Chemical Engineering Journal 76 (2000) 169–177

Transient analysis of isothermal gas flow in pipeline network

S.L. Ke, H.C. Ti ∗
Department of Chemical and Environmental Engineering, National University of Singapore, Singapore
Received 11 January 1999; received in revised form 6 September 1999; accepted 21 September 1999

Abstract
Conventional methods for transient analysis of pipeline network are normally applied to find the numerical solution of the two partial
differential equations (continuity and momentum), which are complex and cumbersome. Following the success of the steady state analysis
of pipeline network, this study extends the usage of the electrical analogy method by combining resistance with the theoretically derived
models of capacitance and inductance. This method leads to a set of first-order ordinary differential equations for transient analysis of
isothermal gas flows in pipeline network. Solving the proposed first-order ordinary differential equation is definitely much simpler than
solving the set of partial differential equations. The computational advantages of the present method are demonstrated by comparing them
with the conventional methods when applied to a range of pipe network simulation examples. ©2000 Elsevier Science S.A. All rights
reserved.
Keywords: Electrical analogy; Gas pipeline; Capacitance, inductance and resistance; Steady and transient analysis; Isothermal flows

1. Introduction In which ρ is the fluid density, v(x, t) the fluid velocity,

P(x, t) is the pressure, ϕ f is Fanning friction factor, D is the
The analysis of flows and pressure drops in piping systems pipe diameter, g is the gravitational acceleration constant,
has been studied by many workers and is usually based upon and θ is the angle between the horizon and the longitudinal
the consideration of steady state conditions. However, the direction of the pipe.
steady state analysis of pipeline network is less applicable, Many algorithms such as finite difference methods or
in the design of actual transmission systems, as unsteady method of characteristic (MOC) have been used to solve the
(transient) state is more often encountered. above partial differential equations [1,8,13]. These methods
The analysis of unsteady state is much more difficult than have been shown to be efficient in solving unsteady flow
that of steady state. The reason for this difficulty is that equations. However, it is not the objective of the present
the unsteady state system is typified by variables which are study to follow the conventional route for the analysis of
functions of time and space (or position). In contrast, for transient flow in pipe distribution system, but to apply elec-
steady state analysis, the variable in the system is only a trical analogies to simulate the same transient flow problem.
function of space. The introduction of the concept of time
variable adds on a new dimension to the mathematical model
of the transient flow in a pipe distribution system, and results 2. Electrical analogy
in computational difficulty. The mathematical model of pipe
distribution system is derived from the two conservation The analogy between fluid network and electrical network
equations (mass and momentum), which have the following has been successfully applied in the simulation of steady
forms [8]: state pipeline networks by many workers [2,3,11]. From the
∂ρ ∂(ρv) electrical circuit theory, the three basic elements which relate
+ =0 (1a) to voltage and current are viz. resistance, capacitance and
∂t ∂x
inductance. Based on the analogy of voltage and current in
∂(ρv) ∂(ρv 2 ) ∂P hϕ i an electrical circuit network with that of pressure drop and
f
+ + + 2ρv 2 + ρg sinθ = 0 (1b) flow in the fluid network, all the three basic elements should
∂t ∂x ∂x D
also be present in the fluid distribution systems [5–7,10].
∗ Corresponding author. Fax: +65-779-1936 The resistance effect of a pipeline, which has been stud-
E-mail address: chetihc@nus.eng.sg (H.C. Ti) ied in the steady state analysis of pipeline network, is due to

1385-8947/00/$ – see front matter ©2000 Elsevier Science S.A. All rights reserved.
PII: S 1 3 8 5 - 8 9 4 7 ( 9 9 ) 0 0 1 2 2 - 9
170 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177

