DESIGN OF A CENTRIFUGAL PUMP-PIPE SYSTEM

Alexander Aueron, Robert Graham, Mark James, Purvil Patel, Steven Rosenberg,

Blake Singer, Gabrielle Steinberg, Zaid Syed-Ali

(Abstract)

The goal of this report was to design a pump-pipe system in order to transport water at a rate of

0.5 m3/s through 200 meters of schedule 40 pipe to an elevation of 100 meters for the lowest

cost. To fully define the final system, it was necessary to identify the impeller blade angles, pipe

diameter and material, and impeller diameter and speed. This was achieved through an analysis

of system head losses and pump performance. From these parameters, the system could then be

analyzed on the basis of cost to determine the least expensive and therefore ideal design option.

Design options were considered over a range of diameters, with each option being optimized on

the basis of cost. These diameters were then analyzed to ensure that appropriate suction head

could be achieved in order to avoid cavitation in the pumps. The final design option was then

analyzed to ensure that flow pressure did not exceed the strength limitations of the selected

material, PVC pipe. The optimal design, on the basis of cost, was found to use a pipe diameter of

0.236 m and a scaled-up pump with an impeller diameter of 0.504 m, operating at 1390 rpm.

This centrifugal pump utilizes a backward-swept impeller with inlet and outlet blade angles of

21.7º and 13.5º, respectively. This system marks a 40 percent increase in scaling from the initial

pump system and yields a total cost of $8,170,000.

INTRODUCTION TO DESIGN PROCESS

System Parameters and Requirements

The given task was to design a system to transport water from system inlet to outlet, given a

centrifugal pump system, with the following design characteristics:

1. Volumetric flow rate of

2. Elevation increase of

3. Total pipe length of of schedule 40 pipe

4. Ten pipe bends

It is also given that both inlet and outlet are at atmospheric temperature.

As an additionally design requirement, the final system will use a scaled-up pump with the

original pump providing the following characteristics:

Inlet and outlet impeller radii of and

Inlet and outlet impeller widths of and

For a given flow rate of ⁄ , the pump head rise is for an impeller speed of

1720 rpm

Flow enters the impeller parallel to the blade such that ⁄

Performance curve given by:

(

)

Where,

flow rate, given in ⁄

Pump power given by:

(

) (

)

Where,

brake horsepower, given in watts

This pump can be scaled up as much as 300%, with a maximum impeller speed of 2500 rpm, for

any allowable diameter.

Design Objective

The final design must identify the following system parameters to define the system.

1. Pipe diameter

2. Impeller outer diameter

3. Impeller speed

4. Inlet and outlet impeller angles

5. Number of pumps in use

This analysis will identify optimum system parameters, dependent on the number of pumps in

use. For these design options, the total system cost must be determined in order to select the least

expensive system configuration.

METHOD OF ANALYSIS

In order to select the optimum system design, it is necessary to determine the optimum pipe

diameter, impeller diameter, and impeller velocity. To achieve this selection, this report will first

determine the total head losses of the pipe system and select a range of appropriate pipe

diameters.

Given that the impeller diameter can be scaled-up by as much as 300%, the full range of

available impeller diameters was considered to determine the appropriate impeller speed, for the

required flow rate of ⁄

From these values, the affinity laws were implemented to determine the head rise per pump in

series and the brake horsepower of each pump. Design options were selected such that the pump

systems meet the head loss experienced from the pumps to the system outlet. An intermediary

cost analysis was performed to select the most viable design options, considering series and

parallel pump configurations. A final cost analysis of each design option was performed to

determine the least expensive system configuration.

Selection of Pipe Diameters

To determine the allowable pipe diameters, the net positive suction head was considered, to

prevent cavitation and ensure optimum pump performance. The net positive suction head

required was given by

(

)

Where,

is given in meters

is given in ⁄

This report will then perform an analysis of the net positive suction head available as a function

of the pipe diameter, which requires an analysis of the head loss from the system inlet to the

pump and the vapor pressure of water at room temperature. The allowable pipe diameters were

given as those which provided greater NPSHA than that required by the pump.

Selection of Pump Systems

Given the head loss experienced for each pipe diameter, pump systems were evaluated on the

basis of series and parallel configurations to determine the appropriate design options. An

intermediary cost analysis was performed for the selection of design options.

Determination of Final Design

From the selected design options, a comparison of total system cost was performed to determine

the least expensive design option. This analysis considered the total cost of pipe, alongside pump

cost and total operational costs for continuous operation over a span of ten years.

