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Development of natural gas flow rate in pipeline networks based on unsteady state Weymouth equation
Journal of Natural Gas Science and Engineering 33 (2016) 427e437
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Development of natural gas flow rate in pipeline networks based on unsteady state Weymouth equation Hossein Amani a, *, Hasan Kariminezhad b, Hamid Kazemzadeh a a b
Faculty of Chemical Engineering, Babol University of Technology, Babol, Iran Department of Physics, Babol University of Technology, Babol, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 March 2016 Received in revised form 9 May 2016 Accepted 13 May 2016 Available online 17 May 2016
The lack of attention to unsteady state condition in pipeline networks results a considerable error for gas researchers. Our work aims to fill this gap for pipeline networks in series, parallel and looped based on unsteady Weymouth equation. For this, we introduced “Gain Coefficient” as the scale of gas flow increases in pipelines. Our results showed the gain coefficient for steady flow in series was a function of diameter and length ratios (D2/D1, L1/L). The value of gain coefficient for unsteady flow closes to steady flow in series until it reaches to 1.97 after 500 h for L1/L ¼ 0.25 and D2/D1 ¼ 2.5. Based on our development, the gain coefficient just was a function of diameter ratio for steady flow in parallel systems. According to our results, the value of gain coefficient for unsteady flow tends towards steady flow in parallel until it reaches to 12.50 for ratio of diameters equal 2.5 after 4000 h. For looped systems, fractions of looping and diameter ratio were main parameters of gain coefficient. Also, for all diameter ratios, a significant growth occurs in gain coefficient for unsteady flow when fraction of looping exceeds from 0.75. Based on our results, for fraction of looping equal to 0.75, gain coefficients for steady and unsteady flows converged to 1.94 for diameter ratio equal to 2.5 after 200 h. From our results, parallel system has a considerable preference in comparison with series and looped systems, but, it needs a large amount of cost. This paper as a basic report tries to help engineers of gas industry to design more accurate gas pipeline networks. © 2016 Elsevier B.V. All rights reserved.
Keywords: Natural gas Pipeline network Unsteady flow Weymouth equation
1. Introduction There is a forthcoming need to transport huge quantity of natural gas from reservoirs to consumption centre. The challenging question is how to expand and operate the network in order to facilitate the transportation of specified gas quantities at minimum cost. It is possible to solve this problem using optimization of pipeline networks. Transportation of compressible natural gas through pipelines has been studied by several researchers under steady-state condition (Weymouth, 1912; Kolomogorov and Fomin, 1957; Stoner, 1969; Ikoku, 1984; Katz and Lee, 1990; Tian and Adewumi, 1994; Zhou and Adewumi, 1998; Ferguson, 2002; nchez and Ríos-Mercado, 2005; RíosOhirhian, 2002; Borraz-Sa Mercado et al., 2006; Wu et al., 2007; Adeosun et al., 2008; Bermúdeza et al., 2015). Practical transportation equations for
* Corresponding author. E-mail address: [email protected] (H. Amani). http://dx.doi.org/10.1016/j.jngse.2016.05.046 1875-5100/© 2016 Elsevier B.V. All rights reserved.
steady-state gas flow are those of Mueller, Panhandle A, Panhandle B, American Gas Association (AGA), and Weymouth (Coelho and Pinho, 2007). Most of the flow equations have been derived from Bernoulli’s equation (White, 1999). As the Weymouth equation is commonly used for high pressure, high flow rate, and large diameter gas gathering systems, therefore, in this research we would only consider Weymouth equation. There are some assumptions in Weymouth equation such as no kinetic energy change, no mechanical work, isothermal flow, constant compressible factor, and steady flow (Osiadacz and Chaczykowski, 2001; Langelandsvik et al., 2005; Shashi Menon, 2005; Davidson et al., 2006; Coelho and Pinho, 2007; Kabirian and Hemmati, 2007; Abbaspour and Chapman, 2008; Chaczykowski, 2009, 2010; Hamedi et al., 2009; Olatunde et al., 2012; Farzaneh-Gord and Rahbari, 2016). These consumptions can affect the accuracy of a result. Tian and Adewumi (1994) and Zhou and Adewumi (1998) modified Weymouth equation through filling some gaps in momentum equation. Tian and Adewumi (1994) presented an analytical steady-state flow equation considering the kinetic energy term in the momentum
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for gas networks in series, parallel and looped pipelines. The results of this paper are useful for engineers to make more accurate flow capacity forecasts in natural gas networks.
Nomenclature Pi Pf Tb Pb T f Z D L
Dt gg QN Q1 Gs Gus
Q1;us
Inlet pressure, psia Outlet pressure, psia Base temperature, R Base pressure, psia Average flowing temperature, R Moody friction factor Gas deviation factor at average flowing temperature and average pressure Inside diameter of pipe, in Length of pipe, miles Change in time Gas specific gravity (air ¼ 1) 3 Volumetric gas flow rate for pipeline network, fth at Pb and Tb 3 Volumetric gas flow rate for a single pipeline, fth at Pb and Tb Gain Coefficient for steady flow Gain Coefficient for unsteady flow
2. Fundamental gas flow equations for a single horizontal pipeline Fig. 1 represents a schematic of a single horizontal pipeline which transports natural gas to market. For steady flow, Weymouth equation is the most fundamental equation and is expressed as (Adeosun et al., 2009):
Q1;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 p2 18:062Tb u u pi f ¼ u pb t gg Tz L16 D3
where Tb is base temperature, pb is base pressure and Q1;s is steady volumetric gas flow rate in the pipeline. According to Adeosun et al. (2009), the unsteady flow in the pipeline (Q1;us ) with a given pressure drop can be described by the following equation.
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u P 2 P 2 D163 f T u i Dt ¼ 3:23 b t 4 4 pb ZT gg 0:032LDt þ 0:0000157D3 Dt þ 0:083D3 L
equation. Also, Zhou and Adewumi (1998) obtained a steady-state gas flow equation without neglecting any term in the momentum equation. Osiadacz and Chaczykowski (2001) compared isothermal and non-isothermal pipeline gas flow in the unsteady conditions. Chaczykowski (2009, 2010) investigated the effect of thermal model for analyzing unsteady gas pipelines. Olatunde et al. (2012) presented direct calculations method of Weymouth equations for unsteady gas volumetric flow rate with different friction factors in horizontal and inclined pipelines. Also, an analytical approach for simulating a pipeline networks under unsteady conditions presented by Farzaneh-Gord and Rahbari (2016). They considered a one dimensional isothermal compressible viscous flow with kirchhoff’s lows. However, steady flow in pipeline operation seldom exists in actual practice due to variation in input and output gas volume. In fact, deviation from steady flow is a major cause of error in the calculation of gas flow rate in a pipe. Although, there are some reports about unsteady flow in a single gas pipeline, but, there is no report about calculation of gas flow for pipeline networks under unsteady condition. Therefore, this paper attempts to fill this gap
(1)
(2)
3. Fundamental gas flow equations for horizontal pipeline networks Optimization of pipelines mainly focuses on de-bottlenecking of the pipeline network, that is, finding the most restrictive segments and replacing/adding some segments to remove the restriction effect. This requires the knowledge of accurate gas flow in the pipe. As simulation can be useful for designers to optimize the pipeline networks, therefore, our work simulates gas volumetric flow rate for various gas pipeline networks under unsteady flow for real systems (Fig. 2). Also, a comparative study between steady and unsteady flow for pipelines in series, parallel and looped is carried out. The results of this work help the researchers to have a good prediction of gas flow rate under unsteady condition in the networks.