∂(ρv) ∂P hϕ i
several factors, such as the roughness and geometric proper- + + 2ρv 2
f
=0 (5)
ties of a pipe, the viscosity of the fluid and the flow rate. The ∂t ∂x D
capacitance effect of a pipeline can be directly attributed to The discharge Q may be written as
the compressibility of the fluid. The inductance effect of a
pipeline is believed to be due to the kinetic energy of the Q = vA (6)
fluid [12]. Substituting Eq. (6) into Eqs. (4) and (5) gives
∂P ρa 2 ∂Q
+ =0 (7)
3. Derivation of models for basic elements ∂t A ∂x
∂Q A ∂P 2ϕf Q|Q|
The models of resistance and capacitance have been in- + + =0 (8)
∂t ρ ∂x DA
troduced in a previous work [10]. However, it is intended
that the models of the three basic elements be derived from Rearranging Eqs. (7) and (8), transient flow through the
Eqs. (1a) and (1b) directly, such that the formulation of the horizontal pipe can be represented by the following set of
models is explicated theoretically. equations:
Due to pressure change involved in a transient process, ∂Q A ∂P
the control volume may be compressed or expanded, so the =− 2 (9)
∂x ρa ∂t
Continuity equation, that is Eq. (1a), can be changed to the
∂P ρ ∂Q 2ϕf ρQ|Q|
following form [1]: =− − (10)
∂x A ∂t DA2
∂P ∂P ∂v
+v + ρa 2 =0 (2) Taking into account that
∂t ∂x ∂x
where a is the acoustical wave velocity. Mass flow rate = ρQ = ρn Qn
The Momentum equation, that is Eq. (1b), may be rear- where the subscript n refers to quantities at standard condi-
ranged by multiplying A, the cross-sectional area of pipe to tions of pressure Pn ∼
= 0.1 MPa and temperature Tn = 288 K.
give The governing equations become
∂(ρvA) ∂(ρvAv) ∂(P A) ∂Qn A ∂P
+ + =− (11)
∂t ∂x h i ∂x ∂x ρn a 2 ∂t
ϕf
+ 2ρAv 2 + ρAg sinθ = 0 ∂P ρn ∂Qn 2ϕf ρn Qn |Q|
D =− − (12)
∂x A ∂t DA2
The first two terms of the above may be expanded as follows:
Based on Eq. (11), we obtain
∂(ρvA) ∂(ρvAv) ∂(ρA) ∂v ∂(ρvA)
+ =v + ρA +v Qn (x, t) − Qn (x + 1x, t)
∂t ∂x ∂t ∂t ∂x   
∂(v) ∂Qn A1x ∂P
+ρvA = QC = − 1x = (13)
∂x ∂x ρn a 2 ∂t
or where QC is the change in flow rate in a pipe dependent on
 
∂(ρvA) ∂(ρvAv) ∂(ρA) ∂(ρvA) compressibility of the fluid.
+ =v +
∂t ∂x ∂t ∂x Therefore, for the consideration of capacitance effect in
∂v ∂(v) a pipeline network, the relationship between pressure and
+ρA + ρvA flow rate with capacitance can be made analogous to voltage
∂t ∂x
and current relationship across an electric capacitor in the
From the Continuity equation Eq. (1a), the term in bracket following form:
vanishes; therefore, the Momentum equation becomes
dV
∂(vA) ∂(vA) ∂(P A) hϕ i Nodal approach J =G (14)
f dt
ρ + ρv + + 2ρAv 2 Z
∂t ∂x ∂x D 1
+ ρAg sinθ = 0 (3) Mesh approach V = J dt (15)
G
In most engineering applications, the convective acceleration where G is the capacitance, reflecting the capacitance effect
terms, v [∂v/∂x] ; v [∂P /∂x] and the slope term are very in a pipeline. Comparing Eq. (14) with Eq. (13), the capac-
small compared to the other terms in the above equations itance has the following form:
and may be neglected [1]. After dropping these terms from
Eqs. (2) and (3), the following are obtained: A1x Vp
G= = (16)
ρn a 2 ρn a 2
∂P ∂v
+ ρa 2 =0 (4) where Vp is the volume within the pipe.
∂t ∂x
 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177 171

