# A Shortcut to Optimize Pipe Diameters by Economic Criteria

By Srbislav Genic and Branislav Jac´imovic |

*This methodology can be used not only for laminar flow, but also for the complete turbulent flow region*

In recent decades, the impact of energy costs on the transport of fluids has led to a significant reduction in fluid velocity and an increase in the diameter of the pipeline. In this article, an improved shortcut methodology is presented for estimating the optimal diameter of pipelines. Besides the laminar flow regime, the algorithm can be applied, for the first time, to the complete turbulent flow region. Several examples are presented that demonstrate the simplicity of the method and illustrate the impact of changes in economic parameters on the optimal diameter of pipelines. Moreover, a sensitivity analysis was done to show that the optimization model provides the solution in an acceptable range, especially for the conceptual phase of projects, as well as for basic design.

# Nomenclature

*A* Parameter, see Equation (14)

*a* Amortization parmeter, $/yr)

*B* Parameter, see Equation (15)

*b* Maintenance costs, $/yr

*C _{c}* Annual capital cost, $/yr

*C _{e}* Annual operational cost of pipeline, $/(W·h)

*c _{en}* Cost of electric energy, $/(W·h)

*D* Pipe inner diameter, m

*D _{opt}* Optimal pipe inner diameter, m

*E* Overall efficiency of pump, compressor or fan

*F* Parameter for cost of fittings, valves, erection

*G* Mass flowrate of fluid, kg/s

*J* Ratio of minor pressure losses and friction pressure drop

*L* Pipe length, m

*P* Pumping power, W

*P _{c}* Pipeline purchase cost, $

Δ*p* Pressure drop, Pa

Δ*p _{fr}* Friction pressure drop, Pa

Δ*p _{ml}* Minor pressure losses, Pa

Re Reynolds number

*R _{r}* relative pipe roughness

*V* Volumetric flowrate, m^{3}/s

*X* Parameter that depends on pipe material

*x* Parameter that depends on pipe thickness

*Y* Plant attainment, h/yr

η Fluid viscosity, Pa·s

ρ Fluid density, kg/m ^{3}

ξ Friction factor

*v* Average fluid velocity, m/s

_{c} _{e} _{en} _{opt} _{c} _{fr} _{ml} _{r} ^{3} ^{3}

## Background

Engineers that work in the chemical process industries (CPI) in fields such as process engineering, oil-and-gas engineering, heating, ventilation and air conditioning (HVAC), thermal and hydroelectric power and related disciplines, quite often perform calculations for pipeline-system transportation of fluids. Fluids flow through plant equipment including reactors, separators, heat exchangers, boilers, tanks, radiators and other devices that are connected by pipes or channels. In this article, we will consider only continuous steady-state transport of fluids.

In order to ensure adequate transport of fluids through the plant, engineer have to select, determine or define the following:

- The route of the pipelines
- The type and size of each pipe (diameter, length and wall thickness)
- The types and sizes of necessary fittings, valves and other accessories
- The types of flowmeters
- The types and basic parameters of pumps, compressors, blowers or fans
- The types of materials for all elements of the transport system (pipes, ducts, fittings, electrical machinery, valves and so on)
- The type and size of pipeline supports and compensating elements
- Transportation safety measures, which can be reduced to stress calculations, testing before and during the commissioning, implementation of protective measures in order to prevent negative external impacts on the transportation system and action taken in order to protect the environment from possible failures of the transport system
- Economic optimization of transport systems in order to reduce the overall costs

Aforementioned elements of design are usually done after design of the basic equipment needed for the process. Some of the listed parameters will be analyzed in more detail.

