HUNGAR~JOURNAL OF INDUSTRIAL CHEMISTRY VESZPREM Vol. 29. pp. 113-117 (2001) OPTIMIZATION OF PIPELINE NETWORK FOR OIL TRANSPORT A. JOKIC andZ. ZAVARGO (Faculty ofTechnolgy, University ofNovi Sad, Bul CaraLazara 1, 21000 Novi Sad, YU) Received: June 18,2001 In this work the pipeline network for oil transport has been optimized. The network layout has already been given, the flow rates are specified and the inlet and outlet pressures are defined. The capital cost of such a network has been minimized. From many feasible combinations of section diameter distributions the aim was to find the optimal one. The problem has been solved by applying nonlinear programming. Keywords: pipeline network, oil transport, capital cost, optimization, nonlinear programming Introduction The total head loss is h =A~L+L£ L 2g D (1) (2) There are many works dealing with the solution of fluid network problems [1-3, 6, 8, 9]. There are also numerous algorithms for pipeline network optimization by minimizing some cost functions [6, 4]. The network which has to be optimized is the network for oil transport (Fig.l). The network consists of nodes, sections and paths. A pipe section is a pipeline with a constant diameter and no branching. A node is defined as the branching point or the point of inlet or outlet from the section. Finally, a path is defined as the sequence of pipe sections between an inlet (source) and outlet (consumer) from the network. Thus, each section has two nodes, one at each end. Each path has at least one section. The configuration of the network is given so are the flows through each section as well as the inlet and outlets pressures. While the inlet and outlet pressures are fixed, the other node pressures are unknown and subject to change via D. This problem has many feasible solutions. The goal was to find that one which minimizes the capital cost. where A is the Darcy friction factor, while L and Le are pipe length and equivalent length respectively. For a long distance pipeline the effect of Le can be neglected. For hydraulically smooth pipes the friction factor depends only on Reynolds number. In that case Blasius correlation [5] can be used A. =0.3164Re-0'25 Combining Eq. ( 1) and (2) with continuity equation 4Q v= Dk (3) (4) it follows for the case when Blasius Eq. (3) can be used KD-4·15 :::; -AP -I::JJpg Pipe section flow where The oil that flows through the network can be K = 0.242Qt75 p 0·75 p 0·25 L considered as isothermal and incompressible. As the network is in the ground the temperature variances are and L1P = P2- P1: &1. = h2- h1. negligible. Having in mind that the section diameter is constant, the section velocity is also constant. The Bernoulli equation for each pipe section in that case is (5) (6) 114 Node 1 2 3 4 5 6 7 8 9 10 11 12 13 Table 1 Pipeline network nodes data h[m] 942 P[kPa].l0-5 147.1 unknown unknown unknown 4.9 unknown 4.9 4.9 unknown unknown unknown 4.9 4.9 Most of the pipes in the engineering practice have rough boundaries. In that case the equation of Altshul [5] can be used { 68 ) 0 · 25 A= 0.1 .E_+- D Re (7) where E is the pipe surface roughnes. In this case the D cannot be explicitly expressed as it can be in Blasius Eq. (3). The Eqs. (5) and (6) have now the following form K* n-s = -AP- Mpg (8) (9) Note that K isn't a true constant because it depends on D,. since A depends on D. Pipe cost function The weight per unit length of the pipe is assumed to be an exponential function of the diameter W=aDP (10) The total weight of the pipe section will be WL [kg] and the price of such a pipe can