Control Volume Analysis for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 33

Z U dA ¼ 0

ð 2 : 1 Þ

where dA indicates a vector element of the area on the CV faces. dA is, by
convention, always pointing outwards (at right angles) from the CV. This direction
is called the ''outward facing normal''. It is important to remember this convention
and its critical use in determining the signs of the contributions from each face.
Note also that, as with all conservation equations for steady flow, the only terms in
( 2.1 ) come from the CV faces. This is because no conserved quantity can accu-
mulate within the CV for a steady flow. In words: at every instant, the amount of
air entering the CV per unit time must be balanced by the same amount leaving
from a different part of the CV, again per unit time. (The molecules comprising
these amounts are, of course, different. If you have trouble with this concept, think
of the water entering a hose from the tap, with the same amount leaving the end of
the hose. The molecules leaving at any time are not those entering at the same
time.)
For the above CV, air enters from the upstream face, causing a negative con-
tribution to ( 2.1 ), because dA is in the direction opposite to U, and leaves from the
downstream face in the far-wake, giving a positive contribution, as then dA is in
the same direction as U. There is also a positive contribution from the cylindrical
face at radius R cv .
At the upstream face, the magnitude of the velocity is constant and equal to U 0 .
To reiterate: this velocity is in the opposite direction to the outward pointing
normal, so that U dA will be negative and have the value of -U 0 dA where dA is
now a scalar element of area. Thus the contribution to the integral in ( 2.1 )is-
U 0 pR c 2 . (Note that the result of a vector dot product is a scalar.) Similarly, the
contribution from the face in the far-wake is U 0 p(R c 2 - R 2 ) ? U ? pR 2 . All these
terms have the units of velocity 9 area or m 3 /s, and are usually termed ''volume
flow rates'' because they give the volume of air that passes the particular face
every second. Usually volume flow rates are given the symbol Q, but this symbol
is used in this text for torque. A Q with a subscript will represent a volume flow
rate for this and the next sections only. If Q R represents the flow rate out of the
cylindrical face of the CV, then Eq. 2.1 gives

þ U 1 pR 2 1 þ Q R ¼ 0

U 0 pR 2 þ U 0 p R cv R 2 1

or

Þ pR 2 1 Q R ¼ U 0 U 1 ð ð 2 : 2 Þ

Q R must be due to a radial velocity. The average value of that velocity, V R ,
multiplied by the flow area, will equal Q R . If the length (in the wind direction) of
the CV is X, say, the flow area is 2pR CV X,soV R can be made arbitrarily small by
increasing R CV . In fact, the following analysis requires R CV R, in order to make
V R negligible and Q R independent of R CV .