Fan Power, Energy, and Temperature

Table of Contents

Nomenclature
Fan Power Output
Fan Power Input
Fan Mechanical (Total) Efficiency
Fan Air Temperature Rise
Fan Motor Temperature Rise
Fan Blade Temperature Rise
Fan Friction Temperature Rise
Fan Heat Gain
Fan Static Efficiency
Fan Energy Consumption
Pressure Losses in Fan and Duct Systems

Nomenclature

2547 = Conversion factor for Btu/h to hp
Cp = Specific heat of air, Btu/lb-oF

D = Duct diameter, in
Ef = Fan energy consumption, kWh

f = Friction factor, dimensionless
gc = Gravitational constant, 32.2 lbmft/lbf-s2

K = Loss coefficient of fitting, dimensionless

L = Length of straight duct, ft

ma= Mass flow rate, lb/h

Ps = Fan static pressure, in. wg or Pa

Pt = Fan total pressure, in. wg or Pa

Pv = Velocity pressure, in. wg

Q = Volume flow rate, cfm or m3/s

qf = Fan heat gain, Btu/h

V = Average velocity of air in duct, ft/s

Wf = Fan power output, hp or Watt

Δhf = Specific enthalpy change across the fan, Btu/lb

ΔPf = Friction loss, in. wg

ΔPk = Total pressure loss in fitting, in. wg

ΔPt = Fan total pressure rise, in. wg or Pa

Δtb = Blade temperature rise, oF

Δtf = Temperature rise across the fan, oF

Δtff = Fan friction temperature rise, oF

Δtm = Motor temperature rise, oF

ηf = Fan total efficiency, decimal

ηm = Motor efficiency, decimal

ηs = Fan static efficiency, decimal
ρ = Air density, lb/ft3

θ = Time of operation, h

Fan Power Output

Fan power output is the power delivered to the air by the fan. It is a function of the fan air volume flow rate and the fan total pressure.

In IP units,

Wf(hp) = QΔPt/(6350ηf)

In SI units,

Wf(W) = QΔPtf

Fan Power Input

Fan power input is the power supplied to the shaft of the fan.

Fan Mechanical (Total) Efficiency

This is the ratio of the fan power output to the fan power input.

Fan Air Temperature Rise

From the energy equation,

Wf(Btu/h) = maΔhf

When there is no moisture transfer,

Wf(Btu/h) = maCpΔtf

Or,

Wf(hp) = (60Qρ/2547)CpΔtf

Hence,

(60Qρ/2547)CpΔtf = QΔPt/(6350ηf)

Or,

Δtf (oF) = ΔPt/(150ρCpηf)

For air at Standard Conditions,

Δtf (oF) = ΔPt/(150 × 0.075 × 0.244 × ηf)

Or,

Δtf (oF) = 0.364ΔPtf

This relationship for Δtf applies when the fan motor is outside the airstream. If the motor is within the airstream, as in a direct drive fan, then ηf is replaced by (ηf × ηm).

Fan Motor Temperature Rise

The motor temperature rise is given by:

Δtm (oF) = Δtf (1-ηm)

Fan Blade Temperature Rise

The fan blade temperature rise is given by:

Δtb (oF) = Δtfηm(1-ηf)

Fan Friction Temperature Rise

The fan friction temperature rise is given by:

Δtff (oF) = Δtfηmηf

Fan Heat Gain

The fan heat gain is given by:

qf(Btu/h) = 60ρQCpΔtf

Fan Static Efficiency

The fan static efficiency is given by:

ηs = (ηfPs)/Pt

Fan Energy Consumption

Energy consumption (kWh) = (Fan power output × time, h)/Motor Efficiency

Ef(kWh) = [QΔPt/(6350ηfηm)] × 0.746 × θ

Or,

Ef(kWh) = QΔPtθ/(8512ηfηm)

Note that 0.746 is the number of kW in an hp.

Pressure Losses in Fan and Duct Systems

In a duct network, the fan total pressure comprises:

Friction loss in straight duct is given by the Darcy-Weisbach Equation:

ΔPf (in. wg) = 6fLV2/(Dgc)

Pressure loss in fittings is given by the equation:

ΔPk (in. wg) = KPv