 5 7 8 8
5 6 2 2 2 2
1
1 exp .
pr prpr pr
pr pr pr
pr pr pr
pr pr pr
p pAz A T A A T AT zT z T
A A A p A p A pz T z T z T
� �� � � �� � � � � �� � � � � �
� � �� � � �
� � �� � � � � �
(5)
We observe that there are four nonlinear terms in Eq. (5). In keeping with the
methodology of the ADM, we can find the solution components of Eq. (5) by the
following recursion:
� � � �
� � � �
0
2 24 2 6 3 1
1 ,1 ,22 4 2 3
5 71 5 7 1 5 7 8
,3 ,46 8
1,
, 0,
pr pr pr pr pri i i
pr pr pr pr pr
pr pri i
pr pr
z
A p A p A p A p A pz
T T T T T
A A A p A A A A pi
T T
�
��
��� � � � �� � � � � � � �� � � � � � � ���� � � � ���
(6)
where � �,1i� , � �,2i� , � �,3i� , and � �,4i� are the Adomian polynomials representing the
nonlinear terms involved in Eq. (6). In other words,
� �,10
1i
i z
�
�
� �� , (7)
� �,2 20
1i
i z
�
�
� �� , (8)
� �
28
,3 5 2 20
1 exp pri
i pr
A pz z T
�
�
� �� � �� �
�� , (9)
� �
28
,4 7 2 20
1 exp pri
i pr
A pz z T
�
�
� �� � �� �
�� . (10)
These Adomian polynomials can be calculated in a straightforward manner via Eq. (3);
however, for the convenience of the reader, we list their first several components in
Appendix B.
Now, according to the principle of the ADM, we can calculate an exact explicit
equivalent formula of the HTP correlation as
0i
iz z
�
�
�� , (11)
where the solution components iz are given by Eq. (6).
One can truncate the summation in Eq. (11) after the first n terms to obtain an
approximate formula with an adjustable degree of accuracy, that is
� �0
n
ii
z z n z�
� ��� . (12)
As we will show in the next section, in practice, it usually suffices to choose n less than
ten in order to achieve a completely acceptable accuracy for most engineering purposes.
As a matter of fact, by considering Eq. (6), the least accurate approximation of the z-
factor by our proposed formula, i.e. � �0z z� � , is unity, which corresponds to the special
case of an ideal gas mixture.
3.2. Improving the proposed formula by the Shanks transform
The Shanks transform, due to Daniel Shanks (1917-1996), is a nonlinear transformation
that can effectively covert a slowly converging sequence to a rapidly converging one
(Shanks, 1955). The Shanks transform � �nSh U of the sequence nU is defined as
� �2
1 1
1 12n n n
nn n n
U U USh UU U U
� �
� �
��
� � .(13)
Further increases in the convergence of the sequence nU can be achieved by successive
applications of the Shanks transform, that is the iterated Shanks transforms, as
� � � �� �2n nSh U Sh Sh U� , � � � �� �� �3
n nSh U Sh Sh Sh U� , etc.
Considering Eq. (13), we notice that the Shanks transform involves only elementary
operations and therefore is computationally preferred. Further discussion about the
Shanks transform is outside the scope of this paper but may be found in (Duan et al.,
2013; Fatoorehchi and Abolghasemi, 2013c; Hanna and Sandall, 1995; Homeier, 1993;
Mikhailov and Silva Freire, 2013; Peng et al., 2002; Vahidi and Jalalvand, 2012; Wimp,
1981).
Now, by letting � � 0
nn ii
U z n z�
� ��� , we can devise a modified formula with a higher
rate of convergence than Eq. (12) as
� � � �� �� �0 , , ,kz Sh z z n� � �� where 2n k� . (14)
In other words, by applying the Shanks transform k times, where k is an integer, to the
sequence consisting of the elements � � � �0 , ,z z n� �� , where each is calculated from Eq.
(6), we can compute the z-factor with a significantly improved accuracy.
4. Illustrative examples
In what immediately follows, the usage of Eq. (12) for calculation of the z-factor of
natural gasses will be illustrated through a number of numerical examples.
Example 1.
Calculate the compressibility factor of the natural gas as described in Table 2.
