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le Article AN IMPROVED ALGORITHM FOR CALCULATION OF THE NATURAL GAS COMPRESSIBILITY FACTOR VIA THE HALL-YARBOROUGH EQUATION OF STATE† Hooman Fatoorehchi,1 Hossein Abolghasemi,1,2* Randolph Rach3 and Moein Assar1 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

*Author to whom correspondence may be addressed. Email addresses: abolghasemi.ha@gmail.com; hoab@ut.ac.ir

†This article has been accepted for publication and undergone full peer review but

has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: [10.1002/cjce.22054] Received 23 December 2013; Revised 27 January 2014; Accepted 7 February 2014

The Canadian Journal of Chemical Engineering © 2014 Canadian Society for Chemical Engineering

DOI 10.1002/cjce.22054

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le The Hall-Yarborough equation (H-Y equation) of state has been favoured in natural gas

engineering due to its accuracy and conciseness for many years. In this paper, the

Adomian decomposition method (ADM) is employed to devise a novel algorithm for

calculating the compressibility factors of natural gases through this reliable equation of

state. A convergence accelerator technique, namely the efficient Shanks transform, is also

exploited to further improve our scheme in terms of computational speed. Unlike most of

the previous numerical solution strategies, our algorithm does not require an initial guess

as the starting point and is computationally efficient. The proposed algorithm is found to

be superior over the common Newton-Raphson algorithm, where we have also

demonstrated that the latter can easily lead to grossly erroneous solutions. For the sake of

illustration, a number of real-world case study problems are solved by our algorithm and

relevant comparisons are provided.

Keywords: Hall-Yarborough equation, Adomian decomposition method, Adomian

polynomials, Shanks transform, gas compressibility factor

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le INTRODUCTION

The compressibility factor is a key parameter required for most natural gas engineering

calculations. A number of these calculations include gas metering, gas compression,

design of processing units, and design of pipeline systems. The gas compressibility factor

can be determined thorough experimental 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 two latter approaches has become increasingly

preferred.[1-5] Some of the most common and popular empirical, and semi-empirical,

correlations for estimation of the natural gas z-factor are the Papay equation (1968), the

Hankinson-Thomas-Phillips correlation (1969), Brill and Beggs’ z-factor correlation

(1974), the Dranchuk-Purvis-Robinson correlation (1974), the Dranchuk and Abu-

Kassem correlation (1975), and the Shell Oil Company correlation (2003).[6-8]

Hall and Yarborough proposed an accurate correlation to estimate the z-factor, the gas

compressibility factor, of natural gas in 1973.[9,10] Since then, their correlation has gained

a dependable reputation owing to its simplicity and ability to fit experimental data with

sufficient accuracy.[11-18] In fact, the H-Y equation is based on the Starling-Carnahan

equation of state but with the simplifying assumption of inelastic spheres rather than real

molecules.[19,20] More recently, in 2007, Hall and Iglesias-Silva have presented a

modification of the original H-Y correlation to represent the z-factor data of the Standing-

Katz chart within a remarkable absolute average percentage deviation of only 0.24.[21]

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le By taking rt as the reciprocal of the pseudo-reduced temperature, i.e., 1r prt T , the H-Y

equation includes the following parameters:

21.2 10.06125 rtrA t e , (1a)

214.76 9.76 4.58r r rB t t t , (1b)

290.7 242.2 42.4r r rC t t t , (1c)

2.18 2.82 rD t , (1d)

from which the gas compressibility factor is obtained as

prApz

Y , (2)

where Y is the reduced density, which is modelled by

2 3 42

3 01

Dpr

Y Y Y Yf Y Ap BY CY

Y

. (3)

In view of the preceding equations, we observe that Equation (3) is strongly nonlinear

and even contains terms with a fractional exponent. In order to calculate Y from Equation

(3), the Newton-Raphson iterative algorithm has been generally used.[7]

In this paper, an efficient algorithm based on the Adomian decomposition method

(ADM) is developed to calculate the reduced density Y from Equation (3) and then to

obtain the gas compressibility factor z from Equation (2). We shall demonstrate in

subsequent sections that the proposed algorithm is decidedly superior to the Newton-

Raphson iterative algorithm in terms of convergence and furthermore does not require an

initial guess as input. A number of illustrative case-study problems are included in the

final section.

