8.7-i APPENDIX 8.7. POWER CONSUMPT ION OF ECM BLOWER MOTORS TABLE OF CONTENTS 8.7.1 INTRODUCTION

................................................... 8.7-1 8.7.2 BLOWER SPEED AND SHAFT POWER

................................ 8.7-1 8.7.3 DEVELOPMENT OF EFFICIENCY EQUATION

......................... 8.7-3 8.7.4 BLOWER MOTOR POWER CONSUMPTION

.......................... 8.7-10

LIST OF TABLES Table 8.7.2.1 Coefficients for Blower Speed Equation

.......................... 8.7-2 Table 8.7.2.2 Coefficients for Blower Shaft Power Equation

..................... 8.7-3 Table 8.7.3.1 Coefficients for Equation of Y at 76% Efficiency as a Function of X

.... 8.7-7 Table 8.7.3.2 Coefficients for Equation of L ogistic Transform of Motor Efficiency... 8.7-10

LIST OF FIGURES Figure 8.7.3.1 Iso-Efficiency Lines for GE ECM 2.3

............................ 8.7-4 Figure 8.7.3.2 Plot of Iso-Efficiency Lines Against Speed and Shaft Power

.......... 8.7-5 Figure 8.7.3.3 Plot of Rotated Iso-Efficiency Lines

............................. 8.7-6 Figure 8.7.3.4 Residuals of Y After Subtracting Predicted 76% Efficiency Y Values . . . 8.7-8 Figure 8.7.3.5 Fit of Transformed Efficiency to Difference between Y and 76% Efficiency Y Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7-9 Figure 8.7.4.1 Comparison of Reported to Estimated Blower Motor Input Power

..... 8.7-118.7-1 APPENDIX 8.7: POWER CONSUMPT ION OF ECM BLOWER MOTORS 8.7.1 INTRODUCTION For the electronically commutated motor (ECM) design option, the Department calculated power consumption of blower motors as a function of furnace static pressure. The Department developed a series of equations to first calculate the blower speed and shaft power from the furnace static pressure. Using the values for shaft power and blower speed, the Department developed a second series of equati ons to calculate the blower motor efficiency. Building on the first and second series of equations, the Department then calculated the blower motor power consumption from the efficiency and shaft power.

8.7.2 BLOWER SPEED AND SHAFT POWER This section explains how the Department determined blower speed and shaft power from the furnace static pressure. For furnaces with ECM motors and standard forward-curved impellors in the blower, the airflow and static pressure at operating conditions are determined from the intersection of the fan curve of the furnace and the system curve of the ducts. Furnaces with these motors are programmed to provide a constant airflow regardless of static pressure. Thus, the fan curves for these furnaces are a vertical line of constant airflow. The Department calculated the furnace static pressure at the nominal air flow for each of the virtual model furnaces for a range of blower speeds. (See Appendix 7.4, Furnace Fan Curves, for an explanation of the furnace static pressure equation.) ( ) P P QN SEF Q

furn blower = −× , 2

where: = furnace static pressure (in.w.g.), P

furn = static pressure across blower alone (in.w.g.), P

blower = airflow (cfm), 

### Q = motor speed (rpm), and 

### N = system effect factor. SEF With the calculated static pressures, the Department developed an equation for blower speed as a function of static pressure at the nominal airflows of the generic model furnaces.
N P furn CCP furn C P furn C P furn ( ) = +× + × + × 01 2 2 3 38.7-2 where: = motor speed (rpm), 

### N = furnace static pressure (in.w.g.), and Pfurn C

0,C 1 ,C 2,C 3 = empirical coefficients from curve fitting. The coefficients for the equations of blower speed as a function of furnace static pressure for generic furnaces with ECM motors and forward-curved impellors are shown in Table 8.7.2.1. Table 8.7.2.1 Coefficients for Blower Speed Equation Q C 0 C 1 C 2 C 3 Cooling

2-ton 800 758.510902 818.311142 -314.27103 88.0707484 3-ton 1200 749.383575 539.726313 -113.76913 32.9063052 4-ton 1600 746.778132 462.47066 -48.570713 0 5-ton 2000 853.498286 446.471489 -43.055706 0 High Fire 2-ton 640 608.29206 994.40398 -510.6498 173.2131 3-ton 960 600.52032 658.79742 -172.3176 59.047199 4-ton 1280 598.34369 567.4097 -77.75611 0 5-ton 1600 682.74296 558.79107 -85.34107 0 Low Fire

