1. Optimization of Pipe Sizes
for Distribution Gas Network
Design
Andrzej J. Osiadacz
Warsaw University of Technology
and
Marcin Gbrecki
Regional Gas Dispatching Center, Warsaw

2. 1. Introduction
Historically, most interest in pipe networks has focused on the development of efficient algorithms
for the analysis of flow, and there are now very useful and efficient computer packages available
for simulation of existing and proposed new schemes. SimNet [6], for example, is widely used in
Poland for analysis and simulation of gas distribution systems.
In contrast, there has been comparatively little research interest into the development of
methodologies aimed at optimising the design of pipe networks. There are very few computer
packages available for commercial use that help the designer to produce truly optimal pipe
network designs. The current available methods for designing networks can be divided into three
groups:
- heuristic methods
- methods which assume a continuous range of diameters is available
- discrete optimisation methods
2. Heuristic methods
Although, these methods have developed through the years from an appreciation of what usually
constitutes a good and economic design, there is no guarantee that the designs produced will in
any sense be optimal. Even for a simple tree-like network there are a very large number of
possible designs that provide feasible solutions. It can be seen that the chances of hitting on the
best solution are very small.
In PILOT method [lo] the pipes are ordered for the tree definition according to the shortest
distance from the source. The program then proceeds in the usual way up to the point of balancing
the flows. The pipe resistances are evaluated, then recalculated using flows obtained by Hardy-
Cross loop balancing method and corresponding diameters are selected. Another Hardy-Cross to
balance the network is performed determining flows and pressures through the system. The
process of scaling resistances and performing Hardy-Cross is repeated until none of the pressures
falls below the minimum design pressure. Once the basic solution has been found the program
aims at further economic improvements.
In FP6 method [9], after all pipes have been set to minimum diameter the loops are defined and
initial flows assigned Hardy-Cross balancing flows. Pipes are then upgraded on maximum
-l-

3. flowpath from source outwards the criterion being the cost benefit. Several flowpaths are
upgraded simultaneously.
Areas of influence are assigned to each governor in CONGAS method [lo].
The tree system is designed for each area with minimum pressure at ends and even pressure drop
loop closing pipes are sized to a minimum and Hardy-Cross flow balancing performed. The
theoretical diameters are substituted by actual diameters and network is rebalanced.
3. Continuous methods
One way of solving the design problem is to initially suppose that any size of diameter is possible.
Doing this allows continuous optimization methods to be employed. The result of such
optimization will be a set of diameters which needs to be corrected to available diameter sizes.
SUMT is an algorithm for sequential unconstrained minimization technique and is a method
devised and programmed by G. Boyne in [2].
In this method the cost is assumed to be given by a function of the form :
C(Q)=c(a+bDp)L (1)
where:
D- is the vector of pipe diameters
L - is the pipe length
a, b, p - are cost parameters
The program minimizes a sequence of functions, where the minimum of one is used as the starting
point in the research for the minimum of the next. The functions to be minimized are:
F, = c(a+bD”)L+R,, c ’
nodes (Pi -Pimu)
(2)
where:
Pi, Pimin - are th e pressure and allowed pressure at node i and pi depends on D
R, - is a weight such that R,+I = R, /I 0
The sequential method works asfollows.
First the diametersD,, minimising Fo are found. Then D, is used as a starting point in the search__ -
for the minimum of Fl. More generally, D. is used as a starting point in the minimization of Fi+l.1
-2-

4. Each of these minimizations is carried out using the conjugate gradient method. The procedure
stops when Di+,is sufficiently close toD, . (Note that R, decreases rapidly with n so the last
solutions are effectively minimising cost).
This process yields a local optimum. The program now randomly generates a new set of diameters
and uses these to begin the whole procedure again and locates another local minimum. This
continues until a “sufficient” number of local optima are found and the cheapest of these is the
desired solution.
In addition this program uses the parametric technique to ,,round” up each local optimum.
Watanatada [l l] has used augmented Lagrangian methods to obtain optimised designs of water
distribution systems. The model he used is asfollows:
Di - diameter of the i - th pipe i E[1,NP]
pj - pressure atj - th node j E [~,NN]
Qk- flow out of the k - th source k E[l,NS]
&tin i, pmirti - the corresponding minimum allowed pressures
The flow equation, relating pressure drop and flow is used to eliminate pipe flows as independent
variables.
New variables are now introduced:
D; = Dmin,+ X; i=l,NP
Pi = Pmini+ ‘Z i=NP+l,NN+NP
Qi = X” i=NN+NP+l,NN+NP+NS
- this last equation means that the total flow from each source must be non-negative
The problem can now be formulated as one of minimizing the cost C(X) subject to the total flow
at each node being zero -
T(X)= CQ-d, =0 i=I,NN (3)
Jlowsat
nodei
- di is the demand at each node, except for sources where it is the total flow Qi
The method used in [ 1l] to solve this problem is to use the augmented Lagrangian function
-3-

5. This function is successively minimized with r = 0,I,2,. ... Initially, EF) = 0, the value of .!,“I
is found from that of Ef) and the corresponding minimising x value, x”‘, by the relationship:
(5)
The value xtr) should then converge to a local minimum.
In [4] the use of the generalised reduced gradient method (GRG) in the design of expansion in
water distribution networks is described. In the optimization model the objective function includes
capital cost of pipelines and operating cost of pumping stations. The main constraints are a set of
non-linear hydraulic equations, upper and low bounds of diameters, and a minimum requirement
of pressure head at nodes. A modified Newton - Raphson technique is use in the hydraulic
simulation to accelerate calculation and the convergence and consistency of the algorithm is
investigated.
For the expansion of an existing water distribution network, consisting of m branches and ~2nodes
the objective function is asfollows:
min COST= K,xLiDe +K2xQiHi (6)
icp, ia
subject to:
g@&,B) = Q
(7)0, 2 Dmini, i Ey,,
Hei,,-
where:
Q - vector of diameters, Q = [D, ,D, ,...D,,,]’
Q - vector ofloads, Q = [Q,,Q,...Q,]'- -
H - vector of pressure heads at nodes, H = [H, ,H, ...HnIT
Dmin- permissible minimum diameter
Hm, - permissible minimum nodal head
s(LL QZ) - vector of functions used in simulation- -
-4-