be easily calculated knowing the price of the pipe per unit weight. It is a well knov..n fact that the ratio &D (pipe wall thickness/pipe diameter) depends upon pressure. Eq. (10) is usually correlated with a given pressure (Pcorr)· In the case of a pipe network it is usually a maximum expected pressure in the network. The higher pressure demands larger q and consequently a larger weight of the pipe. When the pressure in the pipe (P) differs from pressure (Pcorr) for which Eq. ( 10) was correlated, the following relation is proposed [7]. {11) Table 2 Pipeline network sections data Section Input, output node L[m] Q[m3s-l] Sl 1,2 12,000 1.2688 S2 2,3 18,000 1.2688 S3 3,4 150,000 1.2688 S4 4,5 5,000 0.2985 S5 4,6 110,000 0.4478 S6 6,7 65,000 0.0746 S7 6,8 5,000 0.3732 S8 4,9 141,000 0.5224 S9 9,10 5,000 0.1866 SlO 9,11 161,000 0.3358 S11 11,12 5,000 0.1119 S12 11,13 85,000 0.2239 Pipe network optimization For a pipe network that consists of n pipe section there is ann Eqs.(8) i =l,n (12) with 2n unknown variables (Di and L1Pi). To reduce the number of the equations as well as the number of the unknown variables the alternative system of equations can be formulated. For each path we have ±K;Di-4.75:=.-i)MJ,-Llh;pg)=APj-Mzjpg j=l,m (13) i=l i=l where k corresponds to the number of the section that belongs to path j, while m represents the number of paths. In such a way the unknown pressure drops L1Pt are excluded Recall that the paths pressure drops JJ.Pj, from the source to the sink, are fixed. The capital cost function which has to be minimized is n n n WN = L ~L1 == LaLtDf == L.ciDf i=l i=l i=l· (14) By introducing new variables into the Eqs.(l3) and (14), in the case when Blasisus relation can be used, i= l,n we have the following nonlinear cost function (min) WN = i cix;n·ztosp i=l subject to linear constrains k ~ K.x. = t}.P. - Ah .p,ru .L.J ll J JO i=l where ci = aLi. j=l,m (15) (16) (17) For rough pipes the following variables are introduced i=l,n {18) Eqs. (14) and(l3) are now Path P1 P2 P3 P4 P5 P6 Table 3 Pipeline network paths data Sections S1,S2,S3,S8,Sl0,Sl2 S1,S2,S3,S8,S10,S11 S1,S2,S3,S8,S9 S1,S2,S3,S4 S1,S2,S3,S5,S6 S 1 ,S2,S3 ,S5 ,S7 3 Input, output node 1,13 1,12 1,10 1,5 1,7 1,8 Fig. I Pipeline network layout n (min)WN = Lci(x;)-02tl i=l subject to 14.22 14.22 14.22 14.22 14.22 14.22 (19) Table 4 Initial values for smooth pipes Section S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Sll Sl2 D[m] 1.0 1.0 1.0 0.3 0.6 0.5 0.4 0.7 0.3 0.7 0.3 0.6 K 10'6 0.463 0.694 5.782 0.015 0.685 0.018 0.023 1.150 0.007 0.606 0.003 0.157 Table 5 Optimization results for smooth pipes Section Sl S2 S3 S4 S5 S6 S7 S8 S9 SlO Sll S12 D[m] 1.025 1.025 1.025 0.268 0.624 0.393 0.414 0.790 0.255 0.706 0.272 0.638 115 (20) are: a= 1412.15 and f3 =2. The initial guess values f0r D as well as K are given in Table 4 while the optimization results are presented in Table 5. Experimental The layout of the pipeline network for oil transport is shown in Fig.l. The data necessary for the calculations are given in Tables 1, 2 and 3 [7]. Hydraulically smooth pipes For such systems the linear constrains are Eqs. ( 17) Ktxt + K2X 2 + K3x3 + K8:xg + K10x10 + K12x12 = 14,220,000 Klxl + K 2x 2 + K 3x3 + K 8x 8 + K10x10 + K 11xu = 14,220,000 Ktxt + K2 x2 + K3~ + K8:xg + K9~ = 14,220,000 (21) K1x1 + Kzx2 + K3~ + K4x 4 = 14,220,000 Ktxt + KzX2 + K3 Xs + K5x5 + K6 x6 = 14,220,000 Ktxt + Kzx2 + K3x3 + K5x5 + K1 x., = 14,220,000 and the objective function is n (min) WN = L cix-:2105P i=l Thus we have a nonlinear objective function with linear