SYSTEM MODEL

This design assumes that the final pump of the system will be located 50 m below the system

inlet and will be pumped 150 m, vertically, to the system outlet. There is a 36 m drop from the

system inlet to the first pump and a second 14 m drop to the second pump, in order to avoid

cavitation within the pump. The required ten bends will be divided before and after the pump

systems such that half of the bends will be used to deliver water to the pump and the remaining

five will be used to pump the water to the outlet. The only horizontal segments of piping will be

the elbow fittings, such that horizontal flow and major losses can be considered negligible. It is

also assumed that water will enter the system through a square-edged entrance and that the

system will demonstrate constant velocity between the system inlet and outlet.

APPROACH

Selection of Pipe Diameters

For a properly functioning pump, the net positive suction head available (NPSHA) must be

greater than the net positive suction head required (NPSHR). If the NPSHR exceeds the NPSHA,

cavitation occurs in the pump and the liquid can locally flash to vapor. The presence of vapor

alters the local pressure and can result in unsteady flow, inducing vibrations in the pump, leading

to pump damage.

The net positive suction head can be calculated as

Eq. 14-8, pg. 746, Cengal & Cimbala

Where,

The head of the fluid prior to entering the pump can be found by the Bernoulli equation,

Eq. 14-6, pg. 740, Cengal & Cimbala

Where,

(∑ ∑

) Eq. 8-58, pg. 349, Cengal & Cimbala

The total head loss is given by:

(∑ ∑

)

Let ⁄ represent the pump head HP, such that

( )

(∑ ∑

)

Since a required flow rate is given, the velocity is dependent on pipe diameter as shown below:

(

)

(

)

(

)

For a smooth 90 bend connected by a flange, . For a standard 90º elbow fitting,

⁄ . Thus the head at the first pump is given by

( )

It is now necessary to consider the friction factor. Assuming turbulent flow,

( ⁄ )

Using the Colebrook equation,

√ (

√ )

Eq. 8.37, pg. 360, Fox & McDonald

Where,

absolute roughness factor, in meters, dependent upon pipe material

“e/D” is the relative roughness, a dimensionless quantity describing the roughness of a pipe in

relation to its diameter. The absolute roughness is a constant characteristic of the pipe, assuming

the pipe is of homogeneous material characteristics. The absolute roughness of PVC is 1.5

microns (10-6

m) and is used in the calculations for relative roughness and friction factor. This

analysis considered only the use of PVC, due to the low absolute roughness value and

significantly lower material costs when compared to commercial steels, aluminum, and brass.

As shown in the Colebrook equation, the friction factor is dependent upon the flow Reynolds

number. The Reynolds number is defined in terms in terms of fluid density, viscosity, velocity

and pipe diameter.

Adapted from

Table 7-5, Cengel & Cimbala

Within the Reynolds number, the fluid velocity can be defined in terms of volumetric flow rate

and the cross-sectional area (also a function of pipe diameter):

Thus, the Reynolds number can be written as

With the appropriate substitutions, the Colebrook equation can be rewritten as:

√ (

√ )

In this formula, all parameters except pipe diameter, D, are held constant, as absolute roughness

is characteristic of the pipe and volumetric flow is fixed by the customer’s request.

One method to solve this implicit equation is to subtract the right-hand side (RHS) from the left-

hand side (LHS) and then use the Newton-Raphson Method to solve the roots of the resulting

equation. Excel also has iterative capabilities to solve implicit functions. To use this method,

the settings for the spreadsheet are set to solve circular relations in cells using an iterative

approach, with these particular equations using 100 iterations per cell.

With tabulated values for the Darcy friction factor, the NPSHA of the initial pump can now be

calculated as

( )

This can be expanded to analyze the NPSHA at each pump in a system by considering the head

given be preceding pumps, with the Bernoulli equation. The NPSHA was analyzed for the final,

five-pump system to ensure that cavitation did not occur within the system. The result of this

analysis is given in Table 1.

Table 1. The net positive suction head available was determined at each pump in the system. A

36 m drop prior to the first pump and a 14 m drop prior to the second pump were

included to prevent cavitation in the system. Since the net positive suction head

available of each pump was greater than net positive suction head required from each

pump, which was 31.75 m, there was no cavitation in any of the pumps.

Pump NPSHA (m)

1 35.11

2 69.52

3 89.91

4 110.3

5 130.7

Determination of System Head Losses

In order to determine the head rise necessary for system performance, Bernoulli’s equation was

implemented to determine the head loss from the pumps to the system outlet, as follows:

Thus, the head loss can be isolated as

[

] [

]

( )

(

) ( )

For the proposed design, the inlet and outlet are at atmospheric temperature and pressure.