6. L - length of pipe
b, - the set of new branches
p,s - the set of source nodes with pumping station
Kl - a constant related to cost of pipe (weighting factor)
e - exponent related to pipe constant, e E [I,21
Kz - a constant dependent on the unit cost of energy
In the optimization model, the decision variables are the diameters of new pipes Di, i Ebn, the
pressure head H will be solved in the simulation (eq.(7)) if the diameter Q and loads e are
known. H can be called state variable.
There are two problems associated with these methods. The first is how to correct from the ideal
diameter to available diameter sizes. The figures given in [2] suggest that the effects of such
,,rounding” can be made acceptably small. The second problem is that of local optima. A local
optimum is an acceptable network such that a small perturbation of the diameters either causes the
pressure to fall below the minimum or leads to an increased cost. Continuous design methods try
to find local optima because it is known that one of these will be the cheapest design.
4. Discrete methods
Rothfarb at al [8] give a simple and effective method for finding the cheapest design when the
network has no loops.
This method is based on the fact that when a diameter is assigned to a pipe then pressure drop
across that pipe and the cost of the pipe is determined because in a tree the flows are known. Pipes
may be joined in a tree network in one of the two ways - either in series or in a “V”.
(Fig. 1)
1 2
If two pipes are linked in series (Fig. 1) then the list of possible diameter assignments can be
reduced considerably simply by eliminating any pair of diameters of there is another pair which
produces a lower pressure drop at lower cost. This is because if the diameters with the higher
-5-

7. pressure drop are part of a feasible network then they can always be replaced by the pair with
lower pressure drop thereby reducing costs.
Pipes may also be joined in a “V” (see Fig.2).
(Fig.2)
<
This situation differs from the first only in the way the pressure drop is calculated. For any
diameter assignment to these pipes the pressure drop across the pair may be taken as a maximum
of the individual pressure drops. Therefore, provided these pipes are at the downstream end of a
network, any pair of diameters can be eliminated if there is another pair of diameters which, when
assigned to these pipes, produces a lower pressure drop at lower cost. If there are n possible
diameters for each pipe then this “V” elimination will produce at most 2n acceptable pairs. These
methods can be easily extended to networks of pipes. If we have a list of pressure drops and their
costs for arbitrary networks A and B which are joined at node 1 (Fig. 3) then the maximum
pressure drop across the combined system is the maximum of the pressure drop across A and the
pressure drop across B. Again any pressure drop across the combined system can be eliminated if
there is another pressure drop both lower and cheaper .
(Fig.3)
1
Likewise pipe C may be joined to network D (Fig. 4). Again, a list of the system pressures could
be drawn up consisting of pressures which are the obtained by summing one from the C list with
one from the D list. Then these pressures for which there is a lower and cheaper system pressure
-6-

8. would be removed from this list to give a list of pressures (and costs) for the C-D system. This
process ends when a list of acceptable pressures has been built up for the whole system. The
cheapest pressure on this list consistent with the design requirements is then chosen and the
associated diameters found.
(Fig.4)
c
Beale [l] gives a method of optimization which removes non-linearities in the constraints by
introducing piecewise linear approximations. However, the problem as formulated by Beale can
give as solutions designs which are infeasible.
The trouble with his method is the assumption that in the optimization procedure the flow
equation :
can be replaced by the inequality
If, when the problem is solved, there is strict inequality in this expression then the values ofp and
e have no physical significance since they do not satis@ the flow equations. If the diameters we
allowed to be continuous then this problem would not arise because there would always be
equality (inequality implies that k(D) can be increased by a small amount while retaining the
values of e and y thereby giving a cheaper but still feasible solution). In the case of discrete
diameters strict inequality can occur, and as the example shows (see [l]) this produces a set of
pipes satisfying Beales conditions but in some casesdoes violate the true pressure requirements.
-7-

9. 5. Proposed method
In order to optimize a network design it is first necessary to formulate a completely defined
mathematical model of the system. The system has to be optimised from a cost view point and thus
a relationship between the pipe cost and the system variable (i.e. diameter) must be specified. The
objective function which links pipeline cost to diameter for distribution gas network is:
f(Q) = 2.05.[4 L, K LmlT. Di (21) (8)
where:
L - length of the pipe,
D - diameter of the pipe,
m - number of pipes.
The network optimization problem involves minimizing the cost function expressed by above
equation subject to certain operating constraints.
For any network Kirchhoff s laws must apply.
I-st Kirchhoff s Law:
where:
A - nodal-branch incidence matrix, dim _A= (n x m):
aj 7 if branch j entersnode i
aij = -aj , if branch j leavesnodei
0 , if branch j is not connected to node i
(9)
d - vector of loads, dim d = (m x 1).
II-nd Kirchhoff s Law:
-8-

10. where:
I$ - loop-branch incidence matrix, dim B = (u x m), u - the number of independent
loops,
p j , if branch j has the samedirection as loop i
bij = -pi , if branch j hasopposite direction to loop i
0 > if branch j is not in loop i
y - flow equation exponent
In addition, the pressure at all nodes of the system must not fall below a given minimum working
pressure, i.e.:
Pi - Pimin’ O (11)
where:
pi - is the nodal pressure value
Another constraint was imposed on the gas velocity at each branch. It was assumed that
VminI v < vmax (12)
for all branches i.
The above problem has been formulated as a nonlinear programming one with nonlinear
constraints in the form:
min f(z) (13)
subject to equality and inequality constraints:
ci(g)= 0 , i = 1,2,K ,m’
c&20 , i=m’+l,K ,m1
(14)
and the functions f(z) and c;(z), i=1,2,... ,m are real and differentiable. To solve this problem, an
iterative method is used at each iteration which minimizes a quadratic approximation to the
Lagrangian function subject to sequentially linearized approximations to the constraints ([7]). To
begin the calculation a starting point 50 has to be chosen together with positive definite matrix ISo.
-9-