For gas distributing system under isothermal conditions, flow rate is analogous to the Ohm’s law and can be repre-
we can write the equation of state in the form [8] sented by the following equations:
P zRg T Mesh approach V = ZJ (25)
a2 = = (17)
ρ MW
Nodal approach J = YV (26)
Substituting Eq. (17) into Eq. (16), we get
The mathematical model for resistance in a pipeline network
Vp MW
G= (18) depends on the various equations describing the friction fac-
ρn zRg T tor relationship between pressure drop and flow rate. Wey-
Similarly, from Eq. (12), we obtain mouth equation has been used in the previous work [10] on
Osiadacz’s sample networks [8]. Other equations used in the
∂P ρn 1x ∂Qn present study are listed as follows [8]:
P (x, t) − P (x + 1x, t) = −1P = − 1x =
∂x A ∂t Lacey’s equation — (for the pressure range of 0–75 mbar
2ϕf ρn Qn |Q|1x gauge):
+ (19)  
DA2 12
ϕf = 0.0044 1 + (27a)
We assume 0.276D
ρn 1x ∂Qn
−1PL = (20) The Polyflo equation — (for the pressure range of
A ∂t 0.75–7.0 bar gauge):
and s
1
2ϕf ρn Qn |Q|1x = 5.338Re0.076 η (27b)
−1PR = (21) ϕf
DA2
where (−1PL ) is the pressure drop required to accelerate a The Panhandle ‘A’ equation — (for the pressure range above
given mass of fluid [6], which is due to inductance effect and 7.0 bar gauge):
is proportional to the rate of change of flow; and (−1PR ) s
is the pressure drop due to frictional resistance to flow. It 1
= 6.872Re0.073 η (27c)
is noted that Eq. (21) is the same as the Darcy–Weisbach ϕf
formula. Therefore, the total pressure drop of a pipe is the
sum of the pressure drop due to resistance effect and the where η is the efficiency factor accounting for the additional
pressure drop due to inductance effect across the pipe. frictional or drag losses other than losses due to viscous
For the inductance effect in a pipeline network, based on forces.
Eq. (20) and the analogy between electrical and hydraulic
flow, the analogous relationships for inductance relating to
pressure drop and flow rate have the following forms, which 4. Derivation of equations for the transformation
are analogous to voltage and current relationship across an approach
electric inductor:
In order to derive the mathematical model for transient
dJ
Mesh approach V =L (22) flow analysis, some of the fundamental assumptions used in
dt the conventional methods are also applied. These assump-
Z
1 tions are (i) one-dimensional and isothermal flow, and (ii)
Nodal approach J = V dt (23)
L applying the steady state friction factor equation to transient
flow [1,8].
where L is the inductance, reflecting the inductance effect As mentioned above, the change of flow rate across a
in a pipe. Comparing Eq. (22) with Eq. (20), the inductance pipe with time is due to compressibility of the fluid, i.e.
has the following form: the capacitance effect. The pressure drop of a pipe is the
ρn 1x sum of the pressure drop due to resistance effect and the
L= (24) pressure drop due to inductance effect across the pipe. A
A
typical branch of fluid network with transient flow can be
The mathematical representation of the resistance effect in represented as an electrical circuit and be visualised in Fig. 1.
a pipeline network has been employed in the steady state The resistance and inductance are proposed to be connected
analysis. The adaptability of this model has already been in series, and the capacitance and resistance are proposed to
demonstrated by many workers [2,3,10,11]. It can be ex- be connected in parallel.
pressed either in the forms of impedance Z or admittance Y, Based on Fig. 1, the following relationships can be drawn:
which correspond to the mesh or nodal approach. The rela-
tionship between the resistance with the pressure drop and V = V1 + V2 (28)
172 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177

Substituting
V1b = A·s
b V1s (38)
into Eq. (36), the following equation is obtained:
 
bb ·s dV1s
J s = As·b Y bb A·s V
b 1s + G A b
dt
bb ·s dV1s
= As·b Y bb A·s
b V1s + A·b G Ab
s
(39)
dt
When Eqs. (39) and (37) are extended to open path and
closed path frameworks, they become
 o  o   
J A·b bb ·o ·c V1o
= Y [Ab |Ab ]
Jc Ac·b V1c
 o  
A·b bb ·o ·c dV1o /dt
+ G [Ab |Ab ] (40)
Ac·b dV1c /dt
  " ·b #  o 
V2o Co dJ /dt
= L [C b
|C
bb ·o ·c
b
] (41)
V2c Cc·b dJ c /dt

As
V1o = Eo + e1o = Co·b Eb + e1o (42)

V1c = Ec = Cc·b Eb (43)

for an invariant pressure source Eb , we obtain the following
relationships:
dV1o de1o
= (44)
dt dt
dV1c
Fig. 1. Composite branch of a network. =0 (45)
dt
V1 = E + e1 (29) Expanding Eqs. (40) and (41), and incorporating Eqs. (44)
V2 = e2 (30) and (45), we have