Additionally, it is assumed that inlet and outlet velocities are equivalent. This system does not

extract energy through the use of a turbine and thus the head is neglected.

( )

Where,

The height difference between inlet and outlet, 100 m

total losses of the system, in meters

The total system losses can be calculated as

(∑ ∑

)

(∑ ∑

)

Where,

The system design assumes negligible horizontal flow, a square-edged pipe entry, with ten 90

elbow fittings. The sharp-edged entry region has a minor loss coefficient of 0.5 and standard 90

elbow fittings have an equivalent length over diameter of 30.

The system losses are given by:

[ ( )]

The required head rise can then be represented as

[ ( )]

Determination of Pump Impeller Size and Speed

It was given that the impeller diameter can be scaled-up by as much as 300%. With this design

restriction, the full range of impeller diameters was considered, with the required impeller speed

calculated through the use of the affinity law for volumetric flow, as follows:

( )

Eq. 14-38a, pg. 803, Cengel & Cimbala

Where,

A represents the initial design conditions

B represents the scaled-up system conditions

This equation was solved for the scaled-up impeller speed as a ration of the volumetric flow ratio and

impeller diameter ratio, as follows:

(

)

Determination of Pump Head Rise

For the range of impeller diameters and speed, the scaled-up pump head rise was similarly found

through the use of the affinity laws.

(

)

( )

Eq. 14-38b, pg. 803, Cengel & Cimbala

Where,

A represents the initial design conditions

B represents the scaled-up system conditions

(

)

( )

Determination of Brake Horsepower

In order to determine the scaled-up power requirements of the pump system, the affinity laws

were implemented as

(

)

( )

Eq. 14-38c, pg. 803, Cengel & Cimbala

(

)

( )

Assuming incompressible flow,

(

)

( )

Determination of Water Horsepower

The water horsepower represents the energy transferred to the fluid by the pump. This energy is

represented by the following equation:

Eq. 14-3, pg. 765, Cengel & Cimbala

Determination of Pump Efficiency

The efficiency of the pump is given as the ratio of the energy transferred to the fluid to the

energy supplied to the pump. This is represented by the following equation:

Eq. 14-5, pg. 765, Cengel & Cimbala

Determination of Operating Power

The total energy required by the system is given as the water horse power, divided by the

efficiency of both the pump and the motor supplying power to the pump.

Eq. 8-64, pg. 373 Cengel & Cimbala

Where,

Note that motor efficiency is characteristic of the motor and is taken to be 80%.

Determination of Impeller Blade Angles

Given that the head rise of a centrifugal fan is given by the following expression,

( ) Eq. 10.2C, pg 501, Fox & McDonald

And that

Derived from Fig. 10.7, pg 501, Fox & McDonald

The pump head rise can be characterized as

[( ) (

) ( ) (

)]

The design requirements give that flow enters parallel to the blades, the inlet tangential velocity

is zero. The head rise of the pump can be written as

Given that the flow enters parallel to the blade and ⁄ , for the given head rise, fan

speed, and impeller width and diameter, the outlet fan angle can be calculated as

(

(

))

(

(

) (

)

( )(( )

( ) ( ) ( )))

The inlet blade angle is calculated as

[

( )] [

(

)

( )( ) ( )

]

The inlet and outlet blade angles are thus 21.7º and 13.5º, respectively.

SYSTEM SCALING

In order to determine the performance of the scaled-up pump, it is necessary to determine the

scaling factors through the use of the affinity laws. These factors will depend upon the following

initial pump characteristics and system parameters.

Gravity,

Density at room temperature ( ),

Impeller speed, ⁄

( )

( )( )

( ) ( )

Dimensionless Head Coefficient

The head coefficient is given by:

Eq. 14-30, pg. 799, Cengel & Cimbala

( )

( )

This dimensionless head coefficient was disregarded in favor of the affinity law (or similarity

rule) for the head rise such that system parameters can be found in terms of the initial

parameters.