11. The iterations generate a sequence of points B (k=1,2,...) that usually converges to the required
vector of variables and also the generate a sequence of positive definite matrices & (k= 1,2,. ..).
At the beginning of the k-th iteration both a and & are known. The vector d=& is obtained by
minimizing the quadratic function:
subject to linear constraints:
ci(g,)+cf 4%,(x,) = 0 , i = 1,2,K ,m’
c~(~,)+~_~~VC~(X~)>O , i=m’+l,K ,m
(16)
The vector a+1 has the form:
where:
&k+l =&,+akd,
(17)
ak - is a positive step-length
The calculation of & is a quadratic programming problem having the property that, if all
constraints are linear and if f(z) is a quadratic function whose second derivative matrix is E& then
a+& is the required vector of variables. Given that & is the m-component vector of Lagrange
multipliers at the solution of the quadratic programming problem which defines 4, the definition
of &+I must depend on & in order to take account of any constraint curvature.
6. Results of investigations
To prove the correctness of the stated algorithm a non-trivial examples have been solved. We
will show the results for two networks.
The low pressure gas network shown in Fig.5 comprises 108 pipes, 83 nodes and 2 sources. List
of pipes and list of nodes are given in Table 1 and Table 2 respectively.
To calculate flow through each pipe, Pole’s equation [S] has been used:
Ap=
5.117~10-‘3 eL-Q
D5 (18)
where:
Ap - drop pressure along a pipe [Lb/in2],
Q - flow through pipe under standard conditions [cu.ft./h],
D - diameter of pipe [A]
-lO-

12. The results of simulation presented in Table 3 and Table 4 have shown that, network was badly
designed. The gas velocity in many pipes does not exceed lmls (heavy lines in Fig. 5).
The aim of optimization is to minimize objective function (eq.(8)) subject to constraints.
For this case:
Qi = a, .Di2 [CU$/h] (1%
where:
OT,i= 19.6354 X Vi,
Di [inch],
vi [is]
Api =P, .Q-’ [Lb/in21
where:
pi = 1.391X1O-6 X Li X Vi*,
Li [fit]
Vi [R/S]
(20)
y = I in equation (10)
It was assumed that:
16.4 ft / s I vi 2 32.8 ft / s
for each pipe,
pj 2 0.261 Lb / in2
for each node.
Results of optimization are given in Table 5.
The medium pressure gas network shown in Fig. 6 comprises 39 pipes, 36 nodes and 1 source.
List of pipes and list of nodes are given in Table 6 and Table 7 respectively.
For this case Renouard’s equation [6] has been used:
AP = p,! - p,?= 5.01I39 *I o-” *Ps
.L .Q’.”
482s
D.
[(Lb/in2)2J (21)
where:
-ll-

13. pS- density of gas (the subscript s refers to quantities at standard conditions
pS= 14.5 Lb/in2 and temperature T, = 32 “F).
The results of simulation are presented in Table 8 and Table 9.
Branches for which
vi 13.38 fth
or
vi 2 65.4ft / s
are marked by heavy lines in Fig. 6.
For the purpose of optimization:
Q = aj -Q2 [cxjm]
where:
-
CXi= 1.751 X p&s X Vi,
p&s - average absolute pressure in the pipe [psia],
Di [inch],
Vi [fi/S]
Api = pi . &‘.I8 [(Lb / in2)l]
where:
pi = 7.051 X 10m6X Li X p:by X Vii’**,
Li [fil
y = 1.18 in equation (10)
It was assumed that:
32.8 ft / s s vi s45.8 ft / s
for each pipe,
pi 2 14.5 Lb / in2
for each node.
Results of optimization are given in Table 10.
(22)
(23)
-12-

14. 7. Conclusions
In both cases, diameter of pipes were corrected to the closest available diameter sizes.
Investigations have shown, that developed algorithm works properly. Optimization of pipe
diameters for the first network (Fig. 5) gives total profit equal 44603.00 USD. Second case gives
less savings, only 2900.00 USD, because much smaller network was much better designed.
8. References
PI
PI
PI
PI
PI
PI
PI
PI
PI
Beale, E.M. Some Uses of Mathematical Programming Systems to Solve Problems that are
not Linear”. Operational Research Quarterly. 1975, Vol. 26,3. pp. 609-618
Boyne, G.G. ,,The Optimum Design of Fluid Distribution Networks with Particular
Reference to Low Pressure Gas Distribution Networks”. International Journal for Numerical
Methods in Engineering, Vol. 5, 253-270, 1972
Chamberlain, R.M., et al, ,,The Watchdog Technique for Forcing Convergence in Algorithms
for Constrained Optimization”, Report DATMP 80/NA9, University of Cambridge
Guoping Yu and Powell R. ,,A Generalized Reduced Gradient Approach to Expansion of
Water Distribution Networks”. International Conference on Pipeline Systems, 1992,
Manchester, U.K.
Osiadacz, A.J. Simulation and Analysis of Gas Networks”. E&FN Spon Ltd., London 1987
Osiadacz A.J., Zelman H. and Krawczynski T. ,,SimNet SSV - Package for Steady-state
Simulation of any Gas Network” (In Polish), GWiTS, Nr 6, 1995
Powell, M.J.D. ,,Extensions to subroutine VFO2AD”. In System Modelling and Optimization,
Lecture Notes in Control and Information Sciences 38, eds. R.F. Drenick and F. Kozin,
Springer Verlag, 1982
Rothfarb B., Frank M., Rosenbaum D.M., Steiglitz K. and Kleitman D.J. ,,Optimal Design of
Offshore Natural Gas Pipeline Systems”. Operational Research 1980, Vol. 18, Part 6
Stubbs C.W. and Thompson P.D. ,,Designing Gas Distribution Networks by Computer”.
Manchester District Junior Gas Association, April 1979
[lo] Ward E. ,,Design of Gas Distribution Networks Using PILOT”, LRS T195, October 1974
[l l] Watanada T. ,,Least Cost Design of Water Distribution Systems”. Journal of Hydraulics
Division, ASCE, September 1973, HY9
-13-