Solution
Firstly, we need to calculate the pseudo-critical properties of the specified natural gas via
the following empirical correlations (Ahmed, 1989; Guo et al., 2007):
� �2 2 2
678 50 0.5 206.7 440 606.7pc g N CO H Sp y y y�� � � � � � , (13)
� �2 2 2
326 315.7 0.5 240 83.3 133.3pc g N CO H ST y y y�� � � � � � , (14)
where g� denotes the gas specific gravity (air = 1) and iy is the mole fraction of the
species i. Also, the parameters pcp and pcT are in psia and degrees Rankine, respectively.
Therefore, we obtain 691.799 psiapcp � and 375.641 RpcT � � . Consequently,
2.891pr pcp p p� � and 1.756pr pcT T T� � . Next, we select appropriate values for the
coefficients 1A to 8A from Table 1.
By virtue of Eq. (6), we calculate the first several z-factor components as
0 1z � 56 0.82308 10z �� �
1 0.10705z � � 57 0.14982 10z �� �
22 0.52342 10z �� � � 6
8 0.16140 10z �� �
33 0.14963 10z �� � � 7
9 0.15905 10z �� � �
44 0.85836 10z �� �
45 0.33391 10z �� �
Hence, � �9
09 0.88768719i
iz z z
�
� � ��� .
Optionally, we can also use Eq. (14) to find the z-factor. For the required calculations, it
is beneficial to take advantage of tabulation; see Table 3. From the results in this table,
we have computed that � � � �� �� �2 0 , , 4 0.88767z Sh z z� �� �� . Consequently, we can
conclude that in order to achieve a given degree of accuracy, the improved formula, i.e.
Eq. (14), requires the determination of nearly one-half of the number of the z-factor
components via Eq. (6) and thus is more computationally efficient. It is worthwhile to
note that the Newton-Raphson algorithm fails to converge to a solution for Eq. (1) for any
initial guess equal to or greater than 1.798.
Example 2.
Consider a natural gas stream whose properties are listed in Table 2. Calculate the
compressibility factor for this stream via Eq. (12).
Solution
Similarly to Example 1, through Eqs. (13) and (14), we calculate that 7.197prp � and
1.771prT � . Subsequently, we choose the appropriate set of the coefficients 1A to 8A
from Table 1. We proceed with Eq. (6) calculating the first several z-factor components
as
0 1z � 36 0.99756 10z �� �
1 0.14813z � � 37 0.15977 10z �� �
12 0.35732 10z �� � 3
8 0.25924 10z �� � �
23 0.52520 10z �� � 4
9 0.37070 10z �� � �
24 0.46420 10z �� � �
35 0.78356 10z �� � �
Thus, � �9
09 0.88835i
iz z z
�
� � ��� . On the other hand, for this specific example, the
Newton-Raphson algorithm does not converge to a solution of Eq. (1) for any initial
guess equal to or greater than 1.684.
Example 3.
Calculate the compressibility factor of a natural gas at 100 FT � � and 1000 psiap �
characterized by 650 psiapcp � and 427 RpcT � � .
Solution
From the input data it readily follows that 1.538prp � and 1.310prT � . Similarly to the
previous examples, we compute the first several components of the z-factor as
0 1z � 36 0.59780 10z �� � �
1 0.18363z � � 37 0.27705 10z �� � �
12 0.28354 10z �� � � 3
8 0.13228 10z �� � �
23 0.85997 10z �� � � 4
9 0.64569 10z �� � �
24 0.32258 10z �� � �
25 0.13453 10z �� � �
Therefore, � �9
09 0.77376i
iz z z
�
� � ��� .
Similarly to Example 1, we test the efficiency of the Shanks transform in accelerating the
rate of convergence of the sequence of the z-factor components; see Table 3. According
to the results presented in Table 3, it is determined that
� � � �� �� �2 0 , , 4 0.77389z Sh z z� �� �� . In addition, we can easily find that the Newton-
Raphson algorithm fails to provide a convergent solution for this example by any initial
guess equal to or greater than 1.757.
Example 4.
Estimate the compressibility of a natural gas with 7prp � and 3prT � .
