OPTIMAL PIPELINE SIZING TECHNIQUE
By Helmi M. Hathoot/ Member, ASCE, Ahmed I. AI-Amoud/ Associate Member, ASCE,
and Fawzi S. Mohammad3
(Reviewed by the Pipeline Division)
ABSTRACT: In this paper, the writers present a formula for sizing steel pipelines of optimal diameter with
equally spaced, similar pumping units. The general case of an inclined pipeline with fitting is considered in the
analysis. The sizing formula is based on a friction-factor formula that spans through all the flow regimes of the
Moody diagram. Although the formula is implicit, the final solution is attained after a limited number of trial
cycles. A simple computer program carries out design calculations and provides the optimal pipeline size as
well as spacing between pumping units. Solved numerical examples are provided to demonstrate the simplicity
and practicability of the proposed technique.

INTRODUCTION
A number of factors have to be considered in sizing a pipeline of minimum cost. Among these factors are the initial investment cost of pipes and pumps, the annual operating and
maintenance costs for the life of the pipes and pumps, and the
salvage value of the pipeline (Albertson et al. 1960).
In general, a pipe with a large diameter produces a small
friction head loss against which each pump should act; on the
other hand, though a pipe with a smaller diameter is cheaper,
it corresponds to a greater friction head loss (Russel 1963).
Accordingly, the optimal pipeline size is that for which the
total annual cost of pipe, pumps, and power is a minimum
(Hathoot 1984; Cheremisinoff et al. 1988).
In an earlier treatment (Daugherty and Franzini 1977), the
total annual cost of a pipeline (K,) was given by
K, :::: aD2

+

bIDS

(1)

where D = the pipe diameter in meters and a and b are constants. In fact, b is not a constant since it contains the coefficient of friction (f) that varies with both the Reynolds number
and pipe roughness. Hathoot (1980) presented two pipeline
design formulas, one for smooth pipe flow and the other for
completely rough pipe flow, where the actual variation of f is
taken into account. A horizontal pipeline was investigated by
Hathoot (1984), who presented a single design formula that
represents a significant portion of the Moody diagram. Later,
Hathoot (1986) presented a number of pipeline design formulas that represent the Moody diagram and deal with an inclined pipeline (Fig. 1). The objective of this paper is to present a general steel pipeline-sizing technique that covers all
fluid flow regimes. It is worthy to note that SI units are used
throughout this paper.

given by the Darcy-Weisbach equation (Streeter and Wylie
1983) as
(2)

where f = coefficient of friction; L = spacing between two
successive pumping units in meters; V = average pipeline velocity in meters per second; and g = acceleration due to gravity
in meters per square second. For convenience, (2) is put into
this form
hI

=8fLQ hr gD
2

2

5

(3)

where Q = pipeline discharge in cubic meters per second.
Losses in pipe fittings (h fi ,) are generally given by
hilt

= kV 12g
2

(4)

where k = sum of the coefficients representing head loss in
fittings between two successive pumping units. In terms of the
discharge, (4) is put into this form
hflt

=8kQ2/g7r2D

4

(5)

Pumping Energy
The power required per pumping unit (p) is given by
P

='YQHp/TJ

(6)

where 'Y = specific weight of liquid in Newtons per cubic
meter; H p = total head provided per pumping unit in meters;
and TJ = pump efficiency. For an inclined pipeline, the total
head is given by
(7)

where So = slope of pipeline. The positive sign corresponds to

Pipe Losses
Friction losses constitute the major portion of pipe losses.
Between two successive pumping units, friction losses (hI) are
'Visiting Prof., Dept. of Agric. Engrg., Coli. of Agr., King Saud Univ.,
P.O. Box 2460, Riyadh 11451, Saudi Arabia; Prof., Pac. of Engrg., Alexandria Univ., Alexandria, Egypt.
2Assoc. Prof., Dept. of Agric. Engrg., Coli. of Agr., King Saud Univ.,
Riyadh, Saudi Arabia.
'Assoc. Prof., Dept. of Agric. Engrg., CoIl. of Agr., King Saud Univ.,
Riyadh, Saudi Arabia.
Note. Discussion open until November I, 1996. To extend the closing
date one month, a written request must be filed with the ASCE Manager
of Journals. The manuscript for this paper was submitted for review and
possible publication on June 27, 1994. This paper is part of the Journal
of Transportation Engineering, Vol. 122, No.3, May/June, 1996.
©ASCE, ISSN 0733-947X19610003-0254-0257/$4.00 + $.50 per page.
Paper No. 8744.

o
--...

