# Example 7: Modeling pipes of different diameters in series as a single equivalent pipe of constant diameter

Background

The examples provide a comparison of AioFlo results with published data from well known and respected references that are generally accessible to engineers. This will allow prospective AioFlo users to validate its accuracy against a range of typical calculations. The worked examples can also be run by new users as part of their learning process. To learn more about AioFlo click on "Home" in the menu above.

Description

This example considers a typical real world situation with water flowing through a pipeline consisting of two sections of different internal diameter in series. The pipeline includes pipe fittings and a change in elevation. In addition to the pipeline details, the flow rate of the water is known.

The method used illustrates the equivalent pipe method where a single pipe of constant diameter is used to model multiple pipe sections of different diameters in series.

Problem Reference

Flow of Fluids through Valves, Fittings, and Pipe. 1999, Crane Co., TP410M, Page 4-8, Example 4-14

Fluid Details

 Fluid : Water at 15°C Phase : Liquid (incompressible) Density : 999 kg/m³ Viscosity : 1.1 cP Flow rate : 1500 liter/minute

Pipe Details

 Section 1 Pipe size : 4" Sch 40 (ID 102.3 mm) Roughness : 0.05 mm Pipe length : 34 m Fittings : 1 x 4" to 5" reducing elbow (*) Section 2 Pipe size : 5" Sch 40 (ID 128.2 mm) Roughness : 0.05 mm Pipe length : 67 m Fittings : 1 x welding elbow Elevation change : 22 meters

(*) AioFlo does not have data for reducing elbows so this is modeled as a 4" elbow followed by a 4" to 5" pipe reducer.

To be Calculated

Determine the pressure differential between the beginning and end of the pipeline using the equivalent pipe method to model the entire line with the equivalent length of a single constant diameter pipe.

You can run this example in AioFlo by downloading and then opening the data file in AioFlo.

Comparison of Results

Calculated Item Reference AioFlo
Pressure differential 2.6 bar 2.641 bar

Discussion

In this example, where the goal is to calculate the overall pressure differential, it would be easy to calculate the pressure drops for each of the sections and then add them together. However, the technique of modeling pipe sections of different diameters in series as a single equivalent pipe of constant diameter is very useful - especially when it comes to determining flow rates from known pressure drops. The equivalent pipe method is a powerful tool that every piping engineer should have at his/her disposal.

The approaches to the problem taken by Crane and AioFlo are slightly different but the underlying principles are the same. In the Crane example the resistance coefficients of each fitting and pipe section are scaled separately, but in AioFlo all elements of each section are lumped together.

The solution method used here in AioFlo takes advantage of the fact that the pressure drop in turbulent flow through pipes varies inversely with the 5th power of the diameter. If the pressure drop through a pipe of a given length and of inside diameter diam1 is dp1, then the same flow rate through a pipe of the same length and of inside diameter diam2 will result in a pressure drop dp2, i.e.

(dp2 / dp1) = (diam1 / diam2)^5, and

dp2 = dp1 x (diam1 / diam2)^5

Since the pressure drop is linear with pipe length, we can use this same ratio to calculate the equivalent length of the pipe of diameter diam1 that would have the same pressure drop as the original pipe section with diameter diam2.

equivalent length of pipe 2 = Actual length of pipe 2 x (diam1 / diam2)^5

Therefore the actual length (67 m) of the 5" pipe can be modeled as 67 x (102.3 / 128.2)^ 5 = 21.6 m of 4" pipe. The total length of the pipeline is modeled as 55.6 m (= 34 + 21.6) of 4" pipe. Obviously the 4" pipe could be modeled as 5" pipe to give an equivalent total length of 5" pipe. How do we decide which diameter to use as the basis? The diameter of the pipe section that would result in the highest pressure drop should be used as the basis because then any error in modeling the other section would result in any modeling errors being applied to the smaller pressure drop, giving the best overall accuracy. In this example we have 34 m of actual 4" pipe and an equivalent length of 21.6 m of 4" pipe representing the 5" pipe section. The 34 m will clearly have a higher pressure drop than the 21.6 m and we select the 4" pipe as the basis.

As shown in the comparison table above, the results from the Reference and AioFlo are very close. If this technique is used for determining the flow through pipelines consisting of several sections of different diameters when the overall pressure drop is known, the final result can be confirmed by re-calculating the pressure drops for each section based on the actual sizes and the calculated flow rate and adding them together. Usually any errors introduced by this modeling technique will be less than the uncertainties in the parameters of the pipeline and the fluid.

In laminar flow the exponent used for calculating the ratio is 4 (in place of the 5 used above for turbulent flow).