Solution
Choosing the appropriate set of the coefficients 1A to 8A from Table 1, we compute the
first several components of the z-factor via Eq. (6) as
0 1z � 36 0.90149 10z �� � �
11 0.92599 10z �� � 3
7 0.50146 10z �� �
12 0.20542 10z �� � � 3
8 0.28809 10z �� � �
23 0.75704 10z �� � 3
9 0.16962 10z �� �
24 0.33985 10z �� � �
25 0.16942 10z �� �
Consequently, we calculate � �9
09 1.0774i
iz z z
�
� � ��� .
We can optionally use Eq. (14) to compute the natural gas compressibility factor as
� � � �� �� �2 0 , , 4 1.07732z Sh z z� �� �� ; see Table 3. Moreover, for this example, the
Newton-Raphson iterative algorithm diverges for any initial guess equal to or greater than
1.891.
Example 5.
Calculate the z-factor for a natural gas stream with 2.6prp � and 1.25prT � .
Solution
Following the same procedure as in the previous examples, it is straightforward to
calculate the first several z-factor components by Eq. (6) as
0 1z � 26 0.32075 10z �� � �
1 0.31892z � � 27 0.10543 10z �� � �
12 0.72508 10z �� � � 3
8 0.70241 10z �� �
13 0.30294 10z �� � � 3
9 0.11170 10z �� �
14 0.14598 10z �� � �
25 0.71407 10z �� � �
Hence, we estimate � �9
09 0.5549i
iz z z
�
� � ��� . Furthermore, it is futile to calculate the z-
factor from Eq. (1) by the Newton-Raphson iterative algorithm while choosing the initial
guess equal to or greater than 1.649.
5. Results and discussion
In order to confirm the validity of our proposed formula, we have compared the estimates
by Eq. (12) against some experimental data and the results calculated by the AGA 8
method of the American Gas Association. To avoid human error, the AGA 8 calculations
were performed by using the FLOWSOLV v4.10.3 software package. As shown by
Figures 1A to 1D, our explicit equivalent formula of the HTP correlation is almost
completely accurate in reproducing the experimental data for a variety of different
temperatures and pressures. We mention that the observed but slight discrepancies, with
absolute relative deviations of less than 2 percent, between the calculated results and the
experimental data in Fig. 1D are due to an intrinsic error in the original HTP correlation
and are not attributable to our proposed formula. This has been partially corroborated by
solving Eq. (1) for the z-factor by using the “fzero” solver in MATLAB. A visual
examination of Figures 1A to 1D indicates that our explicit form of the HTP correlation
fits the experimental data better, at least slightly better, than the AGA 8 procedure. Table
4 proves this conclusion quantitatively based on an error analysis. According to the
results summarized in Table 4, the deviation of the calculated results, both by the AGA 8
method and our scheme, from the experimental results follows a descending trend for
lower pseudo-critical temperatures and a rising trend from 1.7prT � to 3.0prT � . Such a
non-monotonous change can be attributed to the complex behavior of the z-factor for
natural gas mixtures, as indicated by Katz (Katz, 1959). Nevertheless, we can draw a
conservative conclusion that our explicit form of the HTP correlation predicts the z-factor
within an average absolute relative deviation of less than 1% for the pseudo-temperature
range from 1.5 to 3.
Additionally, in order to assess the computational efficiency of the proposed explicit
correlations, i.e. Eqs. (12) and (14), and the Newton-Raphson iterative algorithm, we
have performed a CPU-time analysis using Version 7 of the MATLAB software package,
which is graphically presented in Fig. 2, for our five numerical examples. From Fig. 2,
we deduce that Eq. (14) provides the most efficient formula for calculation of the z-factor
as it benefits from the nonlinear convergence accelerator technique, namely the Shanks
transform. Moreover, for a given degree of accuracy, Eq. (14) yields the z-factor almost
two times faster than Eq. (12). This is because Eq. (14) requires the computation of
almost half of the number of the Adomian polynomials than Eq. (12) requires for a
solution of comparable accuracy. Finally, the Newton-Raphson algorithm has the lowest
computational speed due to its iterative nature requiring an initial guess.