15. TABLE 1.
LIST OF PIPES
Jo oi
y!?
2
3
4
5
6
7
8
9
IO
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
LN. Gi
100 101
101 102
102 103
103 104
107 101
107 108
107 116
108 102
108 109
108 117
109 110
109 118
110 Ill
110 121
Ill 104
Ill 122
114 115
114 131
115 116
115 136
116 132
117 118
117 134
118 119
118 139
119 121
121 122
121 140
131 135
132 133
132 137
133 134
133 158
134 138
135 136
135 145
136 137
136 147
,ength
-El-
240
456
702
653
43
446
66
75
709
266
656
302
33
322
59
315
256
230
249
502
308
656
154
459
328
259
23
305
207
197
318
230
879
174
361
331
197
315
Diameter 1jinch]
6.00
6.00
6.00
6.00
4.00
6.00
4.00
6.00
6.00
8.00
6.00
4.00
8.00
8.00
8.00
4.00
6.00
6.00
4.00
4.00
4.00
3.00
8.00
3.00
4.00
3.00
6.00
8.00
6.00
3.00
4.00
3.00
3.00
8.00
3.00
4.00
3.00
4.00
mm:
150
150
150
150
100
150
100
150
150
200
150
100
200
200
200
100
150
150
100
100
100
80
200
80
100
80
150
200
150
80
100
80
80
200
80
100
80
100
rlo of
3
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
137 148
138 139
138 149
139 140
139 150
140 141
141 144
144 154
145 146
145 147
147 148
148 157
149 150
149 160
149 161
150 151
150 152
152 153
153 154
154 155
156 157
157 158
159 160
162 161
48 49
48 64
49 50
50 51
51 52
52 53
52 54
54 55
55 98
64 65
64 69
65 66
66 67
66 83
,ength
-El-
302
732
390
728
341
52
367
118
295
240
220
312
827
499
400
295
492
230
289
276
322
220
184
164
184
52
308
817
922
102
394
62
886
554
72
256
525
722
Diameter 1
mm
100
100
200
100
100
150
150
150
80
100
100
100
150
300
150
100
150
150
150
150
80
80
300
100
300
150
300
300
300
400
300
300
200
150
150
125
150
100
inch’
4.00
4.00
8.00
4.00
4.00
6.00
6.00
6.00
3.00
4.00
4.00
4.00
6.00
12.oc
6.00
4.00
6.00
6.00
6.00
6.00
3.00
3.00
12.0(
4.00
12.0(
6.00
12.0(
12.0(
12.0(
16.0(
12.0(
12.0(
8.00
6.00
6.00
5.00
6.00
4.00

16. TABLE 1.CONT.
Vo of S.N,
pipe
77 67
78 69
79 69
80 70
81 71
82 72
83 72
84 73
85 73
86 74
87 78
88 78
89 80
90 80
91 81
92 83
Gi
72
78
80
81
72
73
83
74
85
86
79
92
101
81
82
84
,ength
-PI.-
256
230
574
187
289
623
299
525
308
285
226
322
531
256
594
66
i-
Diameter
:mm’
300
150
100
100
100
100
300
100
100
100
80
150
100
100
100
300
[inch?
12.00
6.00
4.00
4.00
4.00
4.00
12.00
4.00
4.00
4.00
3.00
6.00
4.00
4.00
4.00
12.00
INo of
IagE
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
S.N. R.N
84 85
84 95
85 86
85 96
86 97
92 99
93 94
94 100
94 80
95 102
95 96
96 103
96 97
97 98
98 104
99 100
,ength
[rtl
614
269
535
295
285
302
200
361
256
276
689
269
551
46
230
295
Dia
171111:
100
300
80
100
100
150
100
100
100
80
100
80
80
200
150
reter
jnch]
4.00
12.00
3.00
4.00
4.00
6.00
4.00
4.00
4.00
12.00
3.00
4.00
3.00
3.00
8.00
6.00
1

17. TABLE 2.
LIST OF NODES
Node no. ILoad 1 Node no. /Load
102
103
104
107
108
109
110
111
114
115
116
117
118
119
121
122
131
132
133
134
135
136
137
138
139
140
141
144
145
146
147
148
149
150
151
152
153
154
155
156
633.19
926.66
856.73
362.68
134.90
664.62
881.45
523.01
303.35
8.83
391.29
303.35
1140.66
1408.70
705.59
1250.49
246.14
0.00
431.55
974.69
616.59
2142.19
2532.77
536.08
1501.93
1611.41
1350.43
931.95
20.84
313.95
137.37
472.16
716.54
1961.73
2282.74
310.42
929.84
69.92
0.00
236.96
182.22
158
159
160
161
162
48
49
50
51
52
53
54
55
64
65
66
67
69
70
71
72
73
74
78
79
80
81
82
83
84
85
86
~ 92
93
~ 94
95
96
97
98
99
1124.77
0.00
0.00
588.70
41.32
253.91
174.81
229.90
98.88
233.43
0.00
233.43
35.31
125.37
163.51
713.71
666.04
260.27
171.98
228.49
1347.26
836.96
443.20
181.52
156.09
767.74
775.51
686.52
1082.40
1557.38
1867.09
883.22
284.28
165.27
455.21
1216.24
1261.09
821.77
35.31
300.88