FIG. 1.

Pipeline with Equally Spaced Pumping Units

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upward slopes, and the negative sign corresponds to negative
slopes; therefore, the power required per pumping unit is given
by
P = ('YQ/TJ)(hf

+ hOt ± LSo)

A

={2457 m[

W"

+ '027

r

W]

(16)

(8)

The power required per unit length of pipeline (w) may be
written as

and
B

= e7~30Y6

(17)

(9)
Cost of Energy

The levelized net annual cost of pumping energy per unit
pipe length (Ken) (Cheremisinoff et al. 1988) is written as
8jC2'YQ3
K =
en
9TJ,rD'

+

8kC2'YQ3
C2'YQSo
+--9TJ'lT 2W 4 TJ

(10)

where C2 = levelized net annual cost of pumping energy per
watt.

In computing f, A, and B, double precision should be used
since these quantities are calculated to very large and very
small numbers, therefore allowing them to be sensitive to
round-off error (Scaloppi and Allen 1993).
MINIMUM-COST PIPELINE SIZING

For the minimum cost of design, the levelized total annual
cost of pipeline per unit length, K"" should be a minimum. In
other words, the first derivative of K", (with respect to the pipe
diameter D) should equal zero (Hathoot 1986). Reynolds number may be written as

Cost of Pipes

R = 4Q/'lTDu

The levelized net annual cost of pipe per unit pipe length
(Kp ) (Cheremisinoff et al. 1988) is given by

where \l = kinematic viscosity of liquid. Substitution of R as
given in (18), into (15), (16), and (17) yields

(II)

_ 8

where t = pipe wall thickness in meters; 'Yp = specific weight
of pipe material; and C 1 = levelized net annual cost of pipes
per unit weight of pipe material. In practice, C1 is constant for
a suitable range of pipe diameters. The pipe wall thickness t
is roughly proportional to the pipe diameter (Davis and Sorensen 1969; Russel 1966) so that
t=

(12)

CD

j -

[(2'lTvD
Q )

K,.

8jC2'Y~

= gTJ'lT2D'

+

C2'YQSo

8kC2'YQ3
2LD4 ± - - gTJ'lT
TJ

+

CC

D2
l'Yp'lT

Coefficient of Friction

j =8

[(~)

12

where A and B are given by

]1/12

+ (A +1 B)I.'

+ 0.27 ( ~ )]}16
D

B

=

(9'

382d'lTDu)16

(20)

(21)

For convenience, (14) is put into the following form:
Mj

N

2

K,"

= D' + D4

M

= 8C 'YQ 3; N = 8kC2~Q3

± T + UD

(22)

where
Z

gTJ'lT2

T

(23,24)

gTJ'lT L

C2'YQSo
=--and U = CC1'Yp'lT

(25,26)

TJ

Substitution of j in (19) into (22), differentiating (22) with
respect to D, and equating to zero leads to
-16M
1lI12 6
D

(J)

N

- 4-

D'

[2 (!)12 +
R

2.457(Ei'(F)(G) + 2.5A
(A + Bi"

+ 2UD

+

3.5BJ

(27)

where
J=

(~Y2 + (A }

E = 2.457 In(l/F); F = (7/R)0.9
(IS)

(19)

B)I.'

and

0=

Swamee and Jain (1976) presented a coefficient of friction
equation that covers a significant portion of the turbulent zone
in the Moody diagram, 5,000 < R < 108 and 10- 6 < fiD <
10- 2 , where R = Reynolds number and E.= absolute roughness
of the pipe. However, Churchill (1977) provided a more general coefficient of friction equation that satisfactorily covers
both the turbulent and laminar zones of the Moody diagram
(Scaloppi and Allen 1993). According to Churchill (1977), the
coefficient of friction is given by

+

(A

Q

(14)

Since the coefficient of friction, f, is a function of the pipe
diameter, it is important to discuss it prior to any minimum
cost-analysis.

]1112

1

= {2.457 In [( 1.75'lTDv)°,9

Total Cost of Pipeline

According to (10) and (13), the levelized total annual cost
of pipeline per unit length (K,.) is

12 +

1
A

where C = constant of proportionality that depends upon the
expected pressure and diameter ranges of the pipe. Substitution
of (12) in (11) yields
(13)

(18)

G

= 0.9(7/R)0.9 -

B)u

+

0.27(£/D)

0.27(£/D)

(28)
(29,30)
(31)

solving (27) for D
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D=

(...!... { 16M
2U

+

4ND

(J)III12

[2 (!)12 +

2.457(E)"(F)(G)

R

(A

+

TABLE 1.