J =I +i (31) J o = Ao·b Y bb A·o ·b o bb ·c ·b

b (Co Eb + e1o ) + A·b Y Ab Cc Eb

J = J1 + J2 de1o
(32) +Ao·b Gbb A·o
b (46)
dt
For the nodal approach [14]
J1b = Y bb V1b (33) J c = Ac·b Y bb A·o ·b c bb ·c ·b
b (Co Eb + e1o ) + A·b Y Ab Cc Eb
dV1b de1o
J2b = Gbb (34) +Ac·b Gbb A·o
b (47)
dt dt
Based on Eq. (22), dJ o dJ c
e2o = Co·b Lbb C·o
b
+ Co·b Lbb C·cb (48)
dJ b dt dt
V2b = Lbb (35) Rearranging Eq. (46), we have
dt
Applying the transformation theory [14], de1o
  = [Ao·b Gbb A·o −1 o o bb ·o ·b
b ] [J − A·b Y Ab (Co Eb + e1o )
dV1b dt
J s = As·b J b = As·b (J1b + J2b ) = As·b Y bb V1b + Gbb +Ao·b Y bb A·c ·b
dt b Cc Eb ] (49)
(36) Eqs. (47) to (49) are the governing equations for transient
dJ s pipe flow system with a constant pressure source; and they
V2s = Cs·b Lbb C·sb (37) are a set of first-order ordinary differential equations. Hence,
dt
 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177 173

the transient pipe flow problem which is ordinarily governed

by the set of two partial differential equations can now be
solved by a set of first-order ordinary differential equations
which is much easier to handle. Nevertheless, these equa-
tions can not be solved analytically owing to the complicated
relationships involved in the network problem.
For a network with known topology, tensors Ao·b ,
Ab , Ac·b , A·c
·o ·b
b , C·o , C·o , C·c , and Cc and inductance are in-
b b b

dependent of time and can be determined easily; while the

boundary condition of admittance and capacitance can be
calculated based on the results of steady state analysis.
Therefore, the dynamic response of e1o can be determined
through the solution of a series of ordinary differential equa-
tions as Eq. (49). When e1o is found, Jc can be calculated
from Eq. (47). Then, e2o is calculated from Eq. (48). Using
eo and Jc , the dynamic change of branch flow, branch pres-
sure drop and nodal pressure of network at any given time
can be found through the application of the transformation
techniques.

5. Computational scheme

Once the topology of the pipeline network is ascertained,

the steady state analysis of pipeline network can be carried
out to find the branch flow rate and nodal pressure, which
are then used as the initial value to solve the ordinary differ-
ential equation. The steady state analysis of a pipe network
can be based either on the mesh method or its dualistic nodal
method. Once the steady state analysis is completed, the
transient calculation is carried out by using the general solu-
tion method such as the fourth-order Runge–Kutta method Fig. 2. Computational flow chart.
for the first-order ordinary differential equations. The de-
tailed computation scheme is outlined in Fig. 2. The com-
puter program is written in Microsoft FORTRAN and runs
on a Pentium/75 PC.

6. Sample networks

The mathematical model derived in the present study was

tested on three simple gas networks, two of which have been
analysed by Osiadacz [8] and one by London Research Sta-
tion (LRS) [4]. The first sample network is a straight pipeline
(l = 105 m) with a uniform diameter of 0.6 m. The upstream
pressure is maintained at a constant level of 5 MPa. The
flow rate is depicted in Fig. 3. The operating temperature is
278 K; the density of the fluid is 0.73 kg m−3 ; and the spe- Fig. 3. Change of flow with time (boundary condition).
cific gravity of the fluid is 0.6. For the purpose of analysing,
the pipeline is cut into five segments. Thus, the capacitance Node 1 is the pressure source with a constant pressure of
of each segment is five times the capacitance of the pipe. 5 MPa. The loads at Nodes 2 and 3 are known functions of
Osiadacz’s result is presented in Fig. 4. time, which vary according to the curves depicted in Fig. 6.
The second example is a simple network with three nodes In this example, each pipe is cut into four segments. Hence,
and three branches forming one mesh as shown in Fig. 5. the capacitance of each segment is equal to the capacitance
The physical data are shown in Table 1. of the pipe multiplied by 4 [15].
174 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177

Fig. 4. Change of pressure at the outlet for Sample Network 1.

Fig. 6. Changes of load at Nodes 2 and 3 for Sample Network 2.