The most important elements relating to the pipeline transport of fluid are as follows:

- Transportation capacity that should meet the technical and other requirements of all consumers that are supplied with fluids
- Selection of appropriate pumps and other machines needed for the transport of fluids in order to achieve the necessary pressure head
- Choice of materials for the pipeline or channel and pumping devices taking into account the impact of fluid on materials and possible degradation of the material due to erosion, corrosion and cavitation
- Possible accumulation of liquids or solids in the pipeline
- Occurrence of vibration and noise
- Transport safety that can be tantamount to mechanical calculations, tests before and during the commissioning and implementation of protecting measures against external impacts, as well as actions to protect the environment from possible failures of the pipeline transport system
- Economic optimization of the pipeline transport system, which is done in order to reduce transport costs

Economic criteria is, in most cases, crucial for the design of plants, so the optimum size of the series of units that the plant consists of provides the lowest life-cycle cost of any project. The cost of piping typically represents up to 35% of plant capital cost [*1*]. Fluid pumping cost is also an important part of plant operation cost. According to the U.S. Dept. of Energy (DOE), 16% of a typical facility’s electricity costs are involved with pumping of fluids [*2*]. It is therefore useful to design the piping system as close as possible to the optimal value, in order to minimize the sum of the capital and operating expenses.

Mathematically rigorous methods for selecting pipe diameters are time-consuming because they involve detailed (iterative) procedures to determine the minimum capital and operating costs. Simple equations, like the one proposed in this article, can provide reasonably accurate estimates of optimal pipe diameters in the initial stages of a plant design, which are a good starting point for a more rigorous procedure.

The first pipe economic optimization model was published in 1937 for turbulent flow in hydraulically smooth pipes [*3*] and three years later the model was broadened for laminar flow [*4*]. These models are widely cited in literature and they even became classic university lectures [*5*]. Recently, a new model was published for hydraulically rough pipes [*6*].

This article contains new and simple models for economic pipe sizing for laminar, turbulent and transitional flow.

## Pipe optimization model

Pipeline cost consists of two parameters: capital cost and operational cost. The most economic pipe diameter will be the one which gives the lowest annual cost.

Pipe purchase cost ( *P* _{c}) can be expressed as Equation (1):

(**1**)

Where:

*D* = pipe inner dia. (I.D.), m

*L* = pipe length, m

*X* = a parameter that depends on the type of pipe material

*x* = a parameter that depends on the pipe-wall thickness (pipe schedule)

The annual capital cost ( *C*_{c}, $/yr) of a pipeline is calculated using Equation (2):

(**2**)

Where:

*F* = a factor that includes the cost of valves, fittings and erection

*a* = amortization or capital charge (annual)

*b* = maintenance costs (annual)

The power required for fluid pumping (*P*, W) is given by Equation (3):

(**3**)

Where:

*V*, = volumetric flowrate of fluid, m^{3}/s

ρ = fluid density, kg/m^{3}

Δ*p* = pressure drop, Pa

*E* = overall efficiency of the pump, compressor or ventilator

*G* = mass flowrate of fluid, kg/s

(**4**)

Energy loss in the pipeline includes fluid friction loss, but also potential and kinetic energy losses. This model excludes the latter two losses since the pumping height is always a fixed value and fluid density is considered to be constant. This means that the pressure drop can be calculated as the sum of the friction pressure drop (Δ*p*_{fr}, Pa) and minor pressure losses (Δ*p*_{ml}, Pa):

(**5**)

Weisbach [*7*] proposed in 1845 the equation that is still in use for pressure drop calculations:

(**6**)

Where:

ξ = friction factor coefficient

*v*, = average fluid velocity, m/s

Minor pressure losses can be estimated either as head losses or by using equivalent lengths. In further analysis, the minor pressure losses will simply be taken into account as:

(**7**)

where *J* is the ratio of minor pressure losses and friction pressure drop.

Pressure drop then becomes:

(**8**)

Strictly speaking, Equation (8) applies only to incompressible isothermal flow. In engineering practice, this equation can be accepted for the compressible flow if the total pressure drop is less than 10% of the initial pressure [*8*].