+ 2.5A + 3.5B]
B)2.5

D (m)

})'n

(32)

Eq. (32) is the governing equation for minimum cost pipeline sizing. The preceding equation is of the implicit type and
is to be solved through a trial-and-error procedure.
Two unknowns exist in (32): D and L, and (8) is to be
solved simultaneously with (32). The power per pumping unit,
W, [contained in (8)] should be known in advance for solving
such problems.
COMPUTER PROGRAM

The flowchart in Fig. 2 illustrates a simple computer program for estimating the most economical pipeline diameter. At
the beginning of the calculations, a rational average velocity
is assumed, 1.0 mls say, and the corresponding diameter is
estimated. The spacing L is then calculated by applying (8).
For the second trial, the pipe diameter is estimated by applying (32). A new trial-cycle begins after calculating the spacing L and considering the last calculated pipe diameter. A third
diameter is then calculated. Trial-cycles continue until the difference between two successive calculated values of D decreases. The following are numerical examples when the
aforementioned program is applied.
NUMERICAL EXAMPLE 1

It is necessary to determine a size of pipeline that can deliver 1.2 m 3/s of liquid up a slope of 1:20,000 for the following
data: C = 0.012; C 1 = 0.0026 $/N; C2 = 0.11 $IW; 'Y = 9,810
N/m 3 ; u = 1.01 X 10- 6 m%; 'Yp = 76,518 N/m 3 ; and £ = 7.2
X 1O-~ m. Pumping units are 80 kW each with an efficiency
of 0.67. Losses in fittings are such that k = 10.
Solution

Trial cycles results are listed in Table 1.
In this case, the diameter is taken to be D = 1.0 m. According to (8), the corresponding spacing between pumps L =

[Eq.23 .... M; [Eq.2S]"TT i
[Eq.26] _
U ; Area' 0/1.0;

o • 2 (Area In

q.1S]_R; [Eq.30]
F;
[Eq.20] ..... A ; [Eq.2 I
B;
[Eq.2S]
J; [Eq.19]_ I I
[Eq.S]_ L

J

yes

t
E NO

FIG. 2.

(1 )

L (m)
(2)

1.2361
1.0098
1.0305
1.0273
1.0278

7,110.67
2,306.45
2,611.96
2,562.41
2,569.71

TABLE 2.

)

Flowchart for Computer Program

e/D

R
(3)

1.223839
1.498061
1.467918
1.472569
1.512760

(4)
x
x
x
x
x

10·
10·
10·
10·
10·

5.825
7.130
6.987
7.009
7.005

x
x
x
x
x

10-'
10- 5
10- 5
10- 5
10- 5

Results of li'lal Cycles of Example 2

D (m)

L (m)

(1)

(2)

R
(3)

0.6676
0.8320
0.8553
0.8584
0.8588

1,909.71
4,888.07
5,504.07
5,591.60
5,603.37

1,056.26
847.48
824.44
821.43
821.03

e/D
(4)
4.4940
3.6057
3.5077
3.4949
3.4932

x
x
x
x
x

10- 4
10-4
10- 4
10- 4
10- 4

2,170 m. The flow is turbulent with R = 1.51 X 106 ; £/D =
7.2 X 1O-~; and V = 1.53 mls.
NUMERICAL EXAMPLE 2
It is necessary to determine the size of a pipeline that can
deliver 0.35 m 3/s of a liquid (u = 6.32 X 10- 4 m2/s and 'Y =
12,500 N/m 3 ) down a constant slope of 1:10,000 using equally
spaced pumping units each producing 60 kW with an efficiency of 0.66. A system of fittings is used so that k = 8. The
following data are available: C = 0.01; C1 = 0.002 $/N; C2 =
0.15 $IW; 'Yp = 73,575 N/m3 ; and £ = 3.0 X 10-4 m.

Solution

Trial-cycle results are shown in Table 2.
For convenience, the chosen diameter is D = 0.85 m and
application of (8) yields L =5,359 m. In this case, the flow is
laminar with R = 829.55 and V = 0.62 mls.
CONCLUSIONS

The minimum cost pipeline sizing formula presented in this
paper spans all flow regimes, namely laminar and turbulent.
Although the pipeline sizing formula is implicit, its convergent characteristics make the final solution attainable after a
limited number of trial cycles. The computer program provided by the writers proves to be simple and practical. The
solution of two practical examples yields reasonable pipeline
sizing results.
APPENDIX I.