3.1. Pipeline network gain coefficient First, we define pipeline network gain coefficient (G) as the ratio of gas flow rate in the pipeline network to gas flow rate in a single pipeline at the same pressure drop. Therefore, we have:
G¼
Fig. 1. A sketch of a single pipeline.
QN Q1
(3)
where G is gain coefficient of pipeline network. Also, Q1 and QN are volumetric gas flow rate for a single pipeline and the pipeline network, respectively. However, choose of an appropriate value of G is very important for designers to achieve the maximum gas flow rate.
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
p22 p23 ¼
gg TzL2 16 3
D2
QN;s pb 18:062Tb
429
2 (5)
Therefore, steady flow rate ðQN;s Þ is obtained by adding these two equations as below
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p21 p23 18:062Tb u u 0 1 ¼ u pb u L2 A 1 tgg Tz@ L16 þ 16
QN;s
D13
(6)
D23
In addition, steady flow rate for a single pipeline (Q1;s ) with diameter D1 is given by
Q1;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 18:062Tb u p21 p23 u 0 1 ¼ u pb u tgg Tz@ L16 A
(7)
D13
Therefore, G for steady flow (Gseries ) is obtained by dividing Eq. s (6) by Eq. (7). This factor is given by following equation
Gseries s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 0 1 u u @ L A u 16 u D13 u 1 ¼ u0 u u L1 L 2 @ t 16 þ 16 A D13
(8)
D23
For unsteady flow, applying Eq. (2) to each two segments gives
p21
p22
0 1 QN;us pb 2 @ZT gg A ¼ 16 3:23Tb D13 1 0 4 4 0:032L1 Dt þ 0:0000157D31 Dt þ 0:083D31 L1 A @ Dt
Fig. 2. Schematic diagrams of (a) a series pipeline, (b) a parallel pipeline and (c) a looped pipeline.
3.2. Pipelines in series
(9)
p22
p23
Fig. 2a shows a two-segment gas pipeline in series with a total length L. Based on Weymouth equation for each segment of the pipelines at steady state condition we have (Guo et al., 2007):
p21 p22 ¼
QN;us
gg TzL1 16 3
D1
QN;s pb 18:062Tb
0 1 QN;us pb 2 @ZT gg A ¼ 16 3:23Tb D23 1 0 4 4 0:032L2 Dt þ 0:0000157D32 Dt þ 0:083D32 L2 A @ Dt
2
(10) (4)
Therefore, unsteady flow rate ðQN;us Þ in a two segments pipeline in series is obtained by adding these two equations.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p21 p23 3:23Tb u u 0 1 ¼ 4 4 4 4 pb u 3 3 3 3 u 0:032L D tþ0:0000157D D tþ0:083D L 0:032L D tþ0:0000157D D tþ0:083D L 1 2 1 1 1 2 2 2A t ZT gg @ þ 16 16 D13 Dt
D23 Dt
(11)
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H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Adding Eqs. (15) and (16) gives the total gas flow rate of the parallel system at unsteady state condition.
Using Eq. (2) for a single-pipeline with a diameter D1, gain coefficient for unsteady flow (Gseries ) is obtained by the following us equation:
Gseries us
QN;us ¼ Q1;us þ Q2;us
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 u 0:032LDt þ 0:0000157D31 Dt þ 0:083D31 L u 1 ¼u 0 4 4 4 4 u 16 u 3 @0:032L1 Dtþ0:0000157D31 Dtþ0:083D31 L1 0:032L2 Dtþ0:0000157D32 Dtþ0:083D32 L2 A tD1 þ 16 16 D13
As represented in Eq. (18), gain coefficient for the two-segment pipeline in parallel at unsteady state condition is obtained by dividing Eq. (17) by Eq. (2).
A two-segment gas pipeline in parallel has been shown in Fig. 2b. For investigation of gas flow rate in the pipelines, Wey-
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 Gparallel ¼ us
(12)
D23
3.3. Pipelines in parallel
D13
(17)
4
4
0:032DtLþ0:0000157DtD31 þ0:083D31 L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 D23
þ
4
4
0:032DtLþ0:0000157DtD32 þ0:083D32 L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 D13
4
(18)
4
0:032DtLþ0:0000157DtD31 þ0:083D31 L
mouth equation is used for each of the two segments at steady condition. The total gas flow rate (QN;s ) is the sum of the flow in each pipeline.
QN;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 2 p p Tb u 2 ¼ 18:062 t 1 pb gg TzL
rffiffiffiffiffiffiffi 16 3
D1 þ
rffiffiffiffiffiffiffi! 16
D23
3.4. Pipelines in looped A three-segment looped gas pipeline has been shown in Fig. 2c. For steady flow, by applying Eq. (13) to the first two segments (parallel) gives:
(13)
Therefore, as seen in Eq. (14), gain coefficient for steady flow for a two-segment pipeline in parallel is obtained by dividing Eq. (13) by Eq. (7).
QN;s ¼ Q1;s þ Q2;s
parallel (Gs )
Gparallel s
qffiffiffiffiffiffi ffi qffiffiffiffiffiffi ffi 16 16 D13 þ D23 qffiffiffiffiffiffi ffi ¼ 16 D13
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi! u 2 2 16 16 p p Tb u 3 D13 þ D23 ¼ 18:062 t 1 pb gg TzL (19)
After simplification, we have
(14)
QN;s pb 2 2 q ffiffiffiffiffiffi ffi 18:062Tb 16 16 D13 þ D23
gg TzL1
p21 p23 ¼ qffiffiffiffiffiffiffi
For unsteady flow, applying Eq. (2) to each two segments gives (Adeosun et al., 2009):
(20)
In addition, applying the Weymouth equation to the third
Q1;us
Q2;us
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 u u P 2 P 2 D163 1 2 Tb u D t 1 A @ t ¼ 3:23 4 4 pb ZT gg 0:032LDt þ 0:0000157D31 Dt þ 0:083D31 L
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 u u P 2 P 2 D163 1 2 Tb u D t 2 A @ t ¼ 3:23 4 4 pb ZT gg 0:032LDt þ 0:0000157D32 Dt þ 0:083D32 L
(15)
(16)
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
4. Results and discussion
segment (with diameter D3) yields
p23 p22 ¼
gg TzL3 16 3
D3
QN;s pb 18:062Tb
431
2 (21)
Adding Eqs. (20) and (21) results:
0 1 2 B QN;s pb L L C B C p21 p22 ¼ gg Tz Bqffiffiffiffiffiffiffi 1qffiffiffiffiffiffiffi 2 þ 316 C @ 18:062Tb 16 16 3 A D3 D13 þ D23
In this section, increase of flow capacity using pipeline networks for unsteady flow and steady flow is studied. For this purpose, the effects of three types of pipeline networks such as pipelines in series, parallel and looped on gas volumetric flow rate are investigated. For better understanding, all results were compared with the flow rate of a typical single horizontal pipeline with length L ¼ 20 miles, diameter D1 ¼ 10 inches and a known pressure drop.