2-ton 533.3 506.0368 1164.7986 -727.0327 265.41284 3-ton 800.0 505.36184 716.89594 -95.81477 0 4-ton 1066.7 500.31672 662.7684 -106.8632 0 5-ton 1333.3 569.14764 666.9781 -141.5275 0

For the same air flow and speeds that were used to evaluate furnace static pressure, DOE evaluated the blower shaft power using the euqation developed in Appendix 7.5 Blower Fan Curves. From this evaluation, the Department developed an equation for blower shaft power as a function of furnace static pressure at the nominal airflows of the generic model furnaces.
H P furn CCP furn C P furn C P furn ( ) = +× + × + × 0 1 2 3 3 2
where: = blower shaft power (which is also the motor output power) (W), H = furnace static pressure (in.w.g.), and P

furn C

0,C 1 ,C 2,C 3 = empirical coefficients from curve fitting.8.7-3 The coefficients for the equations of blower shaft power as a function of furnace static pressure for generic furnaces with ECM motors and forward-curved impellors are shown in Table 8.7.2.2. Table 8.7.2.2 Coefficients for Blower Shaft Power Equation Q C 0 C 1 C 2 C 3 Cooling

2-ton 800 0.0152708 0.050746 0.0257333 -0.005583 3-ton 1200 0.2104571 0.2417293 0.0078857 0.0053584 4-ton 1600 0.323437 0.337503 0.0327196 -0.000428 5-ton 2000 0.5250273 0.4308035 0.0462496 -0.015896 High Fire 2-ton 640 0.0077478 0.0418898 0.0275575 -0.00666 3-ton 960 0.1078423 0.1919758 0.0143375 0.0067217 4-ton 1280 0.1656094 0.2697001 0.0420681 -0.001972 5-ton 1600 0.2687731 0.3456568 0.0540197 -0.027583 Low Fire

2-ton 533.3 0.00452 0.0355173 0.0293339 -0.007401 3-ton 800.0 0.0624766 0.1584268 0.0229145 0.0063644 4-ton 1066.7 0.0958492 0.2245117 0.0514025 -0.004302 5-ton 1333.3 0.1555447 0.2907602 0.0517586 -0.034138
8.7.3 DEVELOPMENT OF EFFICIENCY EQUATION To find motor efficiency from motor speed and shaft power, DOE started with data from a typical ECM motor used regularly by furnace manufacturers, GE ECM 2.3. Figure 8.7.3.1 shows the iso-efficiency lines for the GE ECM 2.3 series motors. 1 This chart was taken from a brochure published by General Electric and manually converted to 250 data points.8.7-4 Figure 8.7.3.1 Iso-Efficiency Lines for GE ECM 2.3

Although input power is given as one of the parameters in this chart, once the data are digitzed, they can be rearranged and input power can be considered as a function of shaft power and motor efficiency. Shaft power is calculated as a product of input power and motor efficiency:


### H BE moto r = × η where: = shaft power (W), H = input power (W), and 

### BE = motor efficiency. η

motor8.7-5 A plot of transformed iso-efficiency lines in a speed (N) and shaft power (H) plane is shown in Figure 8.7.3.2 below.

250 500 750 1000 1250

0 250 500 750 1000 H, Shaft Power (w)N, Speed (RPM) 76%

Figure 8.7.3.2 Plot of Iso-Efficiency Lines Against Speed and Shaft Power The Department fit a straight line through the minimum speed points of all iso-efficiency lines. The Department rotated the axes so that the fit line was vertical by applying the following transformation to every point on the iso-efficiency lines in the plot. The new axes of this plot ( X and Y ) do not have any physical meaning, but are used in the intermediate calculations to calculate efficiency. After this rotation, an equation can be fit to the data to give an equation of Y as a function of X (see Figure 8.7.3.3). 

## ( ) ( ) () () X Y H N       =       ×

−       cos sin

sin cos θ θ

θ θ
where: = shaft power (W), 

### H = motor speed (rpm), and N 

### 2 = 17.09 o , the angle of rotation. The angle of rotation is the difference in degrees between vertical and the slope of the line fit through the minimum speed points on the iso-efficiency lines.8.7-6 -1000

-800 -600 -400 -200 0 200 400

-400 -200 0 200 400 600 XY 76%

Figure 8.7.3.3 Plot of Rotated Iso-Efficiency Lines The Department fit an equation to the iso-efficiency line at an efficiency of 76 percent because this line had the most data points. The form of the equation is: 

# ( ) YC XC e C X C 76 0 2 1 2 2 3 

## = × ′ + − + ′          
where:
Y 76 = Y ’@η motor = 76%, Y’ = Y - , and



### Y

trans X’ = . 