)1/2. 01. 0 . 1 .0

Results of li'lal Cycles of Example 1

REFERENCES

Albertson, M., Barton, J., and Simons, D. (1960). Fluid mechanics for
engineers. Prentice-Hall, Inc., New York, N.Y.
Cherernisinoff, P. N., Cherernisinoff, N. P., and Cheng, S. L., eds. (1988).
Civil engineering practice. 2-HydraulicslMechanics., Technornic Publishing Co., Lancaster, Pa.
Churchill, S. W. (1977). "Friction-factor equation spans all fluid-flow
regimes." Chern. Engrg., 84(24), 91-92.
Daugherty, R., and Franzini, J. (1977). Fluid mechanics with engineering
applications. 5th Ed., McGraw-Hill Book Co., New York, N.Y.
Davis, C., and Sorensen, K. (1969). Handbook ofapplied hydraulics. 3rd
Ed., McGraw-Hill Book Co., New York, N.Y.
Hathoot, H. M. (1980). "Optimum design of irrigation pipelines." Bull.•
Int. Commission on Irrig. and Drain. (ICID), 29(2), 73-76.
Hathoot, H. M. (1984). "Minimum cost design of horizontal pipelines."
J. Transp. Engrg., ASCE, 110(3),382-389.
Hathoot, H. M. (1986). "Minimum cost design of pipelines." J. Transp.
Engrg., ASCE, 112(2), 465-480.
Russel, G. (1963). Hydraulics. 5th Ed., Holt, Rinehart and Winston, New
York, N.Y.

256/ JOURNAL OF TRANSPORTATION ENGINEERING / MAY/JUNE 1996

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Russel, G. (1966). Standard handbook for mechanical engineering. McGraw-Hili Book Co., New York, N.Y.
Scaloppi, E. J., and Allen, R. G. (1993). "Hydraulics of center-pivot
laterals." J. lrrig. and Drain. Engrg., ASCE, 119(3),554-567.
Streeter, V. L., and Wylie, B. E. (1983). Fluid mechanics. Int. Stud. Ed.,
McGraw-Hili Book Co., New York, N.Y.
Swamee, P., and Jain, A. (1976). "Explicit equations for pipe flow problems." J. Hydr. Div., ASCE, 657-664.

APPENDIX II.

NOTATION

The following symbols are used in this paper:
A = quantity defined by Eq. (16);
a = constant contained in Eq. (1);
B = quantity defined by Eq. (17);
b = constant contained in Eq. (1);
C = constant contained in Eq. (12);
levelized net annual cost of pipes per unit weight of pipe
C1
material ($/N);
levelized net annual cost of pumping energy per watt
C2
($/W);
D = pipe diameter (m);
E = quantity defined by Eq. (29);
F = quantity defined by Eq. (30);
f = coefficient of friction;
G = quantity defined by Eq. (31);
g = acceleration due to gravity (m/s2 );
H p = total head per pumping unit (m);
hI = friction head loss (m);

hR.
J
K. n
Kp

= head loss in pipe fittings (m);
= quantity defined by Eq. (28);
= levelized net annual cost of pumping energy per unit pipe
length ($/m);
= levelized net annual cost of pipe per unit pipe length
($/m);

K,
K,.

k

= total annual cost of pipeline ($);

= levelized total annual cost of pipeline per unit pipe length
($/m);
= sum of coefficients representing head loss in fittings between two successive pumping units;
= spacing between pumping units (m);
= quantity defined by Eq. (23);

L
M
N = quantity defined by Eq. (24);
P = power provided by each pumping unit (w);
Q = pipeline discharge (m3/s);

R
So
T
t
U
V

W

= Reynolds number;

= slope of pipeline;
= quantity defined by Eq. (25);
= wall thickness of pipe (m);
= quantity defined by Eq. (26);
= average velocity of liquid (m/s);
= power required per unit length of pipeline (w/m);

'Y = specific weight of liquid (N/m3);
'Yp
specific weight of pipe material (N/m3);

=

E

TJ

u

= absolute roughness of pipe material (m);
= pumping unit efficiency; and
= kinematic viscosity of liquid (m /s).
2

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