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le BASICS OF THE ADOMIAN DECOMPOSITION METHOD

The ADM is due to the Armenian-American mathematician Professor George Adomian

(1922-1996).[22,23] It can provide exact analytical solutions to a wide class of linear or

nonlinear functional equations, and systems of such equations, such as ordinary

differential equations,[24,25] partial differential equations,[26,27] integral equations,[28-30]

integro-differential equations,[31-33] algebraic equations,[34-36] differential-algebraic,[37,38]

differential-difference equations,[39] etc. The mathematical literature on applications of

the ADM in the study of problems arising from the applied sciences and engineering is

very extensive.[40-54]

In this section, we present a brief review of the basics of the ADM for the convenience of

the reader.

To illustrate the methodology of the ADM, without loss of generality, consider the

following functional equation,

u N u f , (4)

where N is a nonlinear operator which maps a Banach space E into itself, f is a bounded

specified function and u designates an unknown function. The ADM decomposes the

solution u as an infinite summation 0 ii

u u

and the nonlinearity as

0 iiN u A

,

where the iA are called the Adomian polynomials:[41]

0 10 0

1, , ,

!

ik

i i i kik

dA A u u u N u

i d

. (5)

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le By choosing 0u f , the ADM uses the following recursion relation to generate

components of the solution as

0

1

,

, 0.i i

u f

u A i

(6)

The convergence and reliability of the ADM have been ascertained in prior research.[55-58]

Elsewhere,[59] Fatoorehchi and Abolghasemi have developed a new improved algorithm

to rapidly generate the Adomian polynomials of any desired analytic nonlinear operator.

Their 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.[60-64]

THE PROPOSED ALGORITHM

Application of the ADM to Equation (3)

In keeping with the principles of the ADM, Equation (3) can be transformed to its

canonical equivalent form as

5 4 3 2

3 2 1

3 1 3 13 1

3 1 3 1 3 1 3 1

3 3

3 1 3 1 3 1 3 1 3 1

pr pr

pr pr pr pr

prD D D D

pr pr pr pr pr

Ap B Ap BB BY Y Y Y Y

Ap Ap Ap Ap

ApC C C CY Y Y Y

Ap Ap Ap Ap Ap

(7)

The ADM suggests the exact solution to this nonlinear equation as 0

,iiY Y

where

the approximate solution is 0

n

iiY n Y

. The solution components are computed as

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le

0

1 ,1 ,2 ,3 ,4

,5 ,6 ,7 ,8

,3 1

3 1 3 13 1

3 1 3 1 3 1 3 1

3 3, 0,

3 1 3 1 3 1 3 1

pr

pr

pr pri i i i i

pr pr pr pr

i i i ipr pr pr pr

ApY

Ap

Ap B Ap BB BY

Ap Ap Ap Ap

C C C Ci

Ap Ap Ap Ap

(8)

where the ,1 ,8, ,i i are the Adomian polynomials representing the nonlinear terms

5 4 3 2 3 2 1, , , , , , , andD D D DY Y Y Y Y Y Y Y , respectively. The first four of these Adomian

polynomials do not depend on the gas conditions and hence we list a few of their

components in Appendix B for convenient reference.

Exploiting the Shanks Transform

The Shanks transform, which was initially proposed by Daniel Shanks (1917-1996), is a

nonlinear transform that can effectively covert a slowly converging sequence to a rapidly

converging one.[65] The Shanks transformation nSh U of a sequence nU is defined as

2

1 1

1 12n n n

nn n n

U U USh U

U U U

. (9)

Further speed-up may be achieved by successive implementation of the Shanks

transformation, that is the iterated Shanks transforms 2n nSh U Sh Sh U ,

3n nSh U Sh Sh Sh U , etc.