4.1. Investigation of gain coefficient for pipelines in series
(22) Therefore, total flow rate of the looped system at steady condition is given by:
QN;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u p1 p22 18:062Tb u u 0 1 ¼ u pb u u B C u 3 C ugg TzB Bqffiffiffiffiffiffi L1qffiffiffiffiffiffi2 þ L16 C u @ 16 16 D3 A t D13 þ
Gseries s
D23
(23)
3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 u 0:032 20Dt þ 0:0000157 103 Dt þ 0:083 103 20 ! ¼u u u 16 0:03215Dtþ0:00001571043 Dtþ0:0831043 15 0:0325Dtþ0:00001571543 Dtþ0:0831543 5 t10 3 þ 16 16 10 3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 18:062Tb u p21 p22 u 0 1 ¼ u pb u tgg Tz@ L16 A
(24)
D33
Therefore, gain coefficient for this three-segment pipeline at steady condition is obtained by dividing Eq. (23) by Eq. (24).
Glooped s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u L u 16 u 3 D 1 ¼u u L1 3 uqffiffiffiffiffiffi qffiffiffiffiffiffi 2 þ L16 t 16 16 D3 D13 þ
D23
(25)
3
For unsteady flow, applying the Eq. (17) for each of the first two parallel segments and after simplification, the gain coefficient for the looped system is given by:
Glooped us
(27)
15 3
From Eq. (1) for a single-diameter (D3) pipeline, total flow rate is given by
Q3;s
Consider a D1 ¼ 10 in pipeline that is 20 mi long (Fig. 2a). According to Eq. (8) for steady gas flow, if we replace a L2 ¼ 5 miles of the 10 in pipeline by a D2 ¼ 15 in pipeline segment, the results give gain coefficient ðGseries Þ tends toward 1.133 or 13.32% increase in s flow capacity. Generally, according to Fig. 3a, b, and c for steady state, the increase of gas capacity depends on diameter ratio (D2/ D1) and length ratio (L1/L). In other side for unsteady flow, time also plays an important role for calculation of gas capacity. For D2/ D1 ¼ 1.5 and L1/L ¼ 0.75 Eq. (12) gives:
Fig. 3a, b, and c show temporal variation of gain coefficient for various ratios of L1/L and D2/D1. As shown in Fig. 3a, gain coefficient of the pipelines in series increased with decreasing of L1/L for both of steady and unsteady flow. Also, this figure shows there is a significant difference between the values of Gseries and Gseries . Howus s ever, the values of Gseries at long times tend to reach to the values of us Gseries . Based on our results, from Fig. 3a, these times are 60, 350 and s 800 h for L1/L ¼ 0.75, 0.5 and 0.25, respectively. Moreover, it can be seen from Fig. 3a, b, and c, gain coefficient increased with increasing diameter ratio (D2/D1) for both steady and unsteady flow. But, when D2/D1 increases, the values of gain coefficient under unsteady flow converge to that of steady flow faster. For example, for length ratio (L1/L) equal to 0.25 and D2/ D1 ¼ 1.5, 2 and 2.5, the convergence time is 800, 400 and 300 h, respectively. As a result, the values of Gseries and Gseries for L1/ s us L ¼ 0.25 and D2/D1 ¼ 1.5, 2 and 2.5 converge to 1.72, 1.95 and 1.97, respectively. However, from these figures it is obvious that there is a difference between the Gseries and Gseries at initial time. For better s us understanding, we summarized the effects of diameter ratio and time on Gseries in a three dimensional figure for L1/L ¼ 0.25, 0.5 and us 0.75, respectively (Fig. 4).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 u 0:032DtL þ 0:0000157DtD33 þ 0:083D33 L u ¼u 16 u D33 u0:032DtL þ 0:0000157DtD43 þ 0:083D43 L þ 0 12 u 3 3 3 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 16 16 u D3 D3 B C u 1 2 þ @ A t 4 4 4 4 0:032DtL1 þ0:0000157DtD3 þ0:083D3 L1 1 1
0:032DtL1 þ0:0000157DtD3 þ0:083D3 L1 2 2
(26)
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H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Fig. 3. Effect of L1/L on gain coefficient for pipelines in series under steady and unsteady flow at different time for (a) D2/D1 ¼ 1.5, (b) D2/D1 ¼ 2 and (c) D2/D1 ¼ 2.5.
4.2. Investigation of gain coefficient for pipelines in parallel According to Fig. 2b, consider a D1 ¼ 10 in pipeline that is 20 miles long. If we place a D2 ¼ 15 in pipe parallel to share gas parallel transmission, from Eq. (14) for steady flow Gs increases to 3.95 or 295% increase in flow capacity. Based on our calculations, the
increase of gas capacity depends only on pipe diameter ratio (D2/ D1) for steady flow. As seen in Fig. 5, the values of gain coefficient for steady flow are 3.95, 7.35 and 12.50 for ratio of diameters equal to 1.5, 2 and 2.5, respectively. In other side, from Eq. (18) for unsteady flow we have:
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Fig. 4. Effects of time and diameter ratio on gain coefficient in series for unsteady flow: (a) L1/L ¼ 0.25, (b) L1/L ¼ 0.5 and (c) L1/L ¼ 0.75.
433
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H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Fig. 5. Effect of diameter ratio on gain coefficient for pipelines in parallel under steady and unsteady flow at different times.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16
parallel Gus
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16
10 3 0:64Dtþ3:3825ð104 ÞDtþ35:7636
15 3 þ 0:64Dtþ5:8079ð10 4 ÞDtþ61:40868 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16
(28)
10 3 0:64Dtþ3:3825ð104 ÞDtþ35:7636
The results from this equation are shown in Fig. 5. Based on Gparallel us
Fig. 5, for the parallel system, values of lead towards the values of gain coefficient for steady flow after 800, 2000, about
4000 h for D2/D1 ¼ 1.5, 2 and 2.5, respectively. Moreover, it is obvious that there is a significant difference between the values of Gparallel and Gparallel at initial time. Based on our calculations, the us s
Fig. 6. Effects of time and diameter ratio on gain coefficient in parallel for unsteady flow.
