Finding optimal design diameter of pipeline.
Today's highly capitalized societies require 'maximum benefit with minimum cost.' When designing the pipeline and fittings needed to supply water to, one of the key decisions is picking the proper diameter of pipe for the desired flow rate. The best pipe size is not always the one with the lowest initial cost. With rising energy costs, the friction loss in the pipe at a given flow rate can affect the long-term cost of the pipeline. The objective of selecting the right size pipe is to minimize the sum of the initial capital investment and the annual energy costs. In general, the cost to install a pipeline consists of capital costs and the annual operating, or variable cost. The capital cost is determined by the installed purchase price of the pipe and fittings, along with the service life of the pipe. The annual operating cost of a pipeline depends on the number of hours pumping water and the friction loss in the pipeline at the design flow rate. At a given flow rate, smaller diameter pipe and fittings will have greater friction loss than larger diameter pipe. The friction loss depends on the pipe diameter, pipe material, length of pipeline and the flow rate passing through the pipeline. As the size of the pipeline becomes lower, the pipe cost decreases, but the energy cost increases due to greater friction losses along the pipeline. On the other hand, higher diameter pipe is having high initial investment compared to lower sized pipe. Thus the objective of the economic design is to determine the adequate pipe diameter of different available pipe sizes such that the total annual cost becomes minimal.
For a given set of economic parameters, Keller  has proposed a method based on the construction of economic pipe selection charts to determine the most economic pipe diameter at each section of the pipeline included between two consecutive outlets. Several analytical [2-6] and computer-aided design techniques [7-10] have been proposed to find the optimal diameter of the pipelines. Keller and Bliesner  have mentioned that although the selection of economical pipe sizes is an important engineering decision, it is often given insufficient attention.
The major cost of piping and pumping system is the initial or capital investment on the piping and pumping units. However, the recurrence expenditure on the power consumption has to be given due consideration because generally the recurrence expenditure over the years far outweigh the capital investment. Hence the major length of the pipeline is designed with a view to minimize both the power requirement and pipe material simultaneously.
There exist an optimal diameter, D for the total annual cost of pipes and energy; such that both pipe material and power requirements (P) are minimized simultaneously, as [PD.sup.2] to attain minimum value.
Let D is the internal diameter of the pipe line and P is the power required to pump the design water discharge, Q over an overall head of H. Material of the pipe line per unit length may be assumed as proportional to [D.sup.2]. Hence it has been proposed to find the diameter D when [PD.sup.2] value attains minimum. The power P can be computed by,
The overall head, H consists of three major components,
H = [h.sub.f] + [h.sub.m] + [h.sub.1], (2)
Where, [h.sub.f] = head loss due to friction,
[h.sub.f] = [[lambda][lu.sup.2]/(2gD)] (3)
Where, [bar.u] = average velocity, l = length of pipe line and [lambda] is the friction factor.
Generally calculation for friction factor is done by using Colebrook-White equation or Moody's diagram (Which is a graphical presentation of Colebrook-white equation).
The U.S. Bureau of Reclamation  reported large amounts of field data on commercial pipes: concrete, continuous-interior, girth-riveted, and full-riveted steel pipes. Due to large variations in the field data, average friction factors were used for simplicity. The researchers of the Bureau of Reclamation  found that some of the field data collected could not be explained by the Colebrook-White equation, since the variation of the data followed the curve of transitional turbulent flow which is omitted in the composition of the Colebrook-White equation. The Bureau of Reclamation report  asserted that the Colebrook-White equation was found inadequate over a wide range of flow conditions. Moreover, several researchers have found that the Colebrook-White equation is inadequate for pipes of very smaller diameter. Wesseling and Homma  suggested using a Blasius-type equation or a power law with minor modifications instead of the Colebrook-White equation. They recommended using larger values of the proportionality factor for smaller-size pipes. Since the mid-1970s, many alternative explicit equations have been developed to avoid the iterative process that is inherent in Colebrook- White equation. These equations give a reasonable approximation; however, they tend to be less universally accepted. Von Bernuth and Wilson  conducted laboratory experiments and attempted to find the optimum value of the roughness height of PVC pipes for the Colebrook-White equation and then the value of the friction factor of PVC pipes. Their computation results were, however, quite different from those obtained in the laboratory when using the Colebrook-White equation. Instead they proposed to employ a Blasius-type equation with minor modifications. The friction factor determined from laboratory data decreases with an increase in the Reynolds number even after a certain critical value, whereas the friction factor of the Colebrook-White equation tends to be a constant with an increase in the Reynolds number. Zagarola  has indicated that the Colebrook-White correlation is not accurate at high Reynolds numbers. Considering above views, in the present manuscript a new resistance model, developed by Rao and Kumar , has been used.