(

)

( )

Eq. 14-38b, pg. 803, Cengel & Cimbala

Dimensionless Power Coefficient

The power coefficient is given by:

Eq. 14-30, pg. 799, Cengel & Cimbala

(

)

( )

This coefficient was also neglected in favor of the affinity law, such that:

(

)

( )

Eq. 14-38c, pg. 803, Cengel & Cimbala

Dimensionless Capacity Coefficient

The capacity coefficient is given by:

Eq. 14-30, pg. 799, Cengel & Cimbala

( ) ( )

The affinity law for flow rate:

( )

Eq. 14-38a, pg. 803, Cengel & Cimbala

Pump Specific speed

Specific speed is given by:

√

( ) Eq. 7.22a, pg. 316, Fox and McDonald

Where,

The specific speed of the pump

The angular velocity of the impeller

The volume flow rate

The head of the system

Acceleration due to gravity

For the optimal case, .

It is given that the flow rate is constant, ⁄ .

The head of the system is determined as .

Therefore,

(

)√

(

)

( )( )

( )( )

While Ns is dimensionless, specific speed is often represented in inconsistent terms.

(

)

To convert the dimensionless specific speed to the more commonly used value, it must be

multiplied by a conversion factor of 2733.

(

)

⁄

⁄

According to Fox and McDonald, specific speeds are considered low when 500 < NScu < 4000,

and high when 10000 < NScu < 15000. For a low specific speed, the most appropriate pump is a

radial pump. For high specific speeds, an axial pump is most appropriate. The specific speed of

this system is an intermediate value. Such a value indicates that either an axial or radial pump is

adequate for this application.

However, while the value is not strictly in either range, it is closer to radial pump range. This

system’s design requirements dictate that the system use centrifugal pumps—a subcategory of

radial pumps. Therefore, this value of specific speed validates the system design based on the

pumps that are used.

DESIGN OPTIONS

In selecting design options, it was noted that one pump would not be capable of supplying the

required head rise for the entire system. Therefore, it was necessary to consider pump systems, in

either parallel or series configurations.

For pumps in series, the characteristic curves are given by

( ) Pg. 537, Fox & McDonald

In effect, the net head rise is a direct sum of the head rise contributed by each individual pump.

For pumps in parallel, the characteristic curves are given by

Pg. 538, Fox & McDonald

Where,

n is the number of pumps in the pump system

From analysis, it was observed that pumps in series provide an increase in the shut-off head,

allowing for greater net heard rise. As such, it was decided to implement a system in series rather

than parallel.

Four design options were considered, for series configurations of three, four, five, and seven

pumps, given below in Table 2.

Table 2. Optimized system configurations for the series pump designs.

System Pumps Operating (rpm) ⁄ Pipe D (m)

1 3 2210 1.2 0.147

2 4 1740 1.3 0.159

3 5 1390 1.4 0.236

4 7 1130 1.5 0.165

ESTIMATION OF DESIGN COSTS

The total cost of the system is given by the sum of the individual costs, as follows:

( )

While each individual cost can be linked, it is also possible to evaluate the costs separately.

The cost of power required to run the system for ten years is given by

( )

Power Cost

The power required can be found through the analysis of pump performance, as follows:

Eq. 8-64, pg 373, Cengel & Cimbala

It will be assumed that the motor operates at an efficiency of 0.8.

Eq. 14-5, pg 765, Cengel & Cimbala

These equations can be combined to yield a more straightforward equation for electrical power

required:

Pipe Cost

The cost of the pipes is given by

(

( )(

) )

( )

The “bends” term denotes that there are 10 bends in the pipe and each costs three times as much

as a foot of equivalent diameter PVC pipe. The cost/ft term is dependent on the diameter of pipe

chosen.

Pump Cost(s)

The cost of the pump(s) is given by

( )

(

)

Where,

Final impeller diameter, in meters

Original impeller diameter, in meters

The cost of the pumps increases with diameter of the impeller; this relation scales up from the

cost of the original impeller to larger diameter ones. It also provides for the possibility of more

than one pump, assuming they are the same size.

All three components of the system cost are a function of pipe diameter: the larger diameter

pipes are more expensive than smaller ones, but smaller pipes will require a larger pump and

more power which adds to the cost of the system. This suggests that larger pipes may ultimately

be less expensive, to some extent. This turns out to be true, but the power cost dwarfs both the

pump and pipe cost, as can be seen in this sample calculation.

FINAL DESIGN SELECTION

Final Design

After determine total costs for each design option, the final design was selected based on

minimum cost. This comparison is given in Table 4. The final design features PVC pipe of

diameter 0.236 m, impeller diameter of 0.504 m, and impeller speed of 1390 rpm.

Table 4. Optimized system configurations for the singular and series pump designs. System 3

yields the lowest total cost, and was selected as the final design. The pump scaling is

the ratio of the new impeller diameter to the old impeller diameter.