Table 2
Pipe data of the third sample network
Length Diameter Initial flow Inlet
(mile) (in.) rate (MSCFH) pressure

LRS 1 80 18 1500 350 psig

LRS 2 20 12 800 180 psig
LRS 3 6 14 300 25 psig
LRS 4 3 14 80 2 psig
LRS 5 0.6 4 4 20 in. water gauge

The compressibility factor in this study is determined by

using the following correlation:
1
z= (50)
Fig. 5. Sample Network 2. 1 + αPave
where α is obtained from linear interpolation of the constant
Table 1 taken from Gas engineers’ Handbook [9] and Pave is the
Pipe data of the second sample networka
average pressure of each pipe section.
Pipe From To Diameter (m) Length (m)

1 1 3 0.6 80000
2 1 2 0.6 90000 7. Results and discussion
3 2 3 0.6 100000
aρ = 0.7165 kg m−3 ; S = 0.6; T = 278 K; tmax = 86,400 s.
The dynamic pressure response at the outlet of the first
sample network as simulated by the derived model is pre-
sented in Fig. 4. The computation time spent for solving the
The last sample network is a straight pipe having a uni- examples ranges from approximately 10 s to 5 min, depend-
form diameter. This sample network was used by LRS in ing on the different time steps used.
demonstrating the versatility of their program PAN. Ini- Simulation results of the second sample network using
tially, the pipe was in a steady state; the demand then in- the present model is compared with the results from the
creased by 50% in a step change and was held constant literature. These are shown in Figs. 7 and 8.
thereafter. The pressure at the inlet, P1 , was held constant, It can be seen that the results obtained by using the present
while the pressure at the outlet, P2 , fell to a new steady state method are comparable with those in the literature. The de-
value. In this case, each pipe is cut into four segments and viation is less than 9% for the first example and 1% for the
the capacitance of each segment is four times the capaci- second example. When the last sample network was anal-
tance of the pipe. The analysis was carried out for five dif- ysed, it was found that the average simulated pressure re-
ferent pressure ranges which corresponded to typical pres- sponse was higher than the LRS result by around an average
sure range for low, medium and high. The operating condi- of 10% (the maximum error is around 27% for the case of
tions are listed in Table 2. The LRS results are depicted in lowest pressure). This result shows that the response was
Figs. 9–13. slower, due to the high value of capacitance. Upon further
 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177 175

Fig. 7. The variation of pressure at Node 2. Fig. 9. Comparison of simulated pressure with the LRS result
(P1 = 350 psig).

Fig. 8. The variation of pressure at Node 3.

investigation, it was found that, when the temperature was Fig. 10. Comparison of simulated pressure with the LRS result
decreased, the capacitance became larger and the response (P1 = 180 psig).
slowed down. As most pipes were buried underground and
the pipe wall temperature subjected to daily temperature
change, the assumption of isothermal condition made in the
derivation of Eq. (18) might not satisfy the LRS conditions.
In order to avoid a more complicated model of considering
the conservation of energy, an efficiency factor is adopted in
the present study to consider the deviation from the assump-
tion of isothermal flow condition. After numerous trials, a
factor of 0.65 was selected based on the first case of LRS
result (P1 = 350 psig). It was found that the same factor of
0.65 would fit the other four cases of LRS with a maximum
error of 7%. The results are depicted in Figs. 9–13.
Convergence is normally a common problem in the anal-
ysis of pipeline network. However, in this present study, it-
eration process is only required in the part of steady state
analysis; and the solution of transient analysis can be ob-
tained without any convergence difficulty. Moreover, the
steady state analysis was carried out using the robust trans-
formation method [3,11], and hence, convergence problem Fig. 11. Comparison of simulated pressure with the LRS result
was manageable. (P1 = 25 psig).
176 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177

fects. Hence, when the boundary conditions of a gas pipeline

network do not change rapidly or the capacity of the pipe is
relatively large, the effect of inductance can be neglected .
Nevertheless, further study on the inductance effect for hy-
draulic transient analysis is recommended.

8. Conclusion

A new mathematical model based on electrical analogy

and transformation theory is developed for transient analy-
sis of isothermal gas flow in pipe networks. The transient
behaviour of a pipe, conventionally taking the form of a
set of second-order partial differential equations, can be ex-
pressed by a set of first-order ordinary differential equations
using the new model. The solutions computed using the new
Fig. 12. Comparison of simulated pressure with the LRS result
(P1 = 2 psig). method are compatible with those using the conventional
methods. The new method is simple, straight forward and
without convergence problem. It is easy to implement on a
small personal computer. Comparing with the conventional
methods, this new method shows a promising feature in sav-
ing computational efforts and may be employed as a pow-
erful means for pipe network design and control.