6. Conclusion
An explicit version of the Hankinson-Thomas-Phillips equation for calculation of the
natural gas compressibility factor was developed by applying the Adomian
decomposition method in this paper. For a further improvement of our proposed formula
in terms of the computational speed, we optionally employed a nonlinear convergence
accelerator technique known as the Shanks transform. The proposed explicit formula was
shown to be capable of simulating the experimental PVT data extremely accurately and
much more efficiently as compared to the classic Newton-Raphson algorithm. Our
formula is also found to be robust in converging to the correct z-factor since, unlike most
previous numerical schemes, it does not depend on any arbitrarily chosen input for an
initial guess.
Acknowledgments
The authors would like to express their sincere gratitude to the editor and anonymous
reviewers for their insightful comments.
Appendix A. An alternative MATLAB code for calculation of the Adomian
polynomials
By letting the symbolic variable 0 1 2 nNON u u u u� � � � �� , the following function in
MATLAB returns the Adomian polynomials of a nonlinear operator acting upon NON.
function sol=AdomPoly(expression,nth) Ch=char(expand(expression)); s=strread(Ch, '%s', 'delimiter', '+'); for i=1:length(s) t=strread(char(s(i)), '%s', 'delimiter', '*()expUlogsinh'); t=strrep(t,'^','*'); if length(t)~=2 p=str2num(char(t)); sumindex=sum(p)-p(1); else sumindex=str2num(char(t)); end list(i)=sumindex; endA=''; for j=1:length(list) if nth==list(j) A=strcat(A,s(j),'+'); end end N=length(char(A))-1;F=strcat ('%',num2str(N),'c%n'); sol=sscanf(char(A),F);
For example, the following code in the command window of MATLAB illustrates the
usage of the preceding AdomPoly function.
>> syms U0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 NON >> NON=U0+U1+U2+U3+U4+U5+U6+U7+U8+U9+U10; >> Adompoly(NON^3,5)
Appendix B. First six components of the Adomian polynomials as used in Eq. (6)
Nonlinearity: � � 1N zz
�
� �0,10
1z
� �
� �1
1,1 20
zz
� � �
� �
21 2
2,1 3 20 0
z zz z
� � �
� �
331 1 2
3,1 4 3 20 0 0
2 zz z zz z z
� � � � �
� �
4 2 21 31 1 2 2 4
4,1 5 4 3 3 20 0 0 0 0
3 2 z zz z z z zz z z z z
� � � � � �
� �
25 3 21 3 2 3 51 1 2 1 2 1 4
5,1 6 5 4 4 3 3 20 0 0 0 0 0 0
4 3 3 2 2z z z z zz z z z z z zz z z z z z z
� � � � � � � � �
Nonlinearity: � � 2
1N zz
�
� �0,2 20
1z
� �
� �1
1,2 30
2 zz
� � �
� �
21 2
2,2 4 30 0
3 2z zz z
� � �
� �
331 1 2
3,2 5 4 30 0 0
4 6 2 zz z zz z z
� � � � �
� �
4 2 21 31 1 2 2 4
4,2 6 5 4 4 30 0 0 0 0
5 12 3 6 2z zz z z z zz z z z z
� � � � � �
� �
25 3 21 3 2 3 51 1 2 1 2 1 4
5,2 7 6 5 5 4 4 30 0 0 0 0 0 0
6 20 12 12 6 6 2z z z z zz z z z z z zz z z z z z z