The annual operational cost of the pipeline (*C*_{e}, $/yr) is:

(**9**)

which yields:

(**10**)

Where:

*Y* = the plant attainment (annual operating hours or hours of operation per year), h/yr

*c*_{en} = cost of pumping energy, $/W·h

Total annual pipe cost is:

(**11**)

and, since *C* depends only on *D*, the optimum economic pipe diameter can be found by:

(**12**)

After differentiation of capital, Equation (2), and operational costs, Equation (10), the general form of the solution for Equation (12) is:

(**13**)

Parameter *A* consists of values that must be presupposed for every single pipeline:

(**14**)

and parameter *B* shows the influence of pipe diameter on the friction factor:

(**15**)

In order to obtain the useful shortcut formula for estimation of *D*_{opt}, the derivative dξ/d*D* has to be determined for laminar, critical and turbulent flow. The pipe friction factor, in general, depends on the Reynolds number (Re):

(**16**)

and relative pipe roughness:

(**17**)

The friction factor for all cases of flow regimes in pipes can be calculated using [*9*]:

(**18a**)

(**18b**)

(**18c**)

Pipe relative roughness, according to open literature data, is in the range *R*_{r} = 0 to 0.0333.

For laminar flow, the explicit solution is *B* = 3.24, and for other flow regimes, the mean value of *B* should be estimated after integration. For critical flow, the following applies:

(**19**)

and for turbulent flow (after setting of the upper limit of the integral to Re = 10^{8}, which is a value that is of importance to engineering practice):

(**20**)

Our estimation for variables that should be known for calculation of *A* are presented in Table 1, and the final value is *A* = 0.00236.

After applying *A* in Equation (13), one can get ( *A ^{.} B*)

^{1/5+}

^{x}= 0.47 for laminar flow and (

*A*)

^{.}B^{1/5+}

^{x}= 0.49 for other flow regimes. Therefore, a value of (

*A*)

^{.}B^{1/5+}

^{x}= 0.48 is adopted for all flow regimes in all of the subsequent calculations.

## Economic flowrate or velocity

Optimal pipe diameter equations can be rewritten in order to estimate the most economic flowrate or velocity for a given pipe diameter. From Equation (13), the optimal mass flowrate is:

(**21**)

the optimal volumetric flowrate is:

(**22**)

and the optimal fluid velocity is

(**23**)

## Algorithm for solving equations

Equation (13) provides an implicit calculation procedure, since the pipe diameter has to be known in order to obtain the friction factor or vice versa. Solving of Equation (21), (22) or (23) also demands the iterative procedure since ξ is a function of fluid velocity. The easiest way for solving the listed equations is to make the assumption of a fluid velocity or friction factor and then apply an iterative procedure. In case of turbulent flow, no more than three iterations are needed. For laminar flow, no more than five iterations are needed.

## Example 1

The authors were recently asked to estimate the optimal diameter of carbon-steel pipeline for liquid methanol at ambient temperature. Methanol flowrate was 500 ton/h, its density is 794 kg/m^{3} and viscosity is 0.000576 Pa·s. We adopted the absolute roughness ε = 1 mm as a guess value for the end of pipeline life cycle of 10 years.

** Solution.** Using the methodology proposed above yields an optimal pipe diameter of

*D*

_{opt}= 341 mm, the corresponding velocity is

*v*= 1.915 m/s and pressure drop is Δ

*p*/

*L*= 164 Pa/m. This corresponds to the total pipe cost of

*C/L*= 138.7 $/(yr·m).

Rules of thumb from various literature sources give the results presented in Table 2. The value calculated by Equation (13) gives the velocity that is significantly lower than the ones from the cited recommendations. The consequential total pipeline cost is about 30% greater than the one calculated by hereby proposed model. This fact is in good agreement with the conclusion from Ref. 10 about the influence of energy cost on pipe diameter in recent decades.