Now, by assigning the partial sum 0

n

n iiU Y n Y

, where the iY are computed by

the recurrence relation (8), we can combine the ADM with the Shanks transform to

further increase the rate of convergence. Considering Equation (9), we notice that the

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le Shanks transformation involves only elementary operations and therefore is

computationally preferred.

Finally, we would like to point to one subtle difference between the Shanks

transformation method and an older series acceleration method known as Aitken’s delta-

squared process,[66] which is due to Alexander Aitken (1895-1967). Fundamentally

similar to each other, the latter operates on a sequence iu while the Shanks transform

operates on the new sequence nU , where 0

n

n jjU u

. In other words, the Shanks

transform is applied to partial summation sequences such as nU . In fact, Shanks was

the mathematician who revived, and also generalized, the classic Aitken method and also

demonstrated that the generalized Shanks transforms are closely related to the Padé

approximants.[67] Further discussion about the Shanks transform is outside the scope of

this paper and can be found in the literature.[68-72]

CASE STUDIES

In this section, we include three numerical examples to illustrate the use of our algorithm.

The input data for these examples is listed in Table 1. Before we proceed, we present the

following correlations for calculation of pseudo-critical quantities when impurities such

as nitrogen, carbon dioxide and hydrogen sulfide exist in the gas mixture:[6,7]

2 2 2

678 50 0.5 206.7 440 606.7pc g N CO H Sp y y y , (10)

2 2 2

326 315.7 0.5 240 83.3 133.3pc g N CO H ST y y y , (11)

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le 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.

Example 1.

Estimate the compressibility factor of the natural gas described in Table 1.

Solution

By Equations (10) and (11), the pseudo-critical properties of the gas mixture are easily

found as 4.7697 MPapcp and 208.6 KpcT . Consequently, 0.617678r pct T T

and 2.891013pr pcp p p . Now, from Equations (1a) to (1d), we obtain the system

parameters as

0.031746A ,

6.472554B ,

26.3902C ,

3.921851D .

By virtue of Equation (8), we can easily compute the solution components as

10 0.7196395406 10Y 2

6 0.1012421255 10Y

11 0.1831175144 10Y 3

7 0.8348791247 10Y

22 0.8239288401 10Y 3

8 0.7059124578 10Y

23 0.4385362084 10Y 3

9 0.5131572561 10Y

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le 24 0.2319919830 10Y 3

10 0.4024897203 10Y

25 0.1851988821 10Y

Therefore, we can approximate 10

010 0.1096431955ii

Y Y Y

. Thus, according to

Equation (2), the compressibility factor of the mentioned natural gas is approximately

0.8371z .

Optionally, we can apply the Shanks transform to the sequence generated by

0

n

n iiU Y n Y

to compute the z-factor faster. The relevant results are listed in

Table 2. From the results in Table 2, we recognize that we can achieve almost the same

estimate for ,Y while only using the first five decomposition solution components, by

successive applications of the Shanks transform.

Examples 2 and 3.

To avoid redundancy, we will not repeat the solution procedure, as described in Example

1, but will instead summarize the results for these two numerical examples, along with

those of Example 1, in Table 3.

RESULTS AND DISCUSSION

Based on the previous examples, a CPU-time analysis was carried out for the ADM, the

ADM combined with the Shanks transform and the Newton-Raphson (N-R) algorithm in

order to compare their efficiencies. All of the computations were performed by using a

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le custom code in the MATLAB 7 software package on a personal computer with a 2.66

GHz processor with 2 GB of RAM until a convergence tolerance of 1010 was achieved.

The outcome of this analysis is depicted graphically in Figure 1. According to Figure 1,

we demonstrate that the combined ADM-Shanks transform method is the most

economical in terms of computational effort for all three numerical examples. This

conclusion was not far from our expectation as it was shown in the previous section that

the combined scheme, compared to the classic ADM, requires almost half of the number

of decomposition components for a solution of equal accuracy. For all of the case study

examples, the solution obtained by the ADM consumes less CPU time than the N-R

algorithm and this is due to the considerably faster convergence inherent in the ADM.

