---

Fan Power, Energy, and Temperature

Nomenclature

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

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

f = Friction factor, dimensionless
g_c = Gravitational constant, 32.2 lb_mft/lb_f-s²

K = Loss coefficient of fitting, dimensionless

L = Length of straight duct, ft

m_a= Mass flow rate, lb/h

P_s = Fan static pressure, in. wg or Pa

P_t = Fan total pressure, in. wg or Pa

P_v = Velocity pressure, in. wg

Q = Volume flow rate, cfm or m³/s

q_f = Fan heat gain, Btu/h

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

W_f = Fan power output, hp or Watt

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

ΔP_f = Friction loss, in. wg

ΔP_k = Total pressure loss in fitting, in. wg

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

Δt_b = Blade temperature rise, ^oF

Δt_f = Temperature rise across the fan, ^oF

Δt_ff = Fan friction temperature rise, ^oF

Δt_m = Motor temperature rise, ^oF

η_f = Fan total efficiency, decimal

η_m = Motor efficiency, decimal

η_s = 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,

W_f(hp) = QΔP_t/(6350η_f)

In SI units,

W_f(W) = QΔP_t/η_f

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,

W_f(Btu/h) = m_aΔh_f

When there is no moisture transfer,

W_f(Btu/h) = m_aC_pΔt_f

Or,

W_f(hp) = (60Qρ/2547)C_pΔt_f

Hence,

(60Qρ/2547)C_pΔt_f = QΔP_t/(6350η_f)

Or,

Δt_f (^oF) = ΔP_t/(150ρC_pη_f)

For air at Standard Conditions,

Δt_f (^oF) = ΔP_t/(150 × 0.075 × 0.244 × η_f)

Or,

Δt_f (^oF) = 0.364ΔP_t/η_f

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

Fan Motor Temperature Rise

The motor temperature rise is given by:

Δt_m (^oF) = Δt_f (1-η_m)

Fan Blade Temperature Rise

The fan blade temperature rise is given by:

Δt_b (^oF) = Δt_fη_m(1-η_f)

Fan Friction Temperature Rise

The fan friction temperature rise is given by:

Δt_ff (^oF) = Δt_fη_mη_f

Fan Heat Gain

The fan heat gain is given by:

q_f(Btu/h) = 60ρQC_pΔt_f

Fan Static Efficiency

The fan static efficiency is given by:

η_s = (η_fP_s)/P_t

Fan Energy Consumption

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

E_f(kWh) = [QΔP_t/(6350η_fη_m)] × 0.746 × θ

Or,

E_f(kWh) = QΔP_tθ/(8512η_fη_m)

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:

• Pressure losses across equipment, and
• Friction losses in the straight ducts and fittings in the air distribution system.

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

ΔP_f (in. wg) = 6fLV²/(Dg_c)

Pressure loss in fittings is given by the equation:

ΔP_k (in. wg) = KP_v

© Copyright 2006. All Rights Reserved. Pontyak Resources Corp.