## Example 2

For the next example, consider the flow of bitumen through a carbon-steel pipeline with *D* = 82.5 mm, with volumetric flowrate of *V* = 20 m^{3} h. At 150°C, the density of bitumen is ρ = 959 kg/m^{3} and the viscosity is η = 0.407 Pa·s. What is the optimal pipe diameter and what savings can be obtained after replacing the existing pipeline with the optimal one if the length of pipeline is L = 1,000 m?

** Solution.**According to Equation (13), the optimal pipe diameter is

*D*

_{opt}= 110 mm (

*v*= 0.585 m/s, Re = 152, ξ = 0.422) and pressure drop is Δ

*p*= 9.44 bars.

For further analysis, we have used the standard steel pipe DN100 (O.D. = 114.3 mm, I.D. = 107.1 mm). Using Equations (2) and (10), capital cost is *C*_{c} = 17,350 $/yr, operational costs are *C*_{e} = $6,820/yr, so the total cost is *C* = $24,170/yr. For *D* = 82.5 mm, the pipeline total cost is *C* = $31,090/yr. This means that, after replacement, each year the savings will be $6,920, and the investment will be paid off after 4.5 yr.

On the other hand, recommended velocities and pressure drops from the literature are: 0.6–1.0 m/s in Ref. 14, 0.9–1.2 m/s and 450 Pa/m in Ref. 13, 0.2–1 m/s in Ref. 15, 0.9–1.5 m/s in Ref. 16 for viscous oils and pipelines with nominal diameter DN80–DN250. It is obvious that these recommendations are not covering the region of laminar flow, and one can make a serious mistake by following them.

## Sensitivity analysis

The economic analysis consists of the following two parts:

- The investment and cash flows are first calculated using the most probable values of the various factors (this establishes the base case for analysis).
- Various parameters in the cost model are then varied, assuming a range of error for each factor in turn, in order to provide how sensitive the cash flows and economic criteria are to errors in the forecast figures.

In other words, the basic economic analysis of a project is based on the best estimates that can be made at the moment of design of the system, and a sensitivity analysis is a way of examining the effects of uncertainties in the forecasts on the viability of a project.

The purpose of a sensitivity analysis is to identify those parameters that have a significant impact on project viability over the expected range of variation of the parameter. Ranges of variation of typical parameters are given in Table 3, which is adopted from Ref. 17.

Data from Table 3 suggest the following variation of parameters:

*G*should be in the range 0.8 to 1.2 of basic*G**c*_{en }should be in the range 0.5 to 2 of basic*c*_{en}*A*should be in the range 0.8 to 1.5 of basic*A*

Furthermore, for the shortcut model presented here, we should include the following:

- The factor of minor losses in the range
*J*= 0 to 2 - Pump efficiency from
*E*= 0.6 up to*E*= 0.85 *a*+*b*in the range 0.1–0.20*R*_{r }should be taken into account for new, as well as for old pipes — for carbon steel pipes ε = 0.05–1 mm

Let’s reconsider Example 1. After consideration of the sensitivity analysis, we get the pipe diameters listed in Table 4.

In this case, the mean value of the pipeline diameter is in the range *D*_{opt} = 316–371 mm, which is a variation of less than±10% of the pipe diameter of *D*_{opt} = 341 mm obtained in Example 1. Following the data from Table 4, for this example, one can conclude that the variation of pipe diameter is not significant due to the (very strong) exponent 5 + *x* = 6.5 in Equation (13).

## Concluding remarks

The simple optimization model of pipe diameter presented here is suitable for both laminar and turbulent flow. The sensitivity analysis shows that the optimization model can provide the solution in an acceptable range that is especially suitable for the conceptual phase of projects, as well as for the basic design. n

## References

1. Capps, R. W., Select the Optimum Pipe Size — Simple Equations can Quickly Optimize Pipe Diameters,

Chem.Eng., vol. 105, no. 7., pp. 165–166, 1995.

