This document contains equations related to air flow generated by fans and nozzles. It includes equations for: 1) The theoretical pressure difference generated by a fan based on changes in absolute and relative velocity. 2) How flow rate, pressure, power, and efficiency of a fan vary with changes in rotational velocity and diameter. 3) How air flow rate through a critical nozzle depends on pressure, temperature, and nozzle characteristics. 4) How the temperature of air increases as it passes through a fan, motor/belt system, and duct due to conversion of pressure and kinetic energy to heat.

1. pt
2
[ (C
2
2 C
2
1 ) (u
2
2 u
2
1 ) (w
2
2 w
2
1 )]
q2 q1
D2
D1
3
n2
n1
26
(1)
(2)
3. Flow generation (fans and nozzles)
Here is given the equation for the theoretical pressure difference for a fan (1), which normally
is deduced from Euler's equations. The often used connections for flow variation (2), pressure
variation (3), power variation (4) and efficiency of fans (5) are also reproduced.
Air movements can also be generated by nozzles with steam or pressurized air. They are not
included because they belong to the technology of pressurized air. Moreover they are given,
not as equations, but as diagrams (See for example Hemeon). Nozzles are sometimes used in
ventilation as flow limiter or as mesurement devices and the equations for critical nozzles are
given (6). In source 7 (equations not reproduced here) is described how very small air flows
(0,1-5 lit/min) can be generated by using critical nozzles.
3.1 Theoretical total pressure rise for a fan
p = total pressure differencet
= air density,
resp = velocity in outlet and inlet, respectively2 1
C = absolute velocity for air
u = velocity of wheel periferi
w = air velocity relative to fan blade
The right hand's first term is a pressure rise from the increased absolute velocity. The
second term is a pressure rise from the centrifugal force (for axial flow, i.e. propeller
fans, u = u ). The third term is a pressure decrease from the lowering of the relative2 1
velocity.
3.2 Flow variation for a fan
q = flow rate
n = rotational velocity
D = fan blade diameter.

2. p2 p1
D2
D1
2
n2
n1
2
2
1
N2 N1
D2
D1
5
n2
n1
3
2
1
t
Q p
P
qm A p1
2
R T1
1
p
p1
2
P
P1
1
27
(3)
(4)
(5)
(6)
(7)
3.3 Pressure varation for a fan
= air density
p = total pressure rise.
3.4 Power variation for a fan
N = power needed.
3.5 Efficiency for a fan
= efficiency for a fant
Q = air flow rate
p = total pressure rise for the fan
P = incoming power to the fan.
3.6 Air flow rate through a critical nozzle
q = air flow rate (mass flow rate)m
A = smallest cross surface
p = pressure before the nozzle1
R = gas constant
T = absolute temperature before the nozzle.1

3. 1
(
2
1
)
1
1
p
p1
(
2
1
)
1
0.528 (chritical pressure relation)
Wchritical 2
1
R T1 equals
Wchritical R T when
p2
p1
p
p1
i.e. when max f
qv2
p1
p2
A
2
1
1
1
2
1
R T1 m 3
/s if
p2
p1
p
p1
qv2
197
p1
p2
(m 3
/s m 2
nozzle surface)
qv2
A
p1
p2
1
2RT1
1
1
p2
p1
1
(m 3
/s)
p2
p1
p
p1
28
(8)
(9)
(10)
(11)
(12)
(13)
= C /C = 1,4 for air,p v
For air is T /T = 0,833 (when the pressure relation equals p /p )* *
1 1
w = maximum velocity through the nozzle.chritical
The flow rate after the nozzle with the volume v , i.e. at pressure p is calculated from2 2
For air with temperature 15 °C this is
This value is multiplied with the efficiency for the nozzle. The efficiency for a very
smooth nozzle is 0,90 - 0,95.
If (sub chritical pressure relation) this will be

4. T1 T
1
v
Pv
P
T1
Pv
1200 v
T2
1
r m
1
Pv
1200 v
T3
Pv Pt
1200
29
(14)
(15)
(16)
(17)
3. 7 Air temperature increase in fan and duct
T = temperature increase in fan due to fan work, K1
T = incoming air temperature, K
= adiabatic exponent for air = 1,4
= efficiency of fanv
P = pressure increase in fan, Pav
P = absolute pressure on suction side of fan, Pa.
The equation can usually be simplified to
When the fan motor is also situated in the air flow an added heating will happen
T = temperature increase from power losses in motor and belt drive, K2
= belt drive efficiencyr
= motor efficiencym
P , — see above.v c
The kinetic energy in the air is transferred by friction to heat and gives
T = temperature increase in duct (before the air is let out), K3
P = total pressure difference in fan outlet (pressure side), Pat
P — see above.v