18. rlo. of
aipe
1
2
3
4
5
6
7
a
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
S.N.
100 101
101 102
102 103
103 104
107 101
107 108
107 116
108 102
108 109
108 117
109 110
109 118
110 111
110 121
111 104
111 122
114 115
114 131
115 116
115 136
116 132
117 118
117 134
118 119
118 139
119 121
121 122
121 140
131 135
132 133
132 137
133 134
133 158
134 138
135 136
TABLE 3.
RESULTS OF SIMULATION
3.N. ,ength
PI
240
456
702
653
43
446
66
75
709
266
656
302
33
322
59
315
256
230
249
502
308
656
154
459
328
259
23
305
207
197
318
230
879
174
361
Diameter
[mm] 1[inch]
150 6.00
150 6.00
150 6.00
150 6.00
100 4.00
150 6.00
100 4.00
150 6.00
150 6.00
200 8.00
150 6.00
100 4.00
200 8.00
200 8.00
200 8.00
100 4.00
150 6.00
150 6.00
100 4.00
100 4.00
100 4.00
80 3.00
200 8.00
80 3.00
100 4.00
80 3.00
150 6.00
200 8.00
150 6.00
80 3.00
100 4.00
80 3.00
80 3.00
200 8.00
801 3.00
:low
$u.ft./h]
3128.53
-1013.53
-1184.81
-3182.21
-3814.34
-3932.64
7613.85
4774.90
-1453.55
-7917.56
-2310.64
-23.66
-5736.52
2905.34
-6652.58
612.36
-2931.12
2923.00
-4630.82
1309.47
2679.33
401.88
-9458.34
14.48
-1043.20
-690.05
-367.27
1330.66
2923.00
-950.67
3199.16
-3235.18
1311.23
-13309.76
270.16
Jeloc.
IfUs]
4.53
1.48
1.71
4.59
12.40
5.68
24.77
6.89
2.10
6.43
3.35
0.07
4.66
2.36
5.41
2.00
4.23
4.23
15.09
4.27
8.73
2.03
7.68
0.07
3.38
3.51
0.52
1.08
4.23
4.82
10.43
16.44
6.69
10.83
1.38

19. 37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
136
136
137
138
138
139
139
140
141
144
145
145
147
148
149
149
149
150
150
152
153
154
156
157
159
162
48
48
49
50
51
52
52
54
55
64
64
145
137
147
148
139
149
140
150
141
144
154
146
147
148
157
150
160
161
151
152
153
154
155
157
158
160
161
49
64
50
51
52
53
54
55
98
65
69
TABLE 3. CONT.
j
197 80
315 100
302 100
732 100
390 200
728 100
341 100
52 150
367 150
118 150
295 80
240 100
220 100
312 100
827 150
499 300
400 150
295 100
492 150
230 150
289 150
276 150
322 80
220 80
184 300
164 100
184 300
52 150
308 300
817 300
922 300
102 400
394 300
62 300
886 200
554 150
3.00
4.00
4.00
4.00
8.00
4.00
4.00
6.00
6.00
6.00
3.00
4.00
4.00
4.00
6.00
12.00
6.00
4.00
6.00
6.00
6.00
6.00
3.00
3.00
12.00
4.00
12.00
6.00
12.00
12.00
12.00
16.00
12.00
12.00
8.00
6.00
ISOl 6.00
:low
:cu.ft./h ]
513.48
-1107.47
157.15
1555.61
1363.50
-16173.43
-59.33
-1230.36
-0.35
-1008.94
-1029.78
137.37
62.15
-252.85
586.58
6087.90
-24849.89
630.01
310.42
2266.50
1337.37
1267.44
236.96
-182.22
-186.46
24850.60
-41.32
-13143.78
12892.34
-13320.00
-13548.84
-13648.07
-27452.24
13573.91
13337.66
13305.17
4843.06
7924.27
Jeloc.
:ft/s]
1.67
5.64
0.52
5.09
4.43
13.12
0.20
4.00
0.00
1.44
1.48
0.69
0.20
0.82
1.90
8.79
8.96
0.92
1.02
3.28
I .94
1.84
0.33
0.92
0.95
8.96
0.13
4.76
18.60
4.79
4.89
4.92
5.58
4.89
4.82
10.79
6.99
11.45

20. TABLE 3. CONT.
74 65 66 256 125
75 66 67 525 150
76 66 83 722 100
77 67 72 256 300
78 69 78 230 150
79 69 80 574 100
80 70 81 187 100
81 71 72 289 100
82 72 73 623 100
83 72 83 299 300
84 73 74 525 100
85 73 85 308 100
86 74 86 285 100
87 78 79 226 80
88 78 92 322 150
89 80 101 531 100
90 80 81 256 100
91 81 82 594 100
92 83 84 66 300
93 84 85 614 100
94 84 95 269 300
95 85 86 535 80
96 85 96 295 100
97 86 97 285 100
98 92 99 302 150
99 93 94 200 100
100 94 100 361 100
101 94 80 256 100
102 95 102 276 300
103 95 96 689 80
104 96 103 269 100
105 96 97 551 80
106 97 98 46 80
107 98 104 230 200
108 99 100 295 150
R.N. ILength 1 Diameter rf
1[ft] [[mm1 iinch]
5.00
6.00
4.00
12.00
6.00
4.00
4.00
4.00
4.00
12.00
4.00
4.00
4.00
3.00
6.00
4.00
4.00
4.00
12.00
4.00
12.00
3.00
4.00
4.00
6.00
4.00
4.00
4.00
12.00
3.00
4.00
3.00
3.00
8.00
6.00
=fow
:cu.ft./h]
4679.90
3026.47
940.78
2361.49
5168.31
2496.40
-171.98
-228.49
789.64
0.35
-79.81
34.61
-523.01
156.09
4831.05
305.83
1634.01
686.52
-0.71
796.35
-2494.98
-223.90
-810.83
-1629.42
4547.12
-165.27
-828.48
209.77
-4019.52
310.06
-1141.02
-619.42
-3070.61
10198.53
4246.59
r/eloc.
ifffs]
9.74
4.36
3.05
0.85
7.45
8.10
0.56
0.75
2.56
0.00
0.26
0.10
1.71
0.79
6.99
0.98
5.31
2.23
0.00
2.59
0.89
1.15
2.62
5.31
6.56
0.52
2.69
0.69
1.44
1.57
3.71
3.15
15.58
8.30
6.14