� � � � � � � � �
Nonlinearity: � �2
85 2 2
1 exp pr
pr
A pN z
z z T� �
� �� � �
� �
28
0,3 5 2 20 0
1 exp pr
pr
A pz z T
� �� � �� �
�
� �
2 2 28 8 1 81
1,3 6 2 2 8 2 2 20 0 0 0
5 exp 2 exppr pr pr
pr pr pr
A p A p z A pzz z T z T z T
� � � �� � � � � �� � � �
� �
� �
2 2 2 2 4 2 228 1 8 2 8 1 81 2
2,3 7 9 2 6 8 2 11 4 2 20 0 0 0 0 0
15 13 5 2 2 exppr pr pr pr
pr pr pr pr
A p z A p z A p z A pz zz z T z z T z T z T
� � � �� � � � � � �� � � �
� �
� �
2 3 2 2 4 338 1 8 1 2 8 1 31 1 2
8 10 2 7 9 2 12 4 6 20 0 0 0 0 0 8
3,3 2 22 2 4 3 6 308 3 8 1 2 8 1
8 2 11 4 14 60 0 0
35 49 30 26 16 5
exp42 43
pr pr pr
pr pr pr pr
prpr pr pr
pr pr pr
A p z A p z z A p z zz z zz z T z z T z T z A p
z TA p z A p z z A p zz T z T z T
� �� � � � � ��
� �� � � �� � � �� � � ��
�
� �
2 4 2 2 2 4 42 48 1 8 1 2 8 11 2 1
11 2 10 2 8 9 13 40 0 0 0 0
2 2 2 2 4 2 3 6 48 2 8 1 3 8 1 2 8 1
9 2 9 2 12 4 15 60 0 0 0
4,3 2 28 4 8
8 20
145140 147 105 702
3813 26 483
2 4
pr pr pr
pr pr pr
pr pr pr pr
pr pr pr pr
pr
pr
A p z A p z z A p zz z zz T z T z z z T
A p z A p z z A p z z A p zz T z T z T z T
A p z A pz T
� � � � �
� � � �
� �
� �
282 24 2 4 2 201 3 8 2 1 32 4
11 4 11 4 7 7 60 0 0 0 0
3 6 2 4 8 48 1 2 8 1
14 6 17 80 0
exp
2 15 30 5
243
pr
prpr pr
pr pr
pr pr
pr pr
A pz Tz z A p z z zz z
z T z T z z z
A p z z A p zz T z T
� �� � � � � � �� �� � � �� � � �� � � � � �� �
� �
2 4 5 2 5 2 358 1 8 1 8 1 251
10 14 4 6 12 2 11 20 0 0 0 0
2 2 2 2 2 4 3 3 6 58 1 2 8 1 3 8 1 2 8 1
10 2 10 2 13 4 16 60 0 0 0
28 2 3
90
5,3
489126 5 336 5602
147 147 290 67
26
pr pr pr
pr pr pr
pr pr pr pr
pr pr pr pr
pr
pr
A p z A p z A p z zzzz z T z z T z T
A p z z A p z z A p z z A p zz T z T z T z T
A p z zz T
� � � � �
� � � �
�
� �
2 4 2 2 2 4 28 2 1 8 1 4 8 1 3
2 12 4 9 2 12 40 0 0
3 6 3 4 8 5 2 2 48 1 2 8 1 8 5 8 1 4
15 6 18 8 8 2 11 40 0 0 0
2 4 3 6 2 38 2 3 8 1 3 8
11 4 14 60 0
48 26 48
152 22 2 43 3
4 4 4
pr pr pr
pr pr pr
pr pr pr pr
pr pr pr pr
pr pr
pr pr
A p z z A p z z A p z zz T z T z T
A p z z A p z A p z A p z zz T z T z T z T
A p z z A p z z Az T z T
� � �
� � � �
� � �
282 20
6 2 4 8 31 2 8 1 2
14 6 17 80 0
5 10 5 23 28 1 1 3 2 31 2 1 2
20 10 9 8 8 70 0 0 0 0
1 470
exp
83
4 280 105 105 3015
30
pr
pr
pr pr
pr pr
pr
pr
A pz T
p z z A p z zz T z T
A p z z z z zz z z zz T z z z z
z zz
� �� � � � � � � � � � � �� �� � � �� �
�� � � � � � � � �� � � �� �
Nonlinearity: � �2
87 2 2
1 exp pr
pr
A pN z
z z T� �
� �� � �
� �
28
0,4 7 2 20 0
1 exp pr
pr
A pz z T
� �� � �� �
�
� �
2 2 28 8 1 81
1,4 8 2 2 10 2 2 20 0 0 0
7 exp 2 exppr pr pr
pr pr pr
A p A p z A pzz z T z T z T
� � � �� � � � � �� � � �
� �
� �
2 2 2 2 4 2 228 1 8 2 8 1 81 2
2,4 9 11 2 8 10 2 13 4 2 20 0 0 0 0 0
28 17 7 2 2 exppr pr pr pr