[h.sub.m]=head loss due to pipe fittings (Minor-losses) which can be computed by the formula:
[h.sub.m] = [[k[bar.u].sup.2]/(2g)] (4)
Where, K is the coefficient of pipe fitting and hl = Static lift (the physical lift of water).
Detailed discussion on Resistance model can be found in Rao and Kumar . A brief summary is presented here:
The resistance model developed by Rao and Kumar (16) is of the following form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Where a and b are constants, u* is the bed shear velocity, k is the physical roughness height and [phi] is a correction function if any. Above equation can be converted to a resistance model in terms of friction factor, [lambda] = 8[([u.sub.*]/[bar.u]).sup.2], where u is the average velocity in the pipe:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The constants and correction function are solved by using Nikuradse's data  on sand roughened pipes, as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Where, R* = [ku.sub.*]/v, is the friction Reynolds number.
A thermal power plant example has been explained here. The design parameters of major piping system are as follows:
* Flow discharge, Q= 36,000 [m.sup.3]/hour or 10 [m.sup.3]/second.
* Internal diameter of Pipe: [D.sub.e] = 3.5m,
* Pipe thickness is 16mm
* Pipe material is of steel with Young's Modulus of Elasticity [approximately equal to] 1.95x[10.sup.8] kpa with poisons ratio [approximately equal to] 0.305;
* The Overall length of the pipe, l = 1225m;
For the present problem the static lift is assumed as, [h.sub.1] = 10.6 m
The values of the coefficient, K for various pipe fittings as per known standards that are used in the piping system are listed in Table1.
It can be seen in Table 1 that the overall value of coefficient of all pipe fittings, K= 193, the overall length of the pipe is about 1225m i.e. l=1225m and the design discharge of the entire piping and pumping system, Q = 36000 [m.sup.3]/hr. As explained earlier, for the purpose of deciding economic pipe diameter, D the various design values of variables and computed parameters are given in Table 2. The diameter D which gives minimum value of [PD.sup.2] is chosen as the economic diameter, D as shown in the Figure 2. The recurring expenditure on the power consumption per annum is computed based on the assumption of Rs 4 per every energy unit (kW-hour) consumed, which is shown in the last column of Table 2 in crores of Rupees.
[FIGURE 1 OMITTED]
A new method has been devised to find an optimal diameter of the single line pipe. This method is very quick in calculation and considers relatively less parameters than that found in the literature.
Notation D the internal diameter of the pipe line P the power required to pump Q discharge H head [h.sub.f] head loss due to friction [h.sub.m] head loss due to pipe fittings K the coefficient of pipe fitting [h.sub.l] Static lift (the physical lift of water). u velocity [bar.u] average velocity L length of pipe line [lambda] friction factor g acceleration due to gravity; [u.sub.*] bed Shear velocity; R Reynolds number; [upsilon] Kinematic viscosity; [R.sub.*] Friction Reynolds number.
 Keller, J.,1975, "Economic pipe size selection chart," Proc., ASCE Irrigation and Drainage Division Specialty Conf., Utah State Univ, Logan, Utah.
 Solomon, K., and Keller, J., 1978, "Trickle irrigation uniformity and efficiency," J. Irrig. Drain. Div., 104(3), 293-306.
 Wu, I. P., 1992, "Energy gradient line approach for direct hydraulic calculation in drip irrigation design," Irrig. Sci., 13, 21-29.
 Wu, I. P., 1997, "An assessment of hydraulic design of micro irrigation systems," Agric. Water Manage., 32, 275-284.
 Valiantzas, J. D., 1998, "Analytical approach for direct drip lateral hydraulic calculation," J. Irrig. Drain. Eng., 124(6), 300-305.
 Valiantzas, J. D., 2002, "Hydraulic analysis and optimum design of multidiameter irrigation laterals," J. Irrig. Drain. Eng., 128(2), 78-86.
 Bralts, V. F., and Segerlind, L. J., 1985, "Finite element analysis of drip irrigation submain units," Trans. ASAE, 28(3), 809-814.
 Bralts, V. F., Kelly, S. F., Shayya, W. H., and Segerlind, L. J., 1993, "Finite element analysis of microirrigation hydraulics using a virtual emitter system," Trans. ASAE, 36, 717-725.
 Kang, V., and Nishiyama, S., 1996a, "Analysis and design of micro irrigation laterals," J. Irrig. Drain. Eng., 122(2), 75-82.
 Kang, V., and Nishiyama, S., 1996b, "Design of micro irrigation sub main units," J. Irrig. Drain. Eng., 122(2), 83-89.
 Keller, J., and Bliesner, R. D., 1990, "Sprinkle and trickle irrigation," Van Nostrand Reinhold, New York.