21. lode no.
100
101
102
103
104
107
108
109
110
111
114
115
116
117
118
119
121
122
131
132
133
134
132
13E
137
13e
13s
14c
141
144
14:
14E
14i
14E
14s
1%
151
15;
15:
15L
15f
15c
TABLE 4.
RESULTS OF SIMULATION
Vessure
Lb/in21
0.699
0.698
0.698
0.699
0.702
0.696
0.699
0.700
0.702
0.702
0.659
0.661
0.681
0.701
0.700
0.700
0.701
0.701
0.658
0.673
0.675
0.703
0.658
0.657
0.660
0.706
0.701
0.701
0.701
0.701
0.657
0.657
0.657
0.657
0.71s
0.70:
0.70:
0.702
0.702
0.701
0.701
0.657
lode no.
157
158
159
160
161
162
48
49
50
51
52
53
54
55
64
65
66
67
69
70
71
72
73
74
78
79
80
81
82
83
84
85
8E
92
92
94
95
9c
97
9E
95
Vessure
ILb/in2
0.657
0.657
0.725
0.723
0.718
0.718
0.718
0.719
0.720
0.722
0.725
0.725
0.724
0.724
0.714
0.708
0.701
0.698
0.712
0.696
0.698
0.698
0.697
0.697
0.709
0.709
0.698
0.696
0.694
0.698
0.698
0.697
0.697
0.705
0.698
0.698
0.698
0.697
0.7oc
0.702
0.702

22. TABLE5
RESULTSOFOPTIMIZATION
40. of Length
Pipe ml
1 240
2 456
3 702
4 653
5 43
6 446
7 66
8 75
9 709
10 266
11 656
12 302
13 33
14 322
15 59
16 315
17 256
18 230
19 249
20 502
21 308
22 656
23 154
24 459
25 328
26 259
27 23
28 305
29 207
30 197
31 318
32 230
33 879
34 174
35 361
36 331
37 197
old result of new AD=
diameter optimization diameter old-new
[inch] [inch] [inch] [inch]
6.00 3.42333 4.00 2.00
6.00 4.02147 5.00 1.oo
6.00 1.79539 2.00 4.00
6.00 4.92069 5.00 1.00
4.00 0.05000 0.50 3.50
6.00 5.59558 6.00 0.00
4.00 5.55803 6.00 -2.00
6.00 7.06769 8.00 -2.00
6.00 2.33288 2.50 3.50
8.00 3.33702 4.00 4.00
6.00 0.81741 1.oo 5.00
4.00 1.83676 2.00 2.00
8.00 2.19912 2.50 5.50
8.00 1.59502 1.50 6.50
8.00 2.87766 3.00 5.00
4.00 1.58207 1.50 2.50
6.00 4.07753 5.00 1.oo
6.00 4.07416 5.00 1.oo
4.00 4.10231 5.00 -1 .oo
4.00 1.59862 1.50 2.50
4.00 3.09081 3.00 1.oo
3.00 1.68912 2.00 1.oo
8.00 2.17738 2.50 5.50
3.00 1.46055 1.50 1.50
4.00 0.53192 0.50 3.50
3.00 0.23999 0.50 2.50
6.00 1.31860 1.50 4.50
8.00 0.58544 0.50 7.50
6.00 4.07416 5.00 1.00
3.00 2.05621 2.50 0.50
4.00 1.99633 2.00 2.00
3.00 1.68122 2.00 1.oo
3.00 2.00701 2.00 1.oo
8.00 0.05000 0.50 7.50
3.00 2.50377 2.50 0.50
4.00 1.91797 2.00 2.00
3.00 1.36608 1.50 1.50
AF
[zt]
629.69
617.58
3,425.51
884.16
150.75
172.52
219.62
3,090.60
1,472.14
3,800.95
679.07
238.57
2,658.81
396.95
859.79
346.56
311.01
317.90
1,370.29
364.53
700.64
lJ21.28
710.94
1,159.63
609.75
123.07
2,768.70
279.91
108.27
715.97
245.22
938.86
1,577.86
198.49
745.50
304.69