System Pumps Operating

rpm

Pump

Scaling

Pipe D

(m)

Pump

cost ($)

Pipe

cost ($)

Power

cost ($)

Total

cost ($)

1 3 2210 1.2 0.147 2160 5060 9120000 9130000

2 4 1740 1.3 0.159 3380 5630 8840000 8850000

3 5 1390 1.4 0.236 4900 9770 8160000 8170000

4 7 1130 1.5 0.165 7870 5930 8710000 8730000

It should be noted that this analysis does not consider other design factors such as reliability and

maintenance costs. Additional concern should be taken to ensure that the use of PVC will be able

to withstand pressures resulting from the pressure rises.

Confirmation of Realism

In order to evaluate system practicality, and analysis was completed to ensure that the PVC would be

able to withstand the pressure stresses. PVC pipe was modeled as thin-walled pressure vessels such that

(

)

Eq. 7.30. pg. 463, Beer

These stresses can be applied to the Tresca Criterion to estimate if the PVC will yield

1

For principal stresses

It is also assumed that thin-wall pressure vessels experience only planar stress such that . It

should be noted that the pipe will only exhibit outward, tensile stresses, such that need not be

evaluated. Therefore, the shear stress can be represented as

1 Beer, Ferdinand P.; Johnston, E. Russell; Dewolf, John T. (2001). Mechanics of Materials (3rd ed.). McGraw-Hill.

ISBN 978-0-07-365935-0.

Table 3. Net Positive suction head available, pressure acting on pipe before each pump, and the

corresponding maximum shear stress acting on the pipes before each pump. Note that the

maximum shear stresses are much smaller than the maximum tensile stress of PVC, 48 MPa2

This indicates that the PVC pipes are not in danger of failure

Pump NPSHA (m) Pressure before pump (MPa)

Maximum Shear Stress acting on pipe (MPa)

1 35.11 0.463 3.09 2 69.52 0.790 5.27 3 89.91 0.989 6.60 4 110.3 1.19 7.93 5 130.7 1.39 9.26

2 Engineering Toolbox. Accessed 11/26/2012. [Online] Available: http://www.engineeringtoolbox.com/polymer-

properties-d_1222.html

SAMPLE CALCULATIONS

This section of the report is devoted to sample calculations of all quantities previously discussed.

Friction Factor

With a pipe diameter of 23.6 cm (0.236 m), the formula (with correct substitutions) is:

√ (

( )

( ) ( ( )

)

(

)√

)

Using Excel’s iterative capabilities or solving with the Newton-Raphson Method via MATLAB

or Wolfram, the friction factor can be solved:

Net Positive Suction Head Required (NPSHR)

The net positive suction head required by the pump is given by

(

)

(

)(

)

Net Positive Suction Head Available (NPSHA)

( )

Where represents the change in height from one pump to the next. For the first pump, is

the change in height from the inlet. There is an initial drop of 36 m from the inlet to pump 1,

with an initial atmospheric pressure at inlet.

For the selected pipe diameter of 0.236 m, the friction factor f is Given that the

atmospheric pressure is 101325 Pa and the vapor pressure at 25ºC is 3169 Pa, the NPSHA is as

follows:

( ) (

)

(

)

( ) ( )

( ( ))

The previous two calculations verify that cavitation does not occur, since the Net Positive

Suction Head Available is greater than the Net Positive Suction Head Required.

Impeller Speed

(

)

(

)

Pump Head Rise

(

)

( )

(

)

( )

Brake Horsepower

(

)

( )

( ) (

)

( )

Operating Cost

With the above energy rate, the total cost is found as

Pipe Cost

(

( )(

))

( )

A curve fit was applied to the prices of PVC pipe to determine costs for any sizes between those

specified, such that

Where,

pipe diameter, in meters

( ) ( )

(

( ) (

))(

)

Pump Cost

( )

(

)

( )

Total Cost

( )

As can be seen, the operating cost is significantly greater than the design costs of the pump and

pipe system, accounting for more than 99% of the total cost. The pipes account for the second

highest cost, accounting for just 0.12% of the total cost.

Pressure Increase By Pump

Where is the pressure increase across each pump.

(

)(

)

Determination of Pressure at Any Point in System

The following calculations are done for the inlet of the second pump.

Where HD is the vertical distance to the pump below the inlet, npump is the number of pumps the

flow has passed through to reach the current point in the system at which the pressure is being

calculated. The term represents the pressure increase (calculated above) per pump. At the

inlet of the second pump, at which point the flow has passed through only the first pump, the

equation becomes

(

) (

) ( ) (

)

Pa