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437 parallel.
ratio of gain coefficients (Gus
) at initial time are 82%, Gparallel s 68% and 59% for D2/D1 ¼ 1.5, 2 and 2.5, respectively. For a better observation, as represented in Fig. 6, we summarized the effects of in a three dimensional figure. diameter ratio and time on Gparallel us
435
4.3. Investigation of gain coefficient for pipelines in looped looped
According to Eq. (26), gain coefficient for unsteady flow (Gus ) is a function of D2/D3, fraction of looping (L1/L) and time. The effects of fraction of looping and pipe diameter ratio on gain coefficient versus time have been shown in Fig. 7. In this figure, a L ¼ 20 mi
Fig. 7. Effects of fraction of looping and diameter ratio on the gain coefficient in a looped system for unsteady flow at different times. (a) D2/D3 ¼ 1.5, (b) D2/D3 ¼ 2 and (c) D2/ D3 ¼ 2.5.
436
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Fig. 8. Effects of fraction of looping and time on gain coefficient in a looped system for unsteady flow (a): D2/D3 ¼ 1.5, (b): D2/D3 ¼ 2, and (c) D2/D3 ¼ 2.5.
pipeline with D1 ¼ D3 ¼ 10 in was considered. As shown in Fig. 7a, b and c, for unsteady flow, there is a significant growth of gain coefficient for. For example, from Fig. 7a for L1/L > 0.75, the initial
values of gain coefficient increase from 1.76 to 3.49. Furthermore, Fig. 7a, b and c show that gain coefficient increases with rising diameter ratio (D2/D3). Based on our calculations, for L1/L ¼ 0.75,
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
gain coefficients at initial time are 1.76, 1.89 and 1.94 for D2/ D3 ¼ 1.5, 2 and 2.5 for unsteady flow, respectively. However, for a completely paralleled system (L1/L ¼ 1), at the initial time the gain coefficients are 3.25, 5 and 7.25 for D2/D3 ¼ 1.5, 2 and 2.5, respectively. However, there is a significant difference between the values of gain coefficient for parallel system and pipelines in looped. It can be seen from Fig. 7a that gain coefficient is gradually increases from 1.139 to 1.142 after 100 h for diameter ratio D2/D3 ¼ 1.5 and fraction of looping 0.25. According to Fig. 7a, gain coefficient has 0.3%, 1.1%, 3.9% and 37% variation for fraction of looping L1/ L ¼ 0.25, 0.5, 0.75 and 1 during 100 h, respectively. Also as seen in Fig. 7b and c, for fraction of looping less than 0.75, maximum variation of gain coefficient will be 3.3% and 2.2% after 100 h, respectively. Also, for parallel systems, gain coefficient represents 106% and 205% variation for D2/D3 ¼ 2 and 2.5, respectively. Therefore, based on these results, temporal variation of gain coefficient is only considerable for parallel system. Finally, the effects of diameter ratio and time on gain coefficient in a three dimensional plot have been summarized in Fig. 8 for a looped system. In a general conclusion, according to our results, there are significant differences between values of gain coefficient in pipelines networks under steady and unsteady flows. However, as the values of gas flow rate for unsteady condition are more accurate than those of the steady flow, therefore, the lack of attention to this subject in pipeline network leads to a considerable error for engineers. For example, from Fig. 5 for a parallel system, more than 5 months is required to the values of gain coefficient under unsteady flow tend towards the values of steady flow. Furthermore, our results show that use of parallel systems introduces significant advantages in comparison with series and looped systems. Although using parallel pipeline yields notable increase of gas capacity, but, it needs a large amount of cost. Therefore, engineers must optimize fraction of looped pipeline to reach a sufficient growth of flow capacity by considering market consumption and economical support. However, this paper attempted to increase the knowledge of engineers for better prediction of gas flow rate in their models. 5. Conclusion In this paper, we developed an analytical model based on unsteady-state Weymouth equation without neglecting any terms in the fundamental governing momentum equations to study gas flow rate for pipeline networks in series, parallel and looped systems. Also, in this work, a comparative study carried out between these systems. First, we introduced “Gain coefficient” to find out about scale of gas flow increases in pipelines. Our developments showed functional relationships between time, diameter ratio, length ratio of pipeline network and gain coefficient. Our results also showed gain coefficient increases using networks for both steady and unsteady flows with a preference to parallel system. Although, for all systems a significant difference between the values of gain coefficient in pipeline networks for unsteady and steady flows was seen, but, it is observed that all unsteady-state processes tend towards steady-state with time. However, according to our research, time plays an important role in the prediction of more accurate values of gain coefficient for all systems. These differences especially for parallel system were more clearly observed. As gas engineers need to an accurate calculation of flow rate, the
437
lack of attention to applying unsteady models in their calculation leads to a considerable error. This paper as a basic report tried to help engineers of gas industry to design more accurate gas pipeline networks.
References Abbaspour, M., Chapman, K.S., 2008. Non isothermal transient flow in natural gas pipeline. J. Appl. Mech. 75 (3), 031018. Adeosun, T.A., Olatunde, O.A., Aderohunmu, J.O., Ogunjare, T.O., 2009. Development of unsteady-state Weymouth equations for gas volumetric flow rate in horizontal and inclined pipes. J. Nat. Gas Sci. Eng. 1, 113e117. Adeosun, T.A., Olatunde, O.A., Okwananke, A., 2008. Development of unsteady-state equations for the flow of gas through pipelines. J. Adv. Appl. Fluid Mech. ISSN: 0973-4686 lez-Díazb, J., Gonza lez-Die gueza, F.J., Gonza lez-Ruedab, A.M., Bermúdeza, A., Gonza rdobab, M., 2015. Simulation and optimization models of Fern andez de Co steady-state gas transmission networks. Energy Procedia 64, 130e139. nchez, C., Ríos-Mercado, R.Z., 2005. A Hybrid Meta-heuristic Approach for Borraz-Sa Natural Gas Pipeline Network Optimization. Springer-Verlag, Berlin Heidelberg, pp. 54e65. Chaczykowski, M., 2009. Sensitivity of pipeline gas flow model to the selection of the equation of state. Chem. Eng. Res. Des. 87, 1596e1603. Chaczykowski, M., 2010. Transient flow in natural gas pipeline the effect of pipeline thermal model. Appl. Math. Model. 