9. Nomenclature

A transformation tensor or cross-sectional area

C matrix or transformation tensor used in the
mesh approach
D diameter of a pipe
E covariant tensor for pressure source across a
branch or path
e covariant tensor for pressure drop across a branch
Fig. 13. Comparison of simulated pressure with the LRS result (P1 = 20 or path
i.w.g.). G contravariant tensor for capacitance used in
the nodal approach
The present study also investigated the effect of space and I contravariant tensor for flow due to external
time discretization on the accuracy of solution. This is done input–output on a branch or path
by comparing the simulation results of the second sample i contravariant tensor for flow due to other cause
network when all pipes were cut into four and eight equal on a branch or path
parts. It is found that discretization of pipe section does not J contravariant tensor for total flow on a
affect the precision of simulation result for the pipe network branch or path
examined. It is also found that the variation of time step has L contravariant tensor for inductance used in
negligible effect on the simulation result. On the other hand, the mesh approach
both space and time discretization have noticeable effect on l length of a pipe
the computation effort needed, i.e. the finer the space or time MW molecular weight of fluid
steps, the more is the computation time required. For a node 1P pressure drop across a pipe
with a sudden change of high demand, a smaller time space P1 , P2 pressure at the nodes
is recommended. Pave average pressure of a pipe
The inductance effect on the solution of gas transient flow Qn volumetric flow rate of fluid at standard state
is also investigated. Simulation results of pipe flow with the Re Reynolds number
inductance effect considered were compared to that when Rg gas constant
inductance effect was neglected. It is found that the effect S specific gravity of gas
of inductance which corresponds to kinetic energy of gas is T absolute temperature (K)
negligible on the simulation results for the gas pipe networks V contravariant tensor for pressure drop due to
examined compared to the resistance and capacitance ef- impedance, where V = E + e
 S.L. Ke, H.C. Ti / Chemical Engineering Journal 76 (2000) 169–177 177

VP volume within pipe [2] B. Gay, P. Middleton, The solution of pipe network problems, Chem.
v fluid velocity Eng. Sci. 26 (1971) 109–123.
Y contravariant tensor for admittance used in [3] B. Gay, P.E. Preece, Matrix methods for the solution of fluid network
problems Part I. Mesh methods, Trans. Inst. Chem. Engrs. 53 (1975)
the nodal approach 12–15.
Z contravariant tensor for impedance used in [4] London Research Station, Proceedings of PAN Presentation, British
the mesh approach Gas Corporation, 27 February 1974.
z compressibility factor [5] S.D. Millston, Electric analogies for hydraulic analysis Part 1. System
η efficiency factor components, Machine Design December (1952) 185–189.
[6] S.D. Millston, Electric analogies for hydraulic analysis Part 2. Tubing
ϕf Fanning friction factor characteristics, Machine Design January (1953) 166–170.
ρ density of fluid [7] G. Murphy, D.J. Shippy, H.L. Luo, Engineering Analogies, Iowa
State University Press, IA, 1963.
9.1. Index symbols [8] A.J. Osiadacz, Simulation and Analysis of Gas Networks, E. & F.N.
Spon, London, 1987.
[9] C.G. Segeler, M.D. Ringler, E.M. Kafka, Gas Engineers’ Handbook,
b index used in tensor form, indicating that AGA, NY, 1969.
the tensor is in primitive framework [10] W.Q. Tao, H.C. Ti, Transient analysis of gas pipeline network, Chem.
c index used in tensor form, indicating that Eng. J. 69 (1997) 47–52.
[11] H.C. Ti, K.G. Koh, H.M. Tar, Steady analysis of gas flow in a pipeline
the tensor is in closed path framework
network, in: Proceedings of the 4th Asean Council on Petroleum
o index used in tensor form, indicating that (ASCOPE) Conference & Exhibition, November 1989, Singapore,
the tensor is in open path framework pp. 484–489.
s index used in tensor form, indicating that [12] J.F. Wilkinson, D.V. Holliday, E.H. Bakey, K.W. Hannah, Transient
the tensor is in orthogonal framework Flow in Natural Gas Transmission Systems, AGA, NY, 1965.
[13] E.B. Wylie, V.L. Streeter, Fluid Transients, McGraw-Hill, New York,
1978.
[14] G. Kron, Tensor Analysis of Networks, MacDonald, London, 1965.
References
[15] R.G. Meadows, Electric Network Analysis, The Athlone Press,
London, 1972.
[1] M.H. Chaudhry, Applied Hydraulic Transients, Van Nostrand
Reinhold, New York, 1987.