pr pr pr pr
A p z A p z A p z A pz zz z T z z T z T z T
� � � �� � � � � � �� � � �
� �
� �
2 3 2 2 4 338 1 8 1 2 8 1 31 1 2
10 12 2 9 11 2 14 4 8 20 0 0 0 0 0 8
3,4 2 22 2 4 3 6 308 3 8 1 2 8 1
10 2 13 4 16 60 0 0
84 81 56 34 20 7
exp42 43
pr pr pr
pr pr pr pr
prpr pr pr
pr pr pr
A p z A p z z A p z zz z zz z T z z T z T z A p
z TA p z A p z z A p zz T z T z T
� �� � � � � ��
� �� � � �� � � �� � � ��
�
� �
2 42 4 28 11 3 2 1 1 2
9 9 11 10 13 20 0 0 0 0
2 2 2 4 4 2 28 1 2 8 1 8 24
12 2 15 4 8 11 20 0 0 0
4,4 2 2 4 2 3 6 48 1 3 8 1 2 8 1
11 2 14 40 0
56 28 210 252 258
221243 7 172
4634 603
pr
pr
pr pr pr
pr pr pr
pr pr pr
pr pr
A p zz z z z z zz z z z z T
A p z z A p z A p zzz T z T z z T
A p z z A p z z A p zz T z T z
� � � �
� � � �
� �
� � �
282 2208 4
17 6 10 20 0
2 4 2 4 2 3 6 2 4 8 48 1 3 8 2 8 1 2 8 1
13 4 13 4 16 6 19 80 0 0 0
exp
2
24 2 43
pr
prpr
pr pr
pr pr pr pr
pr pr pr pr
A pz TA p z
T z T
A p z z A p z A p z z A p zz T z T z T z T
� �� � � � � � �� �� � � ��� � � � � � � �� �
� �
25 35 1 3 2 31 1 28 12 11 10 90 0 0 0 02 4 3 6 2 3 6 2 4 8 38 2 3 8 1 3 8 1 2 8 1 2
13 4 16 6 16 6 19 80 0 0 0
5 10 5 2 28 1 8 1 3 8
22 10 12 20 0
5,4
7 462 840 252 56
84 4 43
4 243 24315
pr pr pr pr
pr pr pr pr
pr pr
pr pr
z z z z zz z zz z z z zA p z z A p z z A p z z A p z z
z T z T z T z T
A p z A p z z Az T z T
� � � � �
� � � �
� � �
� �
2 2 2 51 2 8 1
12 2 14 20 0
2 3 2 4 5 2 4 3 3 6 58 1 2 8 1 8 1 2 8 1
13 2 16 4 15 4 18 60 0 0 0
2 2 2 4 28 1 4 8 2 3 8 1 2
11 2 11 2 14 40 0 0
825
891 2891140 4422 3
34 34 60 60
pr pr
pr pr
pr pr pr pr
pr pr pr pr
pr pr pr
pr pr pr
p z z A p zz T z T
A p z z A p z A p z z A p zz T z T z T z T
A p z z A p z z A p z zz T z T z T
�
� � � �
� � � �
282 20
2 4 28 1 3
14 40
3 6 3 4 8 5 2 2 48 1 2 8 1 8 5 8 1 4 1 4
17 6 20 8 10 2 13 4 90 0 0 0 0
21 2100
exp
184 26 2 4 563 3
252
pr
pr
pr
pr
pr pr pr pr
pr pr pr pr
A pz T
A p z zz T
A p z z A p z A p z A p z z z zz T z T z T z T z
z zz
� �� � � � � � � � � � � �� ���� �� � � � � � � � � ��
� � � �� �
References
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Captions for Figures
Figure 1) Comparison of the experimental data with the calculated results from our
equation and the AGA 8 method at a) 1.3prT � , b) 1.5prT � , c) 1.7prT � and d)
3.0prT � . Our calculations are based on the tenth-stage approximation of the z-factor and
the experimental data is due to Standing and Katz (Katz, 1959).
Figure 2) An efficiency comparison of our formula based on the Adomian algorithm, our
formula based on the combined Adomian-Shanks algorithm and the approximation by the
Newton-Raphson algorithm as derived from the CPU-time analysis of the five examples
presented in Section 4.