34, 1051e1067. Coelho, P.M., Pinho, C., 2007. Considerations about equations for steady state flow in natural gas pipelines. J. Braz. Soc. Mech. Sci. Eng. 29 (3), 262e273. Davidson, R.A., Lembo, A., Ma, J., Nozick, L.K., O’Rourke, T.D., 2006. Optimization of investments in natural gas distribution networks. J. Energy Eng. 52e60. Farzaneh-Gord, M., Rahbari, H.R., 2016. Unsteady natural gas flow within pipeline network, an analytical approach. J. Nat. Gas Sci. Eng. 28, 397e409. Ferguson, J.A., 2002. Gas flow in long pipelines. Chem. Eng. 56 (11), 112. Guo, B., Lyons, W.C., Ghalambor, A., 2007. Petroleum Production Engineering, a Computer-assisted Approach, first ed. Elsevier Science and Technology Books, pp. 267e280. Hamedi, M., Zanjirani Farahani, R., Husseini, M.M., Esmaeilian, G.R., 2009. A distribution planning model for natural gas supply chain: a case study. Energy Policy 37, 799e812. Ikoku, C.U., 1984. Natural Gas Production Engineering. John Wiley and Sons Publishing Co., New York. Kabirian, A., Hemmati, M.R., 2007. A strategic planning model for natural gas transmission networks. Energy Policy 35, 5656e5670. Katz, D.L., Lee, R.I., 1990. Natural Gas Engineering Production and Storage. McGrawHill Publishing Co., New York. Kolomogorov, A.N., Fomin, S.V., 1957. Element of the theory of functions and functional analysis. In: Metric and Normed Spaces. Gray Lock Press, Rochester, New York. Langelandsvik, L.I., Postvoll, W., Svendsen, P., Øverli, J.M., Ytrehus, T., 2005. Evaluation of the friction factor formula based on operational data. In: Proceedings of the 37th PSIG Annual Meeting, San Antonio Texas. Ohirhian, P.U., 2002. Direct Calculation of the Gas Volumetric Flow Rate in Horizontal and Inclined Pipes. SPE 37394. Olatunde, O.A., Adeosun, T.A., Usman, M.A., Odunlami, O.M., Olowofoyeku, M.A., Ekakitie, T.E., Mohammed, A.M., 2012. Direct calculation of unsteady-state Weymouth equations for gas volumetric flow rate with different friction factors in horizontal and inclined pipes. Engineering 4, 202e209. Osiadacz, A.J., Chaczykowski, M., 2001. Comparison of isothermal and non isothermal pipeline gas flow models. Chem. Eng. 81 (1e3), 41e51. Ríos-Mercado, R.Z., Kim, S., Boyd, E.A., 2006. Efficient operation of natural gas transmission systems: a network-based heuristic for cyclic structures. Comput. Oper. Res. 33, 2323e2351. Shashi Menon, E., 2005. Gas Pipeline Hydraulics. CRC Press, Boca Raton, Florida. Stoner, M.A., 1969. Steady State Analysis of Gas Production, Transmission and Distribution Systems. SPE 2554. Tian, S., Adewumi, M.A., 1994. Development of Analytical Design Equation for Gas Pipelines. SPE 24861. Weymouth, T.R., 1912. Problems in natural gas engineering. Trans. AIME 34, 185. White, F.M., 1999. Fluid Mechanics, fourth ed. McGraw-Hill Publishing Co., New York. Wu, Y., Lai, K.K., Liu, Y., 2007. Deterministic global optimization approach to steadystate distribution gas pipeline networks. Optim. Eng. 8, 259e275. Zhou, J., Adewumi, M.A., 1998. Gas Pipeline Network Analysis Using an Analytic Steady-state Flow Equation. SPE 51044.
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Development of natural gas flow rate in pipeline networks based on unsteady state Weymouth equation Hossein Amani a, *, Hasan Kariminezhad b, Hamid Kazemzadeh a a b
Faculty of Chemical Engineering, Babol University of Technology, Babol, Iran Department of Physics, Babol University of Technology, Babol, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 March 2016 Received in revised form 9 May 2016 Accepted 13 May 2016 Available online 17 May 2016
The lack of attention to unsteady state condition in pipeline networks results a considerable error for gas researchers. Our work aims to fill this gap for pipeline networks in series, parallel and looped based on unsteady Weymouth equation. For this, we introduced “Gain Coefficient” as the scale of gas flow increases in pipelines. Our results showed the gain coefficient for steady flow in series was a function of diameter and length ratios (D2/D1, L1/L). The value of gain coefficient for unsteady flow closes to steady flow in series until it reaches to 1.97 after 500 h for L1/L ¼ 0.25 and D2/D1 ¼ 2.5. Based on our development, the gain coefficient just was a function of diameter ratio for steady flow in parallel systems. According to our results, the value of gain coefficient for unsteady flow tends towards steady flow in parallel until it reaches to 12.50 for ratio of diameters equal 2.5 after 4000 h. For looped systems, fractions of looping and diameter ratio were main parameters of gain coefficient. Also, for all diameter ratios, a significant growth occurs in gain coefficient for unsteady flow when fraction of looping exceeds from 0.75. Based on our results, for fraction of looping equal to 0.75, gain coefficients for steady and unsteady flows converged to 1.94 for diameter ratio equal to 2.5 after 200 h. From our results, parallel system has a considerable preference in comparison with series and looped systems, but, it needs a large amount of cost. This paper as a basic report tries to help engineers of gas industry to design more accurate gas pipeline networks. © 2016 Elsevier B.V. All rights reserved.
Keywords: Natural gas Pipeline network Unsteady flow Weymouth equation
1. Introduction There is a forthcoming need to transport huge quantity of natural gas from reservoirs to consumption centre. The challenging question is how to expand and operate the network in order to facilitate the transportation of specified gas quantities at minimum cost. It is possible to solve this problem using optimization of pipeline networks. Transportation of compressible natural gas through pipelines has been studied by several researchers under steady-state condition (Weymouth, 1912; Kolomogorov and Fomin, 1957; Stoner, 1969; Ikoku, 1984; Katz and Lee, 1990; Tian and Adewumi, 1994; Zhou and Adewumi, 1998; Ferguson, 2002; nchez and Ríos-Mercado, 2005; RíosOhirhian, 2002; Borraz-Sa Mercado et al., 2006; Wu et al., 2007; Adeosun et al., 2008; Bermúdeza et al., 2015). Practical transportation equations for
* Corresponding author. E-mail address: [email protected] (H. Amani). http://dx.doi.org/10.1016/j.jngse.2016.05.046 1875-5100/© 2016 Elsevier B.V. All rights reserved.
steady-state gas flow are those of Mueller, Panhandle A, Panhandle B, American Gas Association (AGA), and Weymouth (Coelho and Pinho, 2007). Most of the flow equations have been derived from Bernoulli’s equation (White, 1999). As the Weymouth equation is commonly used for high pressure, high flow rate, and large diameter gas gathering systems, therefore, in this research we would only consider Weymouth equation. There are some assumptions in Weymouth equation such as no kinetic energy change, no mechanical work, isothermal flow, constant compressible factor, and steady flow (Osiadacz and Chaczykowski, 2001; Langelandsvik et al., 2005; Shashi Menon, 2005; Davidson et al., 2006; Coelho and Pinho, 2007; Kabirian and Hemmati, 2007; Abbaspour and Chapman, 2008; Chaczykowski, 2009, 2010; Hamedi et al., 2009; Olatunde et al., 2012; Farzaneh-Gord and Rahbari, 2016). These consumptions can affect the accuracy of a result. Tian and Adewumi (1994) and Zhou and Adewumi (1998) modified Weymouth equation through filling some gaps in momentum equation. Tian and Adewumi (1994) presented an analytical steady-state flow equation considering the kinetic energy term in the momentum
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for gas networks in series, parallel and looped pipelines. The results of this paper are useful for engineers to make more accurate flow capacity forecasts in natural gas networks.