constrains. Eq. ( 11) has been correlated for Pr:orr = 159.105 [Pa] and the obtained values for coefficients Pipes with rough boundaries In this case the linear constrains are given by Eqs. ('2(}} K~x; + K;x; + K;x; + K;x; + K~0x;0 + K;2x:z = 14,220,000 K; x; + K;x; + K;x; + K;_x; + K1~x:0 + K;1x:1 = 14.220.000 K; x~ + K;x; + K;x; + K;x; + K;x; = 14,220,000 (22) K; x; + K;x; + K;x; + K;x; = 14,220,000 K~x; + K;x; + K;x; + K;x; + K;x; = 14,220.000 K;x; + K;x; + K;x; + K;x; + K;x; = 14.220,000 with the following objective function (min) WN = !c;(x)-c.lP i=l The initial guess values for D and K"" are presented in Table 6 while the optimization results are given in Table 7. The value for £ :::: 0.2 is taken from literaure [llJ. Since K* values depend on D (see Eqs. 9 and 7) the optimization results cannot be achieved in one step. This means that on the base of the first optimum D values we have to recalculate K* values and repeat the optimization procedure. As it can be seen (Table 5 and Table 6 Initial valurs for rough pipes 116 Section D[m] K*Jo-6 S1 1 0.464 S2 1 0.696 S3 1 5.803 S4 0.268 0.011 S5 0.62 0.613 S6 0.47 0.015 S7 0.5 0.019 S8 0.79 1.085 S9 0.26 0.005 S10 0.72 0.560 Sll 0.3 0.021 S12 0.7 0.144 Table 7) the optimization results are practically the same as in the case of smooth pipes. Conclusion in this work the capital cost of pipeline network for oil transport has been minimized. The configuration of the network was fixed, so were the flow rates through each section as well as the pressure drops in each path. The fluid that flows through the network assumed to be isothermal and incompressible. The objective function which has to be minimized was the equation for the weight of the pipeline network as the function of section diameter D, while the constrains were equations for the pressure drop for each path. Diameter D of each section has been adjusted in sucl~ a way that it could give a minimum weight under the given constrains. The originally nonlinear constrains, with respect to D, were linearized introducing new variables. The objective function remains nonlinear .. The linearly constrained nonlinear objective function has been solved by nonlinear programming. Two relations for Darcy friction factor A, have been used. Blasius formula for hydraulically smooth pipes, Eq. (3), and Altshul correlation for rough pipes, Eq. (7). In the latter, due to the fact that A. is function of D and K is function of A., K has to be recalculated in each step of the procedure. SYMBOLS c coefficient in Eq. ( 16) D pipe diameter, m g acceleration due to gravity, kgm-2 h height~ m llh height difference\ m hL head loss~ m .index. for pipe network sections j index. for pipe network paths lc number of sections in a given path L node height, m L~ equivalent height, m Table 7 Optimization results for rough pipes Section S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Sll S12 D[m] 1.025 1.025 1.025 0.268 0.624 0.393 0.414 0.79 0.255 0.706 0.272 0.638 m number of pipe network paths n number of pipe network sections P pressure, kPa LiP pressure drop, kPa K'Io-6 0.467 0.701 5.838 0.011 0.614 0.014 0.018 1.085 0.005 0.557 0.020 0.141 Pcorr pressure for which Eq. ( 11) has been correlated, kPa Q Flow rate, m3s-1 Re Reynolds dimensionless number W weight per unit length, kgm-1 WS one section weight, kg WN whole network weight, kg x variable defined by Eq. ( 15) x* variable defined by Eq. (18) Greek symbols a coefficient in Eq. (10), kg /3 coefficient in Eq. ( 10) 8 pipe wall thickness, m A. 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