Captions for Tables
Table 1) Numerical values for the coefficients of Eq. (1).
Table 2) Characteristics of the natural gas used in Examples 1 and 2.
Table 3) Table 3) Application of the Shanks transform in pursuit of the compressibility
factor for Examples 1, 3 and 4.
Table 4) Average absolute relative deviations (percent) of the calculated compressibility
factors from experimental data for the four isotherms depicted in Figs. 1A to 1D.
Table 1) Numerical values for the coefficients of Eq. (1). Coefficient 0.4 5.0prp� � 5 15prp� �
1A 0.001290236 0.0014507882
2A 0.38193005 0.37922269
3A 0.022199287 0.024181399
4A 0.12215481 0.11812287
5A 0.015674794� 0.037905663
6A 0.027271364 0.19845016
7A 0.023834219 0.048911693
8A 0.43617780 0.0631425417
Table 2) Characteristics of the natural gas used in Examples 1 and 2. Example 1 Example 2
Temperature 200 F� � Temperature 180 F� �
Pressure 2000 psia� Pressure 5000 psia�Gas specific gravity 0.7� Gas specific gravity 0.65�
20.05Ny �
20.1Ny �
20.05COy �
20.08COy �
20.02H Sy �
20.02H Sy �
Table 3) Application of the Shanks transform in pursuit of the compressibility factor for Examples 1, 3 and 4.
Example 1 n � �nU z n� � � �nSh U � �2
nSh U0 1 � �1 0.89294 0.88743 �2 0.88770 0.88755 0.887673 0.88755 0.88761 �4 0.88764 � �
Example 3 n � �nU z n� � � �nSh U � �2
nSh U0 1 � �1 0.81636 0.78283 �2 0.78800 0.77566 0.773893 0.77941 0.77424 �4 0.77618 � �
Example 4 n � �nU z n� � � �nSh U � �2
nSh U0 1 � �1 1.09259 1.07578 �2 1.07205 1.07758 1.077323 1.07962 1.07728 �4 1.07622 � �
Table 4) Average absolute relative deviations (percent)* of the calculated compressibility factors from experimental data for the four isotherms depicted in Figs. 1A to 1D.
1.3prT � 1.5prT � 1.7prT � 3.0prT �
The AGA 8 Method 3.8582 1.8391 0.7302 1.7591 Our Equation 1.1807 0.67222 0.4177 0.7399
* � � � �� �1
z experimental z calculated 100%z experimental
n i ii
i
AARDn �
�� �
Highlights An explicit form of the H-T-P equation of state based on the Adomian
decomposition.
Extremely accurate in reproducing the natural gas experimental PVT data.
Almost a twofold increase in the computational speed when combined with the
Shanks transform.
0.50.55
0.60.65
0.70.75
0.80.85
0.90.95
1
0 1 2 3 4 5 6 7ppr
zExperimental
Our Equation
AGA 8
0.74
0.79
0.84
0.89
0.94
0.99
0 1 2 3 4 5 6 7 8ppr
z
Experimental
Our Equation
AGA 8
A B
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
0 1 2 3 4 5 6 7 8ppr
z
Experimental
Our Equation
AGA 8
0.99
1.01
1.03
1.05
1.07
1.09
1.11
1.13
0 1 2 3 4 5 6 7 8ppr
z
Experimental
Our Equation
AGA 8
C D
Fig. 1) Comparison of the experimental data with the calculated results from our equation and the AGA 8 method at a) 1.3prT � , b) 1.5prT � , c) 1.7prT � and d) 3.0prT � . Our calculations are based on the tenth-stage approximation of the z-factor and the experimental data is due to Standing and Katz (Katz, 1959).
Figure
0
1
2
3
4
5
6
7
1 2 3 4 5
Example No.
CPU-
time
x 10
[s]
Eq. (12)Eq. (14)N-R Algorithm
Fig. 2) An efficiency comparison of our formula based on the Adomian algorithm, our formula based on the combined Adomian-Shanks algorithm and the approximation by the Newton-Raphson algorithm as derived from the CPU-time analysis of the five examples presented in Section 4.
Figure