Nomenclature Pi Pf Tb Pb T f Z D L
Dt gg QN Q1 Gs Gus
Q1;us
Inlet pressure, psia Outlet pressure, psia Base temperature, R Base pressure, psia Average flowing temperature, R Moody friction factor Gas deviation factor at average flowing temperature and average pressure Inside diameter of pipe, in Length of pipe, miles Change in time Gas specific gravity (air ¼ 1) 3 Volumetric gas flow rate for pipeline network, fth at Pb and Tb 3 Volumetric gas flow rate for a single pipeline, fth at Pb and Tb Gain Coefficient for steady flow Gain Coefficient for unsteady flow
2. Fundamental gas flow equations for a single horizontal pipeline Fig. 1 represents a schematic of a single horizontal pipeline which transports natural gas to market. For steady flow, Weymouth equation is the most fundamental equation and is expressed as (Adeosun et al., 2009):
Q1;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 p2 18:062Tb u u pi f ¼ u pb t gg Tz L16 D3
where Tb is base temperature, pb is base pressure and Q1;s is steady volumetric gas flow rate in the pipeline. According to Adeosun et al. (2009), the unsteady flow in the pipeline (Q1;us ) with a given pressure drop can be described by the following equation.
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u P 2 P 2 D163 f T u i Dt ¼ 3:23 b t 4 4 pb ZT gg 0:032LDt þ 0:0000157D3 Dt þ 0:083D3 L
equation. Also, Zhou and Adewumi (1998) obtained a steady-state gas flow equation without neglecting any term in the momentum equation. Osiadacz and Chaczykowski (2001) compared isothermal and non-isothermal pipeline gas flow in the unsteady conditions. Chaczykowski (2009, 2010) investigated the effect of thermal model for analyzing unsteady gas pipelines. Olatunde et al. (2012) presented direct calculations method of Weymouth equations for unsteady gas volumetric flow rate with different friction factors in horizontal and inclined pipelines. Also, an analytical approach for simulating a pipeline networks under unsteady conditions presented by Farzaneh-Gord and Rahbari (2016). They considered a one dimensional isothermal compressible viscous flow with kirchhoff’s lows. However, steady flow in pipeline operation seldom exists in actual practice due to variation in input and output gas volume. In fact, deviation from steady flow is a major cause of error in the calculation of gas flow rate in a pipe. Although, there are some reports about unsteady flow in a single gas pipeline, but, there is no report about calculation of gas flow for pipeline networks under unsteady condition. Therefore, this paper attempts to fill this gap
(1)
(2)
3. Fundamental gas flow equations for horizontal pipeline networks Optimization of pipelines mainly focuses on de-bottlenecking of the pipeline network, that is, finding the most restrictive segments and replacing/adding some segments to remove the restriction effect. This requires the knowledge of accurate gas flow in the pipe. As simulation can be useful for designers to optimize the pipeline networks, therefore, our work simulates gas volumetric flow rate for various gas pipeline networks under unsteady flow for real systems (Fig. 2). Also, a comparative study between steady and unsteady flow for pipelines in series, parallel and looped is carried out. The results of this work help the researchers to have a good prediction of gas flow rate under unsteady condition in the networks.
3.1. Pipeline network gain coefficient First, we define pipeline network gain coefficient (G) as the ratio of gas flow rate in the pipeline network to gas flow rate in a single pipeline at the same pressure drop. Therefore, we have:
G¼
Fig. 1. A sketch of a single pipeline.
QN Q1
(3)
where G is gain coefficient of pipeline network. Also, Q1 and QN are volumetric gas flow rate for a single pipeline and the pipeline network, respectively. However, choose of an appropriate value of G is very important for designers to achieve the maximum gas flow rate.
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
p22 p23 ¼
gg TzL2 16 3
D2
QN;s pb 18:062Tb
429
2 (5)
Therefore, steady flow rate ðQN;s Þ is obtained by adding these two equations as below
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p21 p23 18:062Tb u u 0 1 ¼ u pb u L2 A 1 tgg Tz@ L16 þ 16
QN;s
D13
(6)
D23
In addition, steady flow rate for a single pipeline (Q1;s ) with diameter D1 is given by
Q1;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 18:062Tb u p21 p23 u 0 1 ¼ u pb u tgg Tz@ L16 A
(7)
D13
Therefore, G for steady flow (Gseries ) is obtained by dividing Eq. s (6) by Eq. (7). This factor is given by following equation
Gseries s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 0 1 u u @ L A u 16 u D13 u 1 ¼ u0 u u L1 L 2 @ t 16 þ 16 A D13
(8)
D23
For unsteady flow, applying Eq. (2) to each two segments gives
p21
p22
0 1 QN;us pb 2 @ZT gg A ¼ 16 3:23Tb D13 1 0 4 4 0:032L1 Dt þ 0:0000157D31 Dt þ 0:083D31 L1 A @ Dt
Fig. 2. Schematic diagrams of (a) a series pipeline, (b) a parallel pipeline and (c) a looped pipeline.
3.2. Pipelines in series
(9)
p22
p23
Fig. 2a shows a two-segment gas pipeline in series with a total length L. Based on Weymouth equation for each segment of the pipelines at steady state condition we have (Guo et al., 2007):
p21 p22 ¼
QN;us
gg TzL1 16 3
D1
QN;s pb 18:062Tb
0 1 QN;us pb 2 @ZT gg A ¼ 16 3:23Tb D23 1 0 4 4 0:032L2 Dt þ 0:0000157D32 Dt þ 0:083D32 L2 A @ Dt
2
(10) (4)
Therefore, unsteady flow rate ðQN;us Þ in a two segments pipeline in series is obtained by adding these two equations.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p21 p23 3:23Tb u u 0 1 ¼ 4 4 4 4 pb u 3 3 3 3 u 0:032L D tþ0:0000157D D tþ0:083D L 0:032L D tþ0:0000157D D tþ0:083D L 1 2 1 1 1 2 2 2A t ZT gg @ þ 16 16 D13 Dt
D23 Dt
(11)
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H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Adding Eqs. (15) and (16) gives the total gas flow rate of the parallel system at unsteady state condition.
Using Eq. (2) for a single-pipeline with a diameter D1, gain coefficient for unsteady flow (Gseries ) is obtained by the following us equation:
Gseries us
QN;us ¼ Q1;us þ Q2;us
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 u 0:032LDt þ 0:0000157D31 Dt þ 0:083D31 L u 1 ¼u 0 4 4 4 4 u 16 u 3 @0:032L1 Dtþ0:0000157D31 Dtþ0:083D31 L1 0:032L2 Dtþ0:0000157D32 Dtþ0:083D32 L2 A tD1 þ 16 16 D13
As represented in Eq. (18), gain coefficient for the two-segment pipeline in parallel at unsteady state condition is obtained by dividing Eq. (17) by Eq. (2).