2. “Improving Pumping System Performance: A Sourcebook for Industry,” Hydraulic Institute ITP, 2006.

3. Genereaux, R. P., Fluid-Flow Design Methods,

Ind. Eng. Chem., vol. 29, pp. 385–388, 1937.

4. Sarchet, B. R., and Colburn, A. P., Economic Pipe Size In The Transportation Of Viscous And Nonviscous Fluids,

Ind. Eng. Chem., vol. 32, pp. 1,249–1,252, 1940.

5. Sinnott, R. K., “Chemical Engineering Design,” Butterworth-Heinemann, 1996.

6. Genic´, S., Jac´imovic´, B., Genic´, V., Economic Optimization of Pipe Diameter for Complete Turbulence,

Energy and Buildings, vol. 45, pp. 335–338, 2012.

7. Weisbach, J., “Lehrbuch der Ingenieur- und Maschinen-Mechanik,” Braunschwieg, 1845.

8. Peters, M. S., Timmerhaus, K. D., “Plant Design and Economics for Chemical Engineer,” McGraw Hill, New York, N.Y., 1991.

9. Genic´, S., Jac´imovic´, B., “Reconsideration of the Friction Factor Data and Equations for Smooth, Rough, and Transition Pipe Flow” presented at 1st International Conference on Computational Methods and Applications in Engineering, Timisoara, Romania, May 2018.

Reconsideration of the Friction Factor Data and Equations for Smooth, Rough and Transition Pipe Flow, Lecture held in Engineering Chamber of Serbia, Belgrade, April 2016.

10. Durand, A. A., and others, Updating the Rules For Pipe Sizing,

Chem. Eng., vol. 116, no. 1, pp. 48–50, 2010.

11. Adams, J. N., Quickly Estimate Pipe Sizing with ‘Jack’s Cube,”

Chem. Eng. Progress,vol. 93, no. 12, pp. 55–59, 1997.

12. “Piping Engineering,” Tube Turns Inc., Louisville, 1986.

13. Walas, S. M., “Chemical Process Equipment – Selection and Design,” Butterworth-Heinemann, Boston, Mass., 1990.

14. Coker, A. K., “Ludwig’s Applied Process Design for Chemical and Petrochemical Plants“, Fourth Edition: Volume 1, Elsevier Inc., 2007.

15. Pavlov, K. F., Romankov, P. G., Noskov, A. A., “Chemical Technology Processes and Equipment: Examples and Tasks,” Leningrad, Nedra Publ., 1987.

16. Datta, A., “Process Engineering And Design Using Visual Basic,” CRC Press, 2008.

17. Genic´, S., Jac´imovi, B., Mitic´, S., Kolendic´, P., “Economic Analysis for Process Engineering” (in Serbian), Association of Mechanical and Electrical Engineers and Technicians of Serbia, Belgrade, 2014.

## Authors

Srbislav Genic´ is full professor of heat transfer operations and equipment, mass transfer operations and equipment and Economic analysis in process engineering at the Dept. of Process Engineering of the Faculty of Mechanical Engineering of the University of Belgrade (Kraljice Marije 16, 11120 Belgrade 35; Phone: +381-11-3302-200; Email: sgenic@mas.bg.ac.rs). His research interests include heat and mass transfer processes and equipment. He holds B.Sc., M.Sc., and Ph.D. degrees in mechanical engineering from the University of Belgrade. He is a court expert and registered Professional Engineer in Serbia.

Branislav Jac´imovic´ is retired full professor of Heat transfer operations and equipment and Mass transfer operations and equipment at the Dept.of Process Engineering of the Faculty of Mechanical Engineering of the University of Belgrade. He is a registered Professional Engineer in Serbia and holds B.Sc., M.Sc., and Ph.D. degrees in mechanical engineering at University of Belgrade. He has over 35 years experience in the field of separation processes (especially distillation) and heat exchangers design.

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