The pattern of pressure drop through a system during off-design conditions is critical for centrifugal-pump and fan applications and even more so for systems employing variable-frequency-drive (VFD) control. Typically, the assumption that, based on the affinity laws, relative electrical-power demand is changed following the variation of relative flow rate, which is raised to the power of 3, is made. However, this assumption is correct only when overall system relative pressure differential changes in direct proportion to relative flow rate, which is elevated to the power of 2.

Overall pressure drop in a system could be presented as consisting of:

• Friction pressure losses.

• Local resistance losses.

• Static- (or head-) pressure losses.

Pressure losses, including friction and local resistance losses, are variable and depend on the velocity of the fluid travelling through a pipe or duct, while static-pressure losses (i.e., vertical elevation loss to lift fluid to a point of utilization and/or minimum required pressure differential at a terminal unit, etc.) could be assumed constant and independent of fluid velocity. We combined friction and local resistance pressure losses into a single category: dynamic pressure losses. This assumption is convenient for the analysis applied in Figure 1.

Figure 1 demonstrates how the ratio between the relative (dimensionless) design dynamic and static-pressure loss impacts overall relative pressure drop in a closed-loop system, which, in turn, influences system power demand. For convenience, we assumed the curve of the system in Figure 1A (for the quadratic relation between pressure drop and flow rate) would go through the origin. The graphs in Figure 1 were built on the assumption the VFD would have a turndown ratio (TDR) of 10 to 1, which is typical of VFDs. This was done by varying the speed of the electrical motor via the changed electrical-power frequency from 60 Hz to 6 Hz. This meant the ratio of current flow rate to design flow rate (the relative flow-rate ratio [RFR]) would vary from 1.0 (design) to 0.1 (minimum). We further assumed the installed horsepower of the electrical motor and VFD would match the system design load. System relative pressure losses (∆PSYS) were calculated from the following equations:

∆PSYS DES = SORPLDES = (DRPLDES + SRPLDES) = 1.0                                                   (1)

∆PSYS CUR = SORPLCUR = (DRPLCUR + SRPLCUR) ÷ SORPLDES                                     (2)

DRPLCUR = DRPLDES × RFRn=2                                                                                         (3)

SRPLCUR = SRPLDES × RFRn=0                                                                                          (4)

where:

∆PSYS DES = system relative pressure losses at design conditions

SORPLDES = system overall relative pressure losses at design conditions

DRPLDES = system relative dynamic pressure losses at design conditions

SRPLDES = system relative static-pressure losses at design conditions

∆PSYS CUR = system relative pressure losses at current conditions

SORPLCUR = system overall relative pressure losses at current conditions

DRPLCUR = system relative dynamic pressure losses at current conditions

SRPLCUR = system relative static-pressure losses at current conditions

n = exponential index parameter varying from 0 for relative static system pressure losses to 2 for relative dynamic system pressure losses

Although dynamic pressure losses follow relative changes in flow rate elevated to the power of 2, system overall pressure losses consisting of both dynamic- and static-pressure components do not. There are, however, two exceptions: the theoretical cases of systems with no dynamic pressure losses (SRPLDES = 1.0; DRPLDES = 0.0) or no static-pressure losses (DRPLDES = 1.0; SRPLDES = 0.0). These cases are shown by Line 6 and Curve 7, respectively, in Figure 1A. The rest of the curves between Line 6 and Curve 7 are related to relative design dynamic pressure losses (DRPLDES) varying from 0.9 to 0.1 and relative design static-pressure losses (SRPLDES) varying from 0.1 to 0.9. Figure 1A indicates the system overall relative pressure drop (∆PSYS) could deviate significantly from the theoretical assumptions. For instance, for the system with DRPLDES of 0.9, ∆PSYS varies from 1 to 0.1 when RFR varies from 1 to 0.1. On the other hand, for the system with DRPLDES of 0.5, ∆PSYS varies only from 1 to 0.5 when RFR varies from 1 to 0.1.