A two-segment gas pipeline in parallel has been shown in Fig. 2b. For investigation of gas flow rate in the pipelines, Wey-
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 Gparallel ¼ us
(12)
D23
3.3. Pipelines in parallel
D13
(17)
4
4
0:032DtLþ0:0000157DtD31 þ0:083D31 L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 D23
þ
4
4
0:032DtLþ0:0000157DtD32 þ0:083D32 L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 D13
4
(18)
4
0:032DtLþ0:0000157DtD31 þ0:083D31 L
mouth equation is used for each of the two segments at steady condition. The total gas flow rate (QN;s ) is the sum of the flow in each pipeline.
QN;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 2 p p Tb u 2 ¼ 18:062 t 1 pb gg TzL
rffiffiffiffiffiffiffi 16 3
D1 þ
rffiffiffiffiffiffiffi! 16
D23
3.4. Pipelines in looped A three-segment looped gas pipeline has been shown in Fig. 2c. For steady flow, by applying Eq. (13) to the first two segments (parallel) gives:
(13)
Therefore, as seen in Eq. (14), gain coefficient for steady flow for a two-segment pipeline in parallel is obtained by dividing Eq. (13) by Eq. (7).
QN;s ¼ Q1;s þ Q2;s
parallel (Gs )
Gparallel s
qffiffiffiffiffiffi ffi qffiffiffiffiffiffi ffi 16 16 D13 þ D23 qffiffiffiffiffiffi ffi ¼ 16 D13
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi! u 2 2 16 16 p p Tb u 3 D13 þ D23 ¼ 18:062 t 1 pb gg TzL (19)
After simplification, we have
(14)
QN;s pb 2 2 q ffiffiffiffiffiffi ffi 18:062Tb 16 16 D13 þ D23
gg TzL1
p21 p23 ¼ qffiffiffiffiffiffiffi
For unsteady flow, applying Eq. (2) to each two segments gives (Adeosun et al., 2009):
(20)
In addition, applying the Weymouth equation to the third
Q1;us
Q2;us
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 u u P 2 P 2 D163 1 2 Tb u D t 1 A @ t ¼ 3:23 4 4 pb ZT gg 0:032LDt þ 0:0000157D31 Dt þ 0:083D31 L
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 u u P 2 P 2 D163 1 2 Tb u D t 2 A @ t ¼ 3:23 4 4 pb ZT gg 0:032LDt þ 0:0000157D32 Dt þ 0:083D32 L
(15)
(16)
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
4. Results and discussion
segment (with diameter D3) yields
p23 p22 ¼
gg TzL3 16 3
D3
QN;s pb 18:062Tb
431
2 (21)
Adding Eqs. (20) and (21) results:
0 1 2 B QN;s pb L L C B C p21 p22 ¼ gg Tz Bqffiffiffiffiffiffiffi 1qffiffiffiffiffiffiffi 2 þ 316 C @ 18:062Tb 16 16 3 A D3 D13 þ D23
In this section, increase of flow capacity using pipeline networks for unsteady flow and steady flow is studied. For this purpose, the effects of three types of pipeline networks such as pipelines in series, parallel and looped on gas volumetric flow rate are investigated. For better understanding, all results were compared with the flow rate of a typical single horizontal pipeline with length L ¼ 20 miles, diameter D1 ¼ 10 inches and a known pressure drop.
4.1. Investigation of gain coefficient for pipelines in series
(22) Therefore, total flow rate of the looped system at steady condition is given by:
QN;s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u p1 p22 18:062Tb u u 0 1 ¼ u pb u u B C u 3 C ugg TzB Bqffiffiffiffiffiffi L1qffiffiffiffiffiffi2 þ L16 C u @ 16 16 D3 A t D13 þ
Gseries s
D23
(23)
3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 u 0:032 20Dt þ 0:0000157 103 Dt þ 0:083 103 20 ! ¼u u u 16 0:03215Dtþ0:00001571043 Dtþ0:0831043 15 0:0325Dtþ0:00001571543 Dtþ0:0831543 5 t10 3 þ 16 16 10 3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 18:062Tb u p21 p22 u 0 1 ¼ u pb u tgg Tz@ L16 A
(24)
D33
Therefore, gain coefficient for this three-segment pipeline at steady condition is obtained by dividing Eq. (23) by Eq. (24).
Glooped s
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u L u 16 u 3 D 1 ¼u u L1 3 uqffiffiffiffiffiffi qffiffiffiffiffiffi 2 þ L16 t 16 16 D3 D13 þ
D23
(25)
3
For unsteady flow, applying the Eq. (17) for each of the first two parallel segments and after simplification, the gain coefficient for the looped system is given by:
Glooped us
(27)
15 3
From Eq. (1) for a single-diameter (D3) pipeline, total flow rate is given by
Q3;s
Consider a D1 ¼ 10 in pipeline that is 20 mi long (Fig. 2a). According to Eq. (8) for steady gas flow, if we replace a L2 ¼ 5 miles of the 10 in pipeline by a D2 ¼ 15 in pipeline segment, the results give gain coefficient ðGseries Þ tends toward 1.133 or 13.32% increase in s flow capacity. Generally, according to Fig. 3a, b, and c for steady state, the increase of gas capacity depends on diameter ratio (D2/ D1) and length ratio (L1/L). In other side for unsteady flow, time also plays an important role for calculation of gas capacity. For D2/ D1 ¼ 1.5 and L1/L ¼ 0.75 Eq. (12) gives:
Fig. 3a, b, and c show temporal variation of gain coefficient for various ratios of L1/L and D2/D1. As shown in Fig. 3a, gain coefficient of the pipelines in series increased with decreasing of L1/L for both of steady and unsteady flow. Also, this figure shows there is a significant difference between the values of Gseries and Gseries . Howus s ever, the values of Gseries at long times tend to reach to the values of us Gseries . Based on our results, from Fig. 3a, these times are 60, 350 and s 800 h for L1/L ¼ 0.75, 0.5 and 0.25, respectively. Moreover, it can be seen from Fig. 3a, b, and c, gain coefficient increased with increasing diameter ratio (D2/D1) for both steady and unsteady flow. But, when D2/D1 increases, the values of gain coefficient under unsteady flow converge to that of steady flow faster. For example, for length ratio (L1/L) equal to 0.25 and D2/ D1 ¼ 1.5, 2 and 2.5, the convergence time is 800, 400 and 300 h, respectively. As a result, the values of Gseries and Gseries for L1/ s us L ¼ 0.25 and D2/D1 ¼ 1.5, 2 and 2.5 converge to 1.72, 1.95 and 1.97, respectively. However, from these figures it is obvious that there is a difference between the Gseries and Gseries at initial time. For better s us understanding, we summarized the effects of diameter ratio and time on Gseries in a three dimensional figure for L1/L ¼ 0.25, 0.5 and us 0.75, respectively (Fig. 4).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 u 0:032DtL þ 0:0000157DtD33 þ 0:083D33 L u ¼u 16 u D33 u0:032DtL þ 0:0000157DtD43 þ 0:083D43 L þ 0 12 u 3 3 3 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 16 16 u D3 D3 B C u 1 2 þ @ A t 4 4 4 4 0:032DtL1 þ0:0000157DtD3 þ0:083D3 L1 1 1
0:032DtL1 þ0:0000157DtD3 þ0:083D3 L1 2 2
(26)
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Fig. 3. Effect of L1/L on gain coefficient for pipelines in series under steady and unsteady flow at different time for (a) D2/D1 ¼ 1.5, (b) D2/D1 ¼ 2 and (c) D2/D1 ¼ 2.5.