### X Xtrans − where: = translated Y value for 76% motor efficiency line, Y

76 = amount Y value was translated, Y

trans = amount X value was translated, and Xtrans C

0,C 1 ,C 2,C 3 = empirical coefficients from curve fitting. This equation has two main parts. The first part is a hyperbola that is used to fit the left side of the curve. The second part is an exponential curve that has the effect of pulling down the8.7-7 right side of the hyperbola to match the data point s. All the coefficients and the translation along the X and Y axes were selected to give the best possible fit to the data. They are listed in Table 8.7.3.1. Table 8.7.3.1 Coefficients for Equation of Y at 76 Percent Efficiency as a Function of X X

trans 281.305 Y

trans 1170.78 C

0 4.066465 C

1 69.497914 C

2 6.732347 C

3 594.895937
The differences in Y between the line at 76 percent efficiency and the other iso-efficiency lines is nearly constant for all values of X (See Figure 8.7.3.4.) The plot shows all the points, with the Y value for the 76 percent efficiency subtracted. The efficiency for data points off the 76 percent iso-efficiency line can now be calculated as a function of the differences in Y from the 76 percent iso-efficiency line. Before doing this, the Department applied a logistic transform to the efficiency values. Because efficiency is limited by definition to between 0 percent and 100 percent, polynomial fits of efficiency are often done more accurately on a logistic transformation. The equation the Department used for the logistic transform is:



### T motor 

### = −       ln 1 1 η
where: = logistic transform of motor efficiency, and T = motor efficiency. η

moto r And the difference in Y from the 76 percent iso-efficiency line is calculated as: 

## Y R Y Y

76 76 = ′

−8.7-8 -1200 -1000

-800 -600 -400 -200 0 200 400 600

-400 -200 0 200 400 600 XY

fit 76% rotated

Figure 8.7.3.4 Residuals of Y After Subtracting Predicted 76 percent Efficiency Y Values where: = translated Y value for 76 percent motor efficiency line, and Y

76 = translated Y value. ′ Y The Department fit a polynomial equation of the difference between Y and the Y of the 76 percent efficiency line to the logistic tran sform of the efficiency (shown in Figure 8.7.3.5).8.7-9 -800 -600 -400 -200 0 200 400 600 800

-400 -200 0 200 400 600 XResidual from Y at 76%

Figure 8.7.3.5 Fit of Transformed Effi ciency to Difference between Y and 76 percent Efficiency Y Values The equation for this fit is:
TC C Y R C Y R C Y R = +× + × +× 0 1 76 2 76 2 3 76 3
where: = logistic transform of motor efficiency, T = difference between Y 

### ' and Y 

### ' for the 76 percent efficiency line, and Y R

76 C

0,C 1 ,C 2,C 3 = empirical coefficients from curve fitting. The coefficients were selected to give the best possible fit to the data. They are listed in Table 8.7.3.2.8.7-10 Table 8.7.3.2 Coefficients for Equation of Logistic Transform of Motor Efficiency C

0 -1.156317102 C

1 -1.075425258 C

2 1.278606109 C

3 -0.947312475. At this point, the efficiency of the motor can be calculated from the transformed efficiency as: 

# ( ) η

motor e T = + 1 1 where: = motor efficiency, and η

moto r = logistic transform of motor efficiency. T

8.7.4 BLOWER MOTOR POWER CONSUMPTION The electrical power consumption of the ECM blower motor is calculated as: 

# ( ) ( )( ) 

# ( ) BE H P furn motor H P furn N P furn = η , where: = blower motor electrical input power (W), 

### BE = shaft power (W), 

### H = motor efficiency, 

# η motor = motor speed (rpm), 

### N = furnace static pressure (in.w.g.), and 

### P

furn = logistic transform of motor efficiency. T A check shows close correlation between the reported and calculated values for BE as seen in Figure 8.7.4.1. The plot compares the motor input power to the motor input power calculated from blower shaft power and motor speed from the digitized data used to develop the equations listed above.8.7-11 0 200 400 600 800 1000 1200

0 200 400 600 800 1000 1200 BE (w)Predicted BE (w)

BE comparison Ideal

Figure 8.7.4.1 Comparison of Reported to Estimated Blower Motor Input Power8.7-12 REFERENCES 1. General Electric Company, General Electric Brochure, (Last accessed August 28, 2002). <http://www.geindustrial.com/products/brochures/GET-8068.pdf>