In the following analysis, we introduced system factor parameter (SFP), which indicates overall functional dependency between system pressure loss and flow rate. The following equation was utilized:

∆PSYS = SORPLCUR ÷ SORPLDES = RFRSFP                                                                     (5)

Equation 5 allows the magnitude of SFP for various combinations of ∆PSYS and RFR, which are shown in Figure 1A, to be found. Figure 1B demonstrates how the actual value of SFP in Equation 5 depends on the system-pressure-drop-distribution pattern between design dynamic and static components of the pressure loss described earlier. SFP represents the average weighted magnitude of the exponent in the equation, which should be applied to relative flow rate to match system relative pressure drop. The top straight line in Figure 1B (No. 6) represents the theoretical case in which system pressure losses are caused by dynamic losses only, which is equivalent to an exponent value of 2 (SFP = 2). This straight line correlates to Line 7 in Figure 1A. The rest of the curves in Figure 1B represent various combinations of system dynamic and static losses and are correlated to curves 2 through 6 in Figure 1A. The Figure 1B curves indicate SFP values could vary substantially to match system relative pressure drop and could be well below the theoretical value of 2, which typically is utilized in engineering designs with VFD applications.

Figure 1B also demonstrates the SFP does not remain constant and varies depending on the ratio of relative design dynamic-pressure losses to static-pressure losses, as well as RFR. Figure 1B shows the usual control strategy of maintaining design pressure differential at an RFR of 1.0 could be improved by resetting the controlled pressure differential as a function of RFR.

Accepting the theoretical case as the design baseline might lead, as shown in figures 1B and 1C, to overestimating magnitudes of potential energy savings associated with VFD applications.

The field areas between Line 6 and the five other curves in Figure 1B show the potential reduction in system relative pressure losses. The greater the magnitude of design system relative static-pressure losses, the larger the field areas.

Similarly, Figure 1C shows system overall relative power-demand reduction ratio (PDRR) as compared with the theoretical value of 3. System PDRR will be in direct proportion to the variation in system relative pressure differential (Figure 1B) and RFR. Line 6 in Figure 1C represents the theoretical PDRR of 3 for the case in which all pressure losses are caused by dynamic losses (correlated with Curve 7 and Line 6 in Figure 1A and 1B, respectively). The other five curves in Figure 1C outline PDRR for various distribution patterns between relative design dynamic and static-pressure losses. Thus, the field areas between Line 6 and each of the other curves represent the reduction in available power-demand savings for various operational loads, which correlate with respective RFR values. Once again, the higher the magnitude of system relative static-pressure losses, the larger the field areas between a theoretical value of 3 and actual PDRR.

Assuming the efficiency of an electrical motor and pump or fan remains constant, the calculation of system relative power demand (RPD) could be simplified. When combined with ∆PSYS from Equation 5, the magnitude of RPD could be presented as follows:

RPD = ∆PSYS × RFR1.0 = RFRSFP × RFR1.0 = RFRPDRF                                                   (6)

where:

PDRF = SFP + 1 = system power-demand reduction factor

Equation 6 allows the magnitudes of PDRF (shown in Figure 1C) for various combinations of ∆PSYS (shown in Figure 1A) and RFR to be found.

For the system with dynamic pressure losses only, the theoretical PDRF will be 3. For the systems having both dynamic and static-pressure losses, the PDRF will be less than 3.

PDRF could vary from 2.8 at an RFR of 0.99 for a DRPLDES of 0.9 (Curve 1 in Figure 1C) to 1.96 at an RFR of 0.1. Thus, for this system, the PDRF magnitude compared with its theoretical value of 3 could be overestimated by as much as 43.3 percent on average over the course of a year, assuming equal-weight-time-distribution occurrences at various loads from an RFR of 1.0 to an RFR of 0.1. Utilizing the same approach for the systems with a DRPLDES of 0.7 (Curve 2 in Figure 1C) and 0.5 (Curve 3 in Figure 1C), the reduction in potential annual power-demand savings as compared with the theoretical case could be as high as 130 percent and 216 percent, respectively.