23. TABLESCONT.
rlo. of
pipe
Length
WI
38 315
39 302
40 732
41 390
42 728
43 341
44 52
45 367
46 118
47 295
48 240
49 220
50 312
51 827
52 499
53 400
54 295
55 492
56 230
57 289
58 276
59 322
60 220
61 184
62 164
63 184
64 52
65 308
66 817
67 922
68 102
69 394
70 62
71 886
72 554
73 72
74 256
75 525
76 722
77 256
old result of new AD=
diameter optimization diameter old-new
[inch] [inch] [inch] [inch]
4.00 1.68126 2.00 2.00
4.00 0.67477 0.75 3.25
4.00 2.68523 3.00 1.oo
8.00 3.12227 3.00 5.00
4.00 1.99334 2.00 2.00
4.00 1.43064 1.50 2.50
6.00 0.35160 0.50 5.50
6.00 1.66458 2.00 4.00
6.00 1.68390 2.00 4.00
3.00 0.65310 0.75 2.25
4.00 1.50908 1.50 2.50
4.00 1.90735 2.00 2.00
4.00 1.36666 1.50 2.50
6.00 4.02766 5.00 1.oo
12.00 5.46139 6.00 6.00
6.00 1.39587 1.50 4.50
4.00 0.98174 1.00 3.00
6.00 2.58368 2.50 3.50
6.00 1.94637 2.00 4.00
6.00 1.88977 2.00 4.00
6.00 0.85776 1.00 5.00
3.00 0.75219 1.50 1.50
3.00 0.73196 0.75 2.25
12.00 5.46583 6.00 6.00
4.00 0.34730 0.50 3.50
12.00 8.26838 10.00 2.00
6.00 8.22056 10.00 -4.00
12.00 8.30113 10.00 2.00
12.00 8.34402 10.00 2.00
12.00 8.36239 10.00 2.00
16.00 10.05071 12.00 4.00
12.00 7.26529 8.00 4.00
12.00 7.25219 8.00 4.00
8.00 7.25189 8.00 0.00
6.00 5.92402 6.00 0.00
6.00 5.62595 6.00 0.00
5.00 5.88101 6.00 -1 .oo
6.00 4.83836 5.00 I .oo
4.00 2.99342 3.00 1.00
12.00 4.61971 5.00 7.00
AF
[a
708.59
1,013.70
864.79
2,624.26
1,638.62
931.44
323.56
1,792.79
576.25
642.65
653.80
494.54
850.83
lJ19.65
4,679.80
2J45.01
934.10
2J46.25
1,120.50
1,408.62
1,596.40
497.66
478.41
1,724.14
579.82
612.64
317.57
1,028.36
2,724.07
3,074.15
728.86
2,548.73
403.55
346.56
710.89
853.16
2.748.03

24. TABLE 5. cont.
No. of
pipe
Length
PI
78 230
79 574
80 187
81 289
82 623
83 299
84 525
85 308
86 285
87 226
88 322
89 531
90 256
91 594
92 66
93 614
94 269
95 535
96 295
97 285
98 302
99 200
100 361
101 256
102 276
103 689
104 269
105 551
106 46
107 230
108 295
old result of new AD=
diameter )ptimizatior diameter old-new
[inch] [inch] [inch] [inch]
6.00 4.12945 5.00 1.oo
4.00 3.77279 4.00 0.00
4.00 0.73075 0.75 3.25
4.00 0.84228 0.75 3.25
4.00 1.22750 1.25 2.75
12.00 3.86555 4.00 8.00
4.00 0.89517 1.oo 3.00
4.00 0.53908 0.50 3.50
4.00 1.47561 1.50 2.50
3.00 0.69617 0.75 2.25
6.00 4.00051 5.00 1.oo
4.00 2.53359 2.50 1.50
4.00 2.25244 2.50 1.50
4.00 1.45999 1.50 2.50
12.00 4.53236 5.00 7.00
4.00 3.93353 4.00 0.00
12.00 5.58385 6.00 6.00
3.00 2.57409 2.50 0.50
4.00 3.86451 4.00 0.00
4.00 3.39790 4.00 0.00
6.00 3.88863 4.00 2.00
4.00 0.71635 0.75 3.25
4.00 1.25257 1.25 2.75
4.00 0.59800 0.50 3.50
12.00 5.78219 6.00 6.00
3.00 2.45573 2.50 0.50
4.00 4.28132 5.00 -1 .oo
3.00 2.55947 2.50 0.50
3.00 3.97317 4.00 -1 .oo
8.00 5.79830 6.00 2.00
6.00 3.76659 4.00 2.00
TOTAL PROFIT [zl] 105,264.42
AF
WI
311.01
628.05
969.63
1,840.90
3,586.68
1,660.62
1,090.05
779.18
492.70
435.42
920.55
443.23
1,621.06
704.62
2,524.63
294.12
793.58
672.13
1,065.79
904.51
2,586.20
378.93
342.99
303.15
54.29
668.41
776.33
TOTAL PROFIT [$] I44.603.57

25. TABLE 6.
LIST OF PIPES
lo. of pipe S.N R.N
1 1 26
2 1 32
3 2 5
4 2 35
5 3 5
6 3 15
7 4 19
8 5 14
9 6 34
10 7 10
II 7 33
12 8 6
13 8 9
14 8 36
15 9 22
16 10 2
17 10 11
18 11 3
19 11 12
20 12 4
21 12 9
22 14 1
23 14 27
24 15 4
25 15 16
26 16 17
27 16 18
28 18 14
29 18 25
30 19 20
31 19 24
32 20 21
33 21 23
34 26 28
35 26 29
36 29 30
37 29 31
38 33 1
39 35 13
Length
-IElI-
1706
525
66
66
197
131
213
295
492
197
482
1247
230
7546
148
591
197
541
279
525
213
180
1378
148
197
197
197
197
394
164
180
197
66
427
295
164
98
469
164
IT Diameter
(mm) (inch)
50 2.00
65 2.50
32 1.25
32 1.25
32 1.25
40 1.50
40 1.50
32 1.25
200 8.00
32 1.25
50 2.00
200 8.00
65 2.50
200 8.00
40 1.50
32 1.25
32 1.25
32 1.25
32 1.25
40 1.50
40 1.50
65 2.50
50 2.00
40 1.50
32 1.25
40 1.50
32 1.25
65 2.50
65 2.50
40 1.50
25 1.00
32 1.25
50 2.00
40 1.50
40 1.50
40 1.50
40 1.50
50 2.00
32 1.25

