The MBR Book: Principles and Applications of Membrane

274 The MBR Book
An air blower is a mechanical process that increases the pressure of air by applying
shaft work. According to the first law of thermodynamics:
�Q � �E��W (A.1)
where Q � heat transferred, E � system energy and W � work done by the system.
The system energy comprises kinetic energy ( 1 ⁄2mu 2 , m and u being mass and velocity,
respectively), potential energy (mgh, h being height and g being acceleration due to
gravity) and internal energy (e). The work done by the system is made up of external
work (W) and work performed against forces due to pressure, p/�. From this information
it is possible to derive Equation (A.2) for a fluid moving at steady flow between points 1
and 2 as illustrated in Fig. A.1.
dQ
u2�u � e2 �e1 �
dm
2
( )
2
(A.2)
This equation is known as the steady-flow energy equation. In this case the change
in internal, kinetic and potential energy of the fluid are negligible, such that associated
terms in Equation (A.2) may be ignored. Ignoring, in the first instance, frictional
losses, manifested as heat, �Q � 0. Because a blower exerts work on the fluid
the work term will be negative in this case and to simplify the equation this will be
ignored. The fluid density in this system is not constant and the conditions are said
to be adiabatic (change in density with no heat transferred to the gas):
dw
dp
� r
(A.3)
where w is the work done per unit mass of fluid. For an ideal gas under adiabatic
conditions
p
� const
rl z 1
u 1 , p 1
( )
1 2
dW
( p2 � p1)
� g( z2 �z1) � �
dm
r
Heat input dQ
u 2 , p 2
z 2
Figure A.1 Steady flow system
Shaft work dw