4.2. Investigation of gain coefficient for pipelines in parallel According to Fig. 2b, consider a D1 ¼ 10 in pipeline that is 20 miles long. If we place a D2 ¼ 15 in pipe parallel to share gas parallel transmission, from Eq. (14) for steady flow Gs increases to 3.95 or 295% increase in flow capacity. Based on our calculations, the
increase of gas capacity depends only on pipe diameter ratio (D2/ D1) for steady flow. As seen in Fig. 5, the values of gain coefficient for steady flow are 3.95, 7.35 and 12.50 for ratio of diameters equal to 1.5, 2 and 2.5, respectively. In other side, from Eq. (18) for unsteady flow we have:
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Fig. 4. Effects of time and diameter ratio on gain coefficient in series for unsteady flow: (a) L1/L ¼ 0.25, (b) L1/L ¼ 0.5 and (c) L1/L ¼ 0.75.
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Fig. 5. Effect of diameter ratio on gain coefficient for pipelines in parallel under steady and unsteady flow at different times.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16
parallel Gus
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16
10 3 0:64Dtþ3:3825ð104 ÞDtþ35:7636
15 3 þ 0:64Dtþ5:8079ð10 4 ÞDtþ61:40868 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16
(28)
10 3 0:64Dtþ3:3825ð104 ÞDtþ35:7636
The results from this equation are shown in Fig. 5. Based on Gparallel us
Fig. 5, for the parallel system, values of lead towards the values of gain coefficient for steady flow after 800, 2000, about
4000 h for D2/D1 ¼ 1.5, 2 and 2.5, respectively. Moreover, it is obvious that there is a significant difference between the values of Gparallel and Gparallel at initial time. Based on our calculations, the us s
Fig. 6. Effects of time and diameter ratio on gain coefficient in parallel for unsteady flow.
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437 parallel.
ratio of gain coefficients (Gus
) at initial time are 82%, Gparallel s 68% and 59% for D2/D1 ¼ 1.5, 2 and 2.5, respectively. For a better observation, as represented in Fig. 6, we summarized the effects of in a three dimensional figure. diameter ratio and time on Gparallel us
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4.3. Investigation of gain coefficient for pipelines in looped looped
According to Eq. (26), gain coefficient for unsteady flow (Gus ) is a function of D2/D3, fraction of looping (L1/L) and time. The effects of fraction of looping and pipe diameter ratio on gain coefficient versus time have been shown in Fig. 7. In this figure, a L ¼ 20 mi
Fig. 7. Effects of fraction of looping and diameter ratio on the gain coefficient in a looped system for unsteady flow at different times. (a) D2/D3 ¼ 1.5, (b) D2/D3 ¼ 2 and (c) D2/ D3 ¼ 2.5.
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H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
Fig. 8. Effects of fraction of looping and time on gain coefficient in a looped system for unsteady flow (a): D2/D3 ¼ 1.5, (b): D2/D3 ¼ 2, and (c) D2/D3 ¼ 2.5.
pipeline with D1 ¼ D3 ¼ 10 in was considered. As shown in Fig. 7a, b and c, for unsteady flow, there is a significant growth of gain coefficient for. For example, from Fig. 7a for L1/L > 0.75, the initial
values of gain coefficient increase from 1.76 to 3.49. Furthermore, Fig. 7a, b and c show that gain coefficient increases with rising diameter ratio (D2/D3). Based on our calculations, for L1/L ¼ 0.75,
H. Amani et al. / Journal of Natural Gas Science and Engineering 33 (2016) 427e437
gain coefficients at initial time are 1.76, 1.89 and 1.94 for D2/ D3 ¼ 1.5, 2 and 2.5 for unsteady flow, respectively. However, for a completely paralleled system (L1/L ¼ 1), at the initial time the gain coefficients are 3.25, 5 and 7.25 for D2/D3 ¼ 1.5, 2 and 2.5, respectively. However, there is a significant difference between the values of gain coefficient for parallel system and pipelines in looped. It can be seen from Fig. 7a that gain coefficient is gradually increases from 1.139 to 1.142 after 100 h for diameter ratio D2/D3 ¼ 1.5 and fraction of looping 0.25. According to Fig. 7a, gain coefficient has 0.3%, 1.1%, 3.9% and 37% variation for fraction of looping L1/ L ¼ 0.25, 0.5, 0.75 and 1 during 100 h, respectively. Also as seen in Fig. 7b and c, for fraction of looping less than 0.75, maximum variation of gain coefficient will be 3.3% and 2.2% after 100 h, respectively. Also, for parallel systems, gain coefficient represents 106% and 205% variation for D2/D3 ¼ 2 and 2.5, respectively. Therefore, based on these results, temporal variation of gain coefficient is only considerable for parallel system. Finally, the effects of diameter ratio and time on gain coefficient in a three dimensional plot have been summarized in Fig. 8 for a looped system. In a general conclusion, according to our results, there are significant differences between values of gain coefficient in pipelines networks under steady and unsteady flows. However, as the values of gas flow rate for unsteady condition are more accurate than those of the steady flow, therefore, the lack of attention to this subject in pipeline network leads to a considerable error for engineers. For example, from Fig. 5 for a parallel system, more than 5 months is required to the values of gain coefficient under unsteady flow tend towards the values of steady flow. Furthermore, our results show that use of parallel systems introduces significant advantages in comparison with series and looped systems. Although using parallel pipeline yields notable increase of gas capacity, but, it needs a large amount of cost. Therefore, engineers must optimize fraction of looped pipeline to reach a sufficient growth of flow capacity by considering market consumption and economical support. However, this paper attempted to increase the knowledge of engineers for better prediction of gas flow rate in their models. 5. Conclusion In this paper, we developed an analytical model based on unsteady-state Weymouth equation without neglecting any terms in the fundamental governing momentum equations to study gas flow rate for pipeline networks in series, parallel and looped systems. Also, in this work, a comparative study carried out between these systems. First, we introduced “Gain coefficient” to find out about scale of gas flow increases in pipelines. Our developments showed functional relationships between time, diameter ratio, length ratio of pipeline network and gain coefficient. Our results also showed gain coefficient increases using networks for both steady and unsteady flows with a preference to parallel system. Although, for all systems a significant difference between the values of gain coefficient in pipeline networks for unsteady and steady flows was seen, but, it is observed that all unsteady-state processes tend towards steady-state with time. However, according to our research, time plays an important role in the prediction of more accurate values of gain coefficient for all systems. These differences especially for parallel system were more clearly observed. As gas engineers need to an accurate calculation of flow rate, the
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lack of attention to applying unsteady models in their calculation leads to a considerable error. This paper as a basic report tried to help engineers of gas industry to design more accurate gas pipeline networks.
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