26. TABLE 7.
LIST OF NODES
Node No. Load
[cu.ft./h]
1 1425.65
2 722.89
3 962.33
4 904.41
5 593.64
6 0.00
7 753.97
~ a 0.00
9 294.88
10 962.33
11 962.33
12 844.73
13 194.23
14 1508.29
15 294.88
~
16 294.88
17 453.79
la 785.05
19 593.64
20 525.13
21 0.00
22 194.23
23 194.23
24 194.23
25 1400.58
26 814.36
27 919.59
28 194.23
29 525.13
30 198.47
31 198.47
32 5297.21
33 0.00
34 183636.44
35 0.00
36 0.00
TABLE 9.
RESULTS OF SIMULATION
qode No.
1
2
3
4
5
6
7
a
9
10
11
12
13
14
15
16
17
la
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
I
I
,
,
,
I
I
/
/
/
I
I
/
Pressure
[Lb/in21
1a.329
21.052
22.785
24.827
21.047
42.781
19.753
43.993
43.116
21.539
24.789
33.294
21.046
la.421
22.983
20.552
20.539
18.457
24.719
24.697
24.692
43.115
24.692
24.703
18.436
17.571
18.266
17.564
17.484
17.481
17.483
la.003
19.045
42.292
21.050
52.214

27. TABLE 8.
RESULTS OF SIMULATION
No.
of pipe
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
S.N. R.N.
1 26
1 32
2 5
2 35
3 5
3 15
4 19
5 14
6 34
7 10
7 33
8 6
8 9
8 36
9 22
10 2
10 11
11 3
11 12
12 4
12 9
14 1
14 27
15 4
15 16
16 17
16 18
18 14
18 25
19 20
19 24
20 21
21 23
26 28
26 29
29 30
29 31
33 1
35 13
Length
WI
1706
525
66
66
197
131
213
295
492
197
482
1247
230
7546
148
591
197
541
279
525
213
180
1378
148
197
197
197
197
394
164
180
197
66
427
295
164
98
469
164
r Diameter
[mm1
50
65
32
32
32
40
40
32
200
32
50
200
65
200
40
32
32
32
32
40
40
65
50
40
32
40
32
65
65
40
25
32
50
40
40
40
40
50
32
[inch]
2.00
2.50
1.25
1.25
1.25
1.50
1.50
1.25
8.00
1.25
2.00
8.00
2.50
8.00
1.50
1.25
1.25
1.25
1.25
1.50
1.50
2.50
2.00
1.50
1.25
1.50
1.25
2.50
2.50
1.50
1.oo
1.25
2.00
1.50
1.50
1.50
1.50
2.00
1.25
Flow Jelocity
Jcu.ft./h] [ftk]
1930.65 11.42
5297.21 18.44
346.44 4.56
194.23 2.56
4721.93 60.83
-2620.70 21.06
1507.23 11.52
4474.73 61.29
183636.44 38.75
-4638.94 61.91
3884.97 22.01
183636.44 38.16
23207.76 45.54
206843.50 39.76
194.23 1.02
1263.56 16.57
-6864.82 85.50
3063.20 37.53
-10890.35 117.42
10983.58 75.75
-22718.65 129.59
4768.54 16.47
919.59 5.38
-8571.94 67.03
5656.00 73.16
453.79 3.87
4907.33 67.65
2721.70 9.38
1400.58 4.82
719.36 5.51
194.23 3.81
194.23 2.33
194.23 0.95
194.23 1.80
922.07 8.63
198.47 1.87
198.47 1.87
3884.97 22.47
194.23 2.56

28. TABLE IO.
RESULTS OF OPTIMIZATION
uo. of
3ipe
1
2
3
4
5
6
7
8
9
IO
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
,ength
:ft]
1706
525
66
66
197
131
213
295
492
197
482
1247
230
7546
148
591
197
541
279
525
213
180
1378
148
197
197
197
197
394
164
180
197
66
427
295
164
98
469
164
old
diameter
[inch]
2.00
2.50
1.25
1.25
1.25
1.50
1.50
1.25
8.00
1.25
2.00
8.00
2.50
8.00
1.50
1.25
1.25
1.25
1.25
1.50
1.50
2.50
2.00
1.50
1.25
1.50
1.25
2.50
2.50
1.50
1.oo
1.25
2.00
1.50
1.50
1.50
1.50
2.00
1.25
lesult of
optimization
‘inch]
0.80193
1.31959
0.83177
0.25144
0.98843
0.82543
0.69913
1.17346
7.64524
1.03909
1.05826
7.62555
2.71055
7.97146
0.24858
0.99321
1.40752
0.9117
1.67957
1.5627
2.33017
1.30756
0.5522
1.36832
1.1382
0.3852
1.05762
0.88437
0.67699
0.48467
0.25162
0.25236
0.25334
0.25684
0.55905
0.26006
0.26001
1.06147
0.2516
new
diameter
jnch]
1.25
1.50
1.oo
0.50
1.oo
0.75
0.76
1.25
8.00
1.25
1.25
8.00
3.00
8.00
0.50
1.oo
1.50
1.oo
2.00
1.50
2.50
1.25
0.75
1.50
1.25
0.50
1.25
1.oo
0.75
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
A D=
old-new
‘inch]
0.75
1.oo
0.25
0.75
0.25
0.75
0.74
0.00
0.00
0.00
0.75
0.00
-0.50 -
0.00
1.oo
0.25
-0.25 -
0.25
-0.75 -
0.00
-1.00 -
1.25
1.25
0.00
0.00
1.00
0.00
1.50
1.75
1.oo
0.50
0.75
1.50
1.oo
1.oo
1.oo
1.00
1.50
0.75
AF
[zll
1,200.04
523.79
13.80
38.15
41.39
82.49
132.46
339.24
126.31
118.81
124.18
43.97
113.84
196.16
212.79
220.36
1,527.67
158.41
281.79
640.33
132.01
66.96
114.44
84.30
343.22
237.62
132.01
79.21
602.76
95.37
TOTAL PROFIT [zt] 6J65.44
TOTAL PROFIT [$] 2,909.08

29. J32 -1337 ,,134
,138 139
146, .
156 157
159 d 160
p=5000 Pa
FIGURE 5 : low pressure gas network

30. 3L 29--
7+ 30
!26
P 28
7
y 6
8
FIGURE 6: medium pressure gas network
36
1
p=360 kPa