**Horsepower vs Speed Characteristics. **There are three basic principles of fan operation which can serve to aid in understanding and finding solutions to motor application problems. These are speed, pressure and air density.

It can be stated that the horsepower input to a fan varies as the cube of the speed, with the other factors held constant.

HP_{2}= HP_{1}x(N_{2})^{3}/N_{1}

Where N=specified speed(RPM)

Thus, if a customer wishes to increase the speed of a fan by 10%, the horsepower required increases by 33%.

Because of the cube law relationship between horsepower and speed, it is important to pay attention to motor full load RPM and make certain that the operating load point between the motor and the fan are correctly matched. Often, belted fan units are shipped with variable pitch motor sheaves (pulleys) so that the fan load can be field adjusted for system balancing. If the sheave is not adjusted when installed (it normally comes in a closed position), the load on the motor can be tremendous. Remember, the motor and fan speeds are proportional to the sheave pitch diameters:

Motor Sheave Dia./Fan Sheave Dia. =Fan RPM/Motor RPM

**Horsepower vs Pressure Characteristics.** No all-encompassing statement can be made about the effect of varying the pressure while holding other variables constant. However, with most centrifugal fans, as the pressure of the system increases (resistance to the air movement increases), the horsepower input decreases. This can be explained by pointing out that as the pressure builds up, the fan moves a smaller volume of air (see Figure 1), and this decrease more than offsets the increase in pressure.

Therefore, less power is required. In all air-moving units, the magnitude of the air-flow rate produced by a unit operating at a constant speed decreases as the pressure needed to overcome the total flow resistance of the system increases, and vice versa.

System pressure (or total pressure) in an air-moving system is the sum of the static and velocity pressures. Static pressure is the compressive pressure in a fluid, and represents its potential energy. Velocity pressure is produced by the velocity of fluid flow, and represents its kinetic energy. In fan systems, static pressure differences are usually below 10 inches (H_{2}O), and the air is considered incompressible. The total absolute pressure at any point in the system is the sum of the total pressure at that point and the atmospheric pressure. (Pressure in an air-moving system is usually measured in inches of water.)

With the axial-flow fan and backward inclined fan, the horsepower input tends to remain relatively constant as the pressure changes.

**Horsepower vs Air Density,** Temperature and Humidity. Again, with other remaining factors constant, the horsepower input to a fan varies directly with the air density. This is to be expected, because if the fan is handling heavier air, it will be doing more work and require more input. This can be carried further and tied in with temperature and relative humidity, since these are two factors which affect the density when pressure is held constant. Since density decreases as temperature and relative humidity increase, it follows that the warmer the air or the more humid the air being handled by a fan, the less horsepower input is required.

**Horsepower Correction for Air Density**. Standard air density is established as .0750 lb/ft3 at 70°F. Finding the density at other temperatures requires converting the new temperature reading to absolute (Ta = TF + 460; note that 70°F equals 530° absolute) and then using the following formula:

530 x .0750 /new T_{a} =39.75/new T_{a}= new density

At 125°F, the new air density is:

530 x .0750 /585 = 39.75/585=.0679 lb/ft^{3}

And at 45°F, the new air density is:

530 x .0750/505 =39.75/505 = .0787 lb/ft^{3}

Table 1 lists a number of air temperatures with the associated air densities and horsepower ratios (i.e. correction factors). All horsepower ratios are based upon a one horsepower motor at 40°C (104°F)

AIR TEMP. °F | AIR DENSITY LB/FT ^{3} | HORSEPOWER RATIO |

-60 -50 -40 | .0994 .0970 .0946 | 1.41 1.38 1.34 |

-30 -20 -10 | .0924 .0903 .0883 | 1.31 1.28 1.25 |

0 10 20 | .0864 .0846 .0828 | 1.23 1.20 1.17 |

30 40 50 | .0811 .0795 .0779 | 1.15 1.13 1.10 |

60 70 80 | .0764 .0750 .0736 | 1.08 1.06 1.04 |

90 100 | .0723 .0710 | 1.03 1.01 |

104(40°C） | .0705 | 1.00 |

110 120 | .0697 .0685 | .99 .97 |

130 140 150 | .0674 .0663 .0652 | .96 .94 .92 |

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