Moreover, one serious drawback of the N-R algorithm in the treatment of Equation (3) is

its requirement of a guess for the initial approximation. Despite any other potential

advantages, the N-R algorithm is highly dependent on such a first guess and, in other

words, different initial guesses may lead to different zeroes of Equation (3).

Unfortunately, there is no systematic way to help us select an appropriate initial guess for

the N-R algorithm and therefore the choice of a successful initial guess can only be left

open to trial and error. This serious disadvantage is better illustrated by the data presented

in Table 4. As it is well known, the high sensitivity of the N-R algorithm to the value of

the initial guess can become quite problematic and cause grossly erroneous estimations of

Y and consequently the z-factor.

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le In order to check the validity of the estimates of our algorithm, we have modelled three

sets of experimental data, which are due to Standing and Katz.[73] In addition, we have

compared the results obtained from our proposed scheme with those provided by the

Newton-Raphson algorithm. Figure 2a-d depicts the results for these comparisons. As it

can be observed, our algorithm is able to predict the values for the z-factor better than, or

at least as good as, the N-R algorithm in all the three cases. Particularly, the proposed

approach excels for lower values of the pseudo-reduced pressure at a given pseudo-

reduced temperature. The absolute relative deviations of the z-factor from the

experimental results, defined by experimental theoretical experimentalz z z , for the three numerical

schemes, namely the N-R algorithm, the ADM, and the ADM combined with the Shanks

transformation, are summarized in Table 5.

As a final comment, we remark that the H-Y correlation is invalid for pseudo-reduced

temperatures less than one.[6]

CONCLUSION

An efficient semi-analytical, semi-numerical algorithm based on the Adomian

decomposition method was proposed for calculation of the gas compressibility factor

through the Hall-Yarborough equation of state. Our scheme was shown to be

conceptually simple and straightforward. The optional application of a nonlinear

convergence accelerator technique, namely the Shanks transform, was shown to almost

double the computational efficiency. In addition, the performance of the proposed

method was found to be superior over the classic Newton-Raphson iterative method both

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le in terms of the CPU-time and robustness. Regarding the previously cited benefits, we

recommend our proposed scheme for its precise simulations of real-world problems

arising from natural gas engineering practice.

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; end A=''; 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);

APPENDIX B: THE FIRST SIX COMPONENTS OF THE ADOMIAN

POLYNOMIALS FOR THE POLYNOMIAL NONLINEARITIES IN EQUATION (8)