 U.S. Bureau of Reclamation, 1965, "Friction factors for large conduit flowing full," Engineering Monograph, No. 7, U.S. Dept. of Interior, Washington, D.C.
 Wesseling, J., and Homma, F., 1967, "Hydraulic resistance of drain pipes," Neth. J. Agric. Sci., 15, 183-197.
 Von Bernuth, R. D., and Wilson, T., 1989, "Friction factors for small diameter plastic pipes," J. Hydraul. Eng., 115(2), 183-192.
 Zagarola, M. V., 1996, "Mean-flow scaling of Turbulent Pipe Flow," Ph.D.thesis, Princeton University, USA.
 Rao, ARK and Kumar, B, 2006, "Friction Factor for Turbulent Pipe Flow," Journal of Indian Water Works Association, 29-36, Oct-Dec issue.
 Nikuradse, J.,1933, "Stroemungsgesetze in rauhen Rohren," Ver. Dtsch.Ing. Forsch, 361.
Achanta Ramakrishna Rao
Professor, Civil Engineering Department,
Indian Institute of Science, Bangalore-12
Bimlesh Kumar (Corresponding Author)
Research Scholar, Civil Engineering Department,
Indian Institute of Science, Bangalore-12
E-mail: firstname.lastname@example.org tel.: +919886357650
Table 1 : Coefficient of K for Different Pipe Fittings Sl Number no Type/unit K of units Total K 1 Pipe Enlarger 0.1 4 0.4 2 T-Joint 1.8 8 14.4 3 900 Bend 0.9 6 5.4 4 Pipe Reducer 0.015 2 0.03 5 Elbow, 1350 0.9 7 6.3 6 Butterfly Valves 1.8 6 10.8 7 Air Valves 1.08 3 3.24 8 Non Return Valves 1.3 2 2.6 9 Condenser * 90 1 90 10 Sprinkler * 35 1 35 11 Sum K [approximately equal to] 168 12 Add 15% for miscellaneous items such as -pipe 25 joints, pipe leakages, curvatures of pipe line and any other unforeseen item or for any omissions and commissions in the piping system 13 Total K for pipe fittings, K [approximately 193 equal to] * Equivalent K- values assumed Table 2 : Design Parameters for Optimal Pipe Diameter Diameter Average Reynolds Friction Head loss due of the pipe velocity number Factor to friction D [bar.u] [10.sup.-6]R [lambda] [h.sub.f] m m/s -- -- m 2.5 2.04 5.09 0.00896 0.91 2.6 1.88 4.90 0.00901 0.75 2.7 1.75 4.72 0.00906 0.63 2.8 1.62 4.55 0.00911 0.53 2.9 1.51 4.39 0.00916 0.44 3 1.41 4.24 0.00921 0.38 3.1 1.32 4.11 0.00926 0.32 3.2 1.24 3.98 0.0093 0.27 3.3 1.17 3.86 0.00935 0.24 3.4 1.10 3.74 0.00939 0.20 3.5 1.04 3.64 0.00943 0.18 3.6 0.98 3.54 0.00947 0.16 3.7 0.93 3.44 0.00951 0.14 3.8 0.88 3.35 0.00955 0.12 3.9 0.84 3.26 0.00959 0.11 4 0.80 3.18 0.00963 0.09 Total Head loss K [[bar.u]. [h.sub.f] + Diameter sup.2]/(2g). [h.sub.m] + of the pipe Minor Losses [h.sub.l], Power D [h.sub.m] H P m m m kW 2.5 40.84 52.3 5134 2.6 34.91 46.3 4537 2.7 30.02 41.2 4045 2.8 25.95 37.1 3636 2.9 22.55 33.6 3295 3 19.69 30.7 3008 3.1 17.27 28.2 2765 3.2 15.21 26.1 2558 3.3 13.45 24.3 2382 3.4 11.94 22.7 2230 3.5 10.63 21.4 2099 3.6 9.50 20.3 1986 3.7 8.51 19.2 1888 3.8 7.65 18.4 1801 3.9 6.90 17.6 1726 4 6.23 16.9 1660 Expenditure on power per annum in crores of Rupees computed by assuming Rs Diameter 4 per unit of the pipe P[D.sup.2] (kW-hour) D -- Crores of m kw-[m.sup.2] Rupees 2.5 32085 17.98 2.6 30668 15.9 2.7 29485 14.18 2.8 28508 12.74 2.9 27709 11.54 3 27070 10.54 3.1 26571 9.68 3.2 26198 8.96 3.3 25939 8.34 3.4 25782 7.82 3.5 25719 7.36 3.6 25740 6.96 3.7 25841 6.62 3.8 26014 6.32 3.9 26254 6.04 4 26556 5.82 Note: kinematic viscosity of the water is assumed as [10.sup.-6] [m.sup.2]/s