Any device has an inertial resistance to change in motion. Pumps are no different - in order to change their speed, a torque must be applied. Instantaneous speed change would require infinite torque. When modeling fluid transients, a more realistic expectation of pressures and flows can be determined when taking this into account. As expected, accounting for inertia requires some additional information.

Inertial Model Theory

Pumps in AFT Impulse are considered completely homologous. This means that the pump curves used are considered to be scalable to any speed with the affinity laws - real pumps will deviate somewhat from this behavior. Accounting for this behavior would require much more data for little benefit. Fortunately, this simplification makes approaching something like an inertial model much more straightforward.

Due to the above simplification we can use standard relationships for power, torque, rotational speed, and rotational acceleration.

where

Pshaft = power required to drive the shaft at a given torque and speed

Tshaft = the torque applied to the impeller shaft.

ω = the angular speed of the shaft, in rad/s

Itotal = the total rotational inertia of the assembly. This must include the inertia of the pump impeller, entrained fluid, the shaft, and the motor rotor - all of the connected rotating components.

The shaft is acted on by the driver (typically a motor), and the pump (the impeller). Positive torque and speed are defined to be in the same direction, with positive relating to normal pumping conditions.

Under steady conditions, there is no torque imbalance meaning that Tshaft and angular acceleration are zero.

There are a couple of common ways the torque becomes imbalanced - when starting up a pump a large torque is applied by the driver, while the pump torque is initially zero. When tripping a pump, the driver torque drops to zero. In both cases the shaft torque is not zero, meaning the speed must change.

We can integrate the torque balance equation over a time step to get a form that is useable in numerical simulation. Torque is a function of time, so we consider the average torque over the time step.

Solving for the new speed:

There are three unknowns in this equation: ωnew, Tdriver,new, and Tpump,new. We need two more equations for a solution.

To represent the driver, we need raw data relating speed ω and torque Tdriver.

This leaves two unknowns: ωnew and Tpump,new. A function (curve fit) relating power and flow is required input - with the affinity laws, this is further a function of speed.

We know that the torque is related to power by a factor of speed from the first equation above, which means torque is also a function of speed and flow.

We have a relationship for ωnew and Tpump,new, but have introduced one more variable - flow. This is solved for with the pump head curve and the Method of Characteristics, as its magnitude depends on the system as well.

With iteration, we can now determine a new speed and new torque for the new time step. The only additional input required is the behavior of Tdriver and the value of Itotal. A pump trip is defined as an event where all driver torque is immediately set to zero. A pump startup is represented as an "uncontrolled" start - driver speed vs torque provides the information required.

Total Rotating Inertia

To model an inertial startup or trip, the total inertia of the rotating system must be known. This must include all rotating components - the motor rotor, shaft, impeller, entrained liquid, and any other connected components.

Obtaining this information is often quite difficult. Determining the rotational inertia of a single material component with a complex shape like an impeller is challenging enough on its own, and this is only part of the puzzle. Instead of calculating a value analytically, it is possible to use test data to determine a value for inertia. The best test data will come directly from the pump/motor assembly of interest, but again, this can be difficult to get.

An alternative method is to use statistical analysis of industry pumps and motor to determine a likely inertia value for a given pump and motor. Thorley Thorley, A.R.D., Fluid Transients in Pipeline Systems, 2nd Ed., D&L George, Ltd. 2004 presents just such analysis of pumps and motors across industries, and AFT Impulse employs this estimate when the appropriate option is selected.

For pumps, Thorley presents the following correlation derived from a linear regression of 284 data points (the formula has been adjusted to use rpm instead of 1,000's of rpm):

where

Ipump = The rotational inertia of only the pump in kg-m2

P = shaft power at rated conditions and BEP in kW. To determine the power at rated conditions, the system fluid must be defined and a Standard Pump Curve with Power/Efficiency data must be available.

N = shaft speed in rpm

Thorley notes that this correlation has a very good correlation coefficient of 0.96. However, he further notes that the actual values of inertia range from +100 to -50% from the predicted value.

Note: Thorley presents two correlations - the one above is applicable to most pumps, however, one type of pump from a particular manufacturer in the study followed a slightly different trend. This is not presented here, and AFT Impulse uses the above correlation in its estimate.

For motors, Thorley presents the following, from 272 data points (the formula has been adjusted to use rpm instead of 1,000's of rpm):

where

Imotor = The rotational inertia of only the motor in kg-m2

P = shaft power at rated conditions and BEP in kW

N = shaft speed in rpm

Similar to pumps, this has a good correlation of 0.97 with a similar spread.

The total rotational inertia is then considered to be the combination of the two estimates:

Note: Remember that this value is only an estimate. As Thorley notes, there is significant spread in the statistical data. If pump speed change is a critical parameter, total rotational inertia is also a critical parameter. In such cases, a sensitivity study should be completed by doubling and halving the estimated inertia value.