Nonlinearity: 5N Y Y

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le 5

00,1 Y

4

0 11,1 5Y Y

3 2 4

0 1 0 22,1 10 5Y Y Y Y

2 3 3 4

0 1 0 1 2 0 33,1 10 20 5Y Y Y YY Y Y

4 2 2 3 2 3 4

0 1 0 1 2 0 2 0 1 3 0 44,1 5 30 10 20 5Y Y Y Y Y Y Y Y YY Y Y

5 3 2 2 2 2 3 3 4

1 0 1 2 0 1 2 0 1 3 0 2 3 0 1 4 0 55,1 20 30 30 20 20 5Y Y Y Y Y YY Y Y Y Y Y Y Y YY Y Y

Nonlinearity: 4N Y Y

4

00,2 Y

3

0 11,2 4Y Y

2 2 3

0 1 0 22,2 6 4Y Y Y Y

3 2 3

0 1 0 1 2 0 33,2 4 12 4Y Y Y YY Y Y

4 2 2 2 2 3

1 0 1 2 0 2 0 1 3 0 44,2 12 6 12 4Y Y Y Y Y Y Y YY Y Y

3 2 2 2 2 3

1 2 0 1 2 0 1 3 0 2 3 0 1 4 0 55,2 4 12 12 12 12 4Y Y Y YY Y Y Y Y Y Y Y YY Y Y

Nonlinearity: 3N Y Y

3

00,3 Y

2

0 11,3 3Y Y

2 2

0 1 0 22,3 3 3Y Y Y Y

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le 3 2

1 0 1 2 0 33,3 6 3Y Y YY Y Y

2 2 2

1 2 0 2 0 1 3 0 44,3 3 3 6 3Y Y Y Y Y YY Y Y

2 2 2

1 2 1 3 0 2 3 0 1 4 0 55,3 3 3 6 6 3YY Y Y Y Y Y Y YY Y Y

Nonlinearity: 2N Y Y

2

00,4 Y

0 11,4 2Y Y

2

1 0 22,4 2Y Y Y

1 2 0 33,4 2 2YY Y Y

2

2 1 3 0 44,4 2 2Y YY Y Y

2 3 1 4 0 55,4 2 2 2Y Y YY Y Y

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Captions for Figures

Figure 1. Results of the CPU-time analysis for the three case study examples.

Figure 2. Comparison of the experimental data with the calculated values of the z-factor

by the Newton-Raphson algorithm and the Adomian decomposition method at a)

1.3rT , b) 1.5rT , c) 1.7rT , and d) 2.0rT .

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le Tables

Table 1) Characteristics of the natural gases used in Examples 1, 2 and 3. Example 1 Example 2 Example 3

Temperature 366.5 K Temperature 355.4 K Temperature 310.9 K Pressure 13.7895 MPa Pressure 34.4737 MPa Pressure 6.8947 MPa Gas specific gravity 0.7 Gas specific gravity 0.65 Pseudo-critical temperature 237.2 K

20.05Ny

20.1Ny Pseudo-critical pressure 4.4815 MPa

20.05COy

20.08COy

20.02H Sy

20.02H Sy

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le Table 2) Application of the Shanks transform for calculation of the reduced density as in Example 1.

n nU Y n nSh U 2nSh U

0 10.7196395406 10

1 10.9027570550 10 0.1052547429

2 10.9851499390 10 0.1078904359 0.1095187219

3 0.1029003559 0.1088969282 4 0.1054333271

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le Table 3) The calculated results for Examples 1, 2 and 3. Example 1 Example 2 Example 3

10

010 ii

Y Y

0.1096431955 0.1840252985 0.0888536970

2

0,…,4n nSh U

0.1095187219 0.1846133236 0.0884727532

z by 10Y Y 0.8371 0.9997 0.7564

z by Y from the N-R algorithm 0.8362 1.0002 0.7557

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le Table 4) The problematic sensitivity of the N-R algorithm on selection of the initial guess; The case with Example 1.

Initial guess Converged root 0.0 not computable a 0.5 0.1097523525 0.9 0.2348061308 1.0 not computable a 1.1 0.09854 0.00707 i 1.2 0.4372250396 1.3 1.5196496311 1.4 0.95582+0.01282 i 1.5 0.1452655846

a due to division by zero.

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le Table 5) Absolute relative deviations (ARD %) between the experimental and calculated results for the gas compressibility factor*

prT 1.3

prP 0.5 1.5 2.5 3.5 4.5 5.5 6.5

N-R 66.84 3.162 3.134 1.783 0.774 0.809 0.639 ADM 1.790 2.216 1.618 1.223 0.313 0.683 0.575 ADM+Shanks 1.601 2.055 1.482 1.169 0.295 0.574 0.566

prT 1.5

N-R 73.03 0.002 4.863 3.165 1.725 1.032 0.663 ADM 1.475 1.638 0.790 1.582 1.473 0.840 0.599 ADM+Shanks 1.398 1.595 0.752 1.551 1.391 0.812 0.570

prT 1.7

N-R 28.15 5.386 6.757 4.413 2.483 1.407 1.144 ADM 0.909 1.048 0.909 0.149 0.217 0.575 0.639 ADM+Shanks 0.783 0.947 0.889 0.135 0.206 0.569 0.622

prT 2

N-R 23.59 15.94 11.48 8.154 5.599 4.017 3.394 ADM 0.819 0.802 0.721 0.565 0.448 0.733 1.165 ADM+Shanks 0.690 0.683 0.704 0.557 0.434 0.721 1.021 * The experimental data is due to Standing and Katz.[72]

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Figure 1

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Figure 2a

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Figure 2b

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Figure 2c

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Figure 2d