# Summary

The moment of inertia of a pump is its resistance to changes in angular velocity as it rotates about its shaft. Knowledge of the moment of inertia of a pump, motor and associated components is typically required for transient analysis of a pumped system. This article presents methods by which pump and motor moment of inertia may be estimated.

# Definitions

: | Acceleration due to gravity | |

: | Differential head of the pump (m) | |

: | Moment of Inertia | |

: | Pump speed (rpm) | |

: | Pump power at the best efficiency point (kW) | |

: | Volumetric flow rate | |

: | Pump start up time (s) | |

: | Pump efficiency |

# Introduction

The moment of inertia of a pump is its resistance to changes in angular velocity as it rotates about its shaft. The inertia is the product of the rotating weight and the square of its radius (or diameter) of gyration.

Pumps with large rotating mass will have higher inertia and therefore take longer to spin down on loss of power and longer to reach full speed during start-up. This is often beneficial for controlling transient pressures as the pump will slowly decelerate after a pump trip, continuing to move the fluid. This fluid movement acts to minimise column separation in downstream piping, which is more likely to occur if the fluid flow abruptly stops at the pump. It is for this reason that flywheels are often installed to increase the overall pump moment of inertia.

The total moment on inertia for a pump is the sum of the moment of inertia for each component. When analyzing a pump trip or shut down, the pump moment of inertia must account for all rotating components:

**Motor inertia:**This is typically the largest component of the pump moment of inertia. Accurate values of the motor moment of inertia are typically available from the motor manufacturer or pump vendor and should be used where possible.**Pump impeller inertia:**This accounts for the rotational mass of the impeller and is typically 10-15% of the motor inertia. Accurate values of the pump impeller inertia are usually available from the vendor and should be used where possible.**Shaft inertia:**This accounts for the inertia of the rotating shaft. It is occasionally included as part of the pump impeller inertia by vendors, however due to its small contribution to pump inertia (typically less than 5% of motor inertia) it is often ignored.**Flywheel inertia:**This accounts for the inertia of any flywheels that may be installed on the pump shaft. As stated previously flywheels are designed to increase inertia and therefore as a key design parameter this value is usually available from the vendor.**Transmission inertia:**This accounts for the inertia of the pump transmission (if equipped). Depending on the transmission design this could significantly increase the pump inertia. Due to the variability of transmission design this value is difficult to estimate and best obtained from the vendor.

During transient analysis it is often most conservative to underestimate the pump moment of inertia, particularly for fluids with high vapor pressures. Pumps with a lower moment of inertia will spin down faster, more abruptly slowing the fluid at the pump outlet while fluid further down the pipe line continues to flow due to momentum. This causes column separation, where a vacuum is formed between the stagnate fluid at the pump outlet and the fluid flowing downstream due to momentum which results in vaporisation of the fluid and subsequent transient pressures as the vacuum collapses.

While it is advisable to always obtain inertia data from vendors, it is not always readily available. In these circumstances the pump moment of inertia may be estimated.

# Estimation of Pump Moment of Inertia

## Pump Impeller Moment of Inertia

The moment inertia of a pump impeller may be estimated using the method relationship proposed by Wylie et al. as shown below.

Here the shaft power of the pump may be calculated as shown below with a more comprehensive discussion of pump power available here.

In addition to the inertia of the pump impeller, inertia of the motor may not be available from the vendor and therefore requires estimation.

## Motor Moment of Inertia

The inertia of the pump motor is typically the largest contributor to the pump moment of inertia. Similarly to the pump impeller it may be estimated using a relationship presented by Wylie et al. as shown below.

The motor inertia is typically the most accessible inertia value when considering pump inertia as it is used during motor design. To ensure maximum accuracy, vendor values for motor inertia should be utilized when possible.

## Flywheel Moment of Inertia

The moment of inertia for a flywheel may be calculated using the general equation for rotational inertia of a rigid body as shown below.

Where m is the mass of the flywheel (kg), r is the radius of gyration (m) and k is an inertial constant to account for the shape of the flywheel. The inertial constant for some common flywheel shapes are listed below.

Flywheel type | k | Notes |
---|---|---|

Spoked wheel loaded at the rim (thin walled, hollow cylinder) | 1 | r is the inner diameter of the cylinder |

Solid cylinder | 0.5 | r is the radius of the cylinder |

Thick walled cylinder | r is the outer radius of the cylinder, t is the wall thickness, | |

Flat solid disc | 0.5 | r is the radius of the disc |

Thin walled, hollow sphere | 2/3 | radius is between the axis of rotation and inside of the sphere |

Solid sphere | 2/5 | r is the inner radius of the sphere |

Rod with rotational axis at rod center | 1/12 | r is the length of the rod |

Rod with rotational axis at rod end | 1/3 | r is the length of the rod |

Square plane | 1/6 | r is the length of the square side |

# Estimation of Total Moment of Inertia From Pump Start Up Time

The total moment and inertia of the pump, coupling and motor may also be estimated from time it takes the pump to start up (i.e. go from 0 to 100% speed) as shown in the equation below.

This is useful where pump start up time can be accurately measured as it will more accurately account for the inertia of all rotating components.

# American and European Conventions

The American convention is to use the radius when calculating moment of inertia as adopted in this article i.e. WK^{2} where K is the radius of gyration. However the European convention is to calculate the moment of inertia on a diameter basis i.e. PD^{2}. This results in a moment of inertia calculated using the European convention to be 4 times that calculated using American convention.

Care should be taken to ensure that the moment of inertia for each pump component is calculated using the same convention. Furthermore to avoid modelling errors, the moment of inertia must use the same conventions as any software adopted for numerical modelling.

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