"Total-to-Total Efficiency" vs. "Bulk Total-to-Total Efficiency"
The plot above shows two methods of expressing efficiency in Concepts
NREC’s Pushbutton CFD®, which the program refers to as “Total-to-Total
Efficiency” and “Bulk Total-to-Total Efficiency.” The results shown are not unusual. Typically, the difference in the two methods is at a maximum in the most distorted region (the trailing edge of the impeller), and they drift closer together as the flow fields mix out and flow unsteadiness, due to periodic rotating wakes, decays.
Although several averaging techniques are available, there is no universally accepted or correct method of averaging. Yet various flow parameters may have preferred methods of averaging, which follows from the structure of terms in basic conservation equations. Area averages are natural for parameters, considered in mass flow balances. Mass flow averages for velocity terms and area averages for pressure are natural for momentum balances (aero loads and mechanical forces). Mass flow averages for enthalpy, entropy, kinetic energy and pressure are preferred for energy balances (power, efficiency, loss).
The three averaging options available in Pushbutton CFD focus on area, mass, and mass-momentum-energy. Area averaging is probably the least often used method. Occasionally, pressure is averaged using area since the integrated pressure force on the surface is conserved with this method. Mass averaging is the most common method used and shows values in the CFD run table. The third option used for averaging is the mass-momentum-energy method (mixing to uniform flow state).
Calculating an average based on mass-momentum-energy is more complicated since an iterative method is needed. Conceptually, during this type of averaging non-uniform flow is converted to a uniform flow state across the averaging surface, while preserving the same mass flow, momentum flux in three directions, and energy flux as the distorted flow.
For compressible fluids, there are actually two such flow states – one supersonic and one subsonic. The actual state reported is selected by calculating the mass-averaged Mach number to determine which side of the sonic point is used.
The mass-momentum-energy value is generally a more pessimistic number, but it is also a more conceptually representative number. Implicitly this number includes the effect of mixing loss that would be incurred when the non-uniform flow mixes out to a uniform state. The next figure shows a simple mixing problem in a frictionless duct. The flow, starting in the distorted state, mixes in the duct, and this irreversible process increases the entropy of the flow in the mass-averaged sense. The mass-momentum-energy averaged method incorporates the final potential mixing loss implicitly. In essence, the method mixes the flow and accounts for this loss from the beginning, so no change in entropy is seen.
A simple mixing problem in a frictionless duct.
Further complicating factors are the two efficiencies quoted in the Pushbutton CFD output. In general, the Bulk Total-to-Total Efficiency value is the recommended one. The simple Total-to-Total Efficiency is basically a mass-averaged value of the efficiency. Bulk efficiency uses the bulk total pressure (from mass-averaging static pressure, static and total enthalpy) to calculate the isentropic total enthalpy required for determining efficiency. This is also the method used in the CFD solver output which appears in the DOS command window when a CFD calculation is run.
So when comparing CFD results to test data, the “correct” averaging method to use is the same method used for measuring and processing the test results. However, regardless of the test method used, it should be noted that in actual applications, the losses due to mixing of distorted flow are almost never avoided. While it is theoretically possible to recover the work from the various stream tubes without mixing them, this is almost never the case in practice. Inevitably, the stream tubes leaving the impeller will mix together and eventually impact performance. Using the mass-momentum-energy averaging method effectively captures this effect.