# Elementary axial fan theory

Figure 3.8 shows an axial flow fan blade section at some partic­ular radius, with its associated velocity triangles. The air enters the impeller axially with a velocity v, = vm1, and leaves with ve­locity v2.

The shape of the triangles is almost identical with those of a backward bladed centrifugal fan, but it should be noted that

U., = u2, and vm1 = vm2. The total pressure developed is given by the same equation as for a centrifugal fan, namely, pu2vu2, vu2 being the rotational component of v2. It should be noted that the expanded form of Euler’s equation no longer includes a forced vortex component since u., = u2 at each radius.

The theoretical characteristics may be derived since:

Vu=u-vmcotp2

P = puvu=pu2-puvm cotp2

O 4Q

= pu — pu-—-p — x C0tp2 Equ3.19

Iid^M-v 1

Where v equals hub to tip ratio D-|/D2. The characteristics are shown in Figure 3.9, and are seen to be very similar to those for a backward bladed centrifugal fan, apart from the stall point.

It is usual to design a blade to give the same axial velocity and pressure development at each radius, in which case P = pcorvu = constant, or rvu = constant. This will be seen to be the condition for a free vortex and permits radial equilibrium of forces on the fluid. It is necessary to have increased blade an­gles at the hub section to achieve the higher values of vu at the smaller radius. Departures from free vortex designs have therefore been made, which limit the blade chord adjacent to the hub. These develop less pressure in this region and are known as arbitrary vortex designs.

 Vm1   Inlet guide vane /v 1, Figure 3.10 Axial flow blade with upstream guide vane Alternative forced vortex designs are also available, where maximum pressure development takes place at the tips of the blades. For good efficiency the tip gap needs to be kept to an absolute minimum.

Since the air leaving the impeller has a rotational component of velocity, vu, there is a loss of total pressure of

 U——— »4-«-vu_^| . 2nd impeller rotation <——  Lp(vz-v^) = ^pvu2 Equ 3.20

If the rotational energy is allowed to be dissipated along the duct system. Downstream guide vanes may be fitted to reduce the velocity to vm and thereby regain static pressure equal to ^pv2.

Even so, many commercial designs are produced without guide vanes to reduce costs, these being known as Tube Axials. The resulting loss in efficiency is relatively unimportant at low fan power.

It is possible to avoid rotational energy loss by having a guide vane upstream of the impeller which pre-rotates the entering air in a direction opposite to that of the impeller rotation. The impel­ler is designed to do sufficient work on the air to remove this ro­tation (Figure 3.10).

Then,

P = Pu2vu2 — РЦЧ.1 = O-pu^-v^) = puvu1 and, at the design point, vui = vm cot Я, — u

Puvu1 =puvm cot в — pu2

Equ 3.21

Equ 3.22

Another type, the contra-rotating fan, makes use of air leaving an impeller with rotation to enter a second impeller rotating in the opposite direction. This second impeller acts in a similar manner to that of an upstream guide vane fan, as can be seen from the velocity triangles, in Figure 3.11. There, the inlet and outlet velocity triangles for each impeller have been combined into a single diagram, made possible since vm and u are the same in each case. Each impeller develops the same pressure if u and vu for each are the same, and the air is discharged axi­ally, that is:

P=2puvu Equ 3.23

A similar arrangement, with both impellers running in the same direction, is possible by using guide vanes between the impel­lers. Whilst this obviates the need for opposite handed impeller, a large angular deflection of the air is necessary. Very careful design of these intermediate guide vanes is required to ensure that flow separation does not occur.

3.2.3.1 Use of aerofoil section blades

 = Pvu(u ± Vu) As with centrifugal fans, the air passing through an impeller constructed with sheet metal blades will not follow the blade profile very accurately unless the number of blades is infinite. Since aerofoil data is available, it is possible to predict the per­formance of an axial flow fan more accurately if blades of aero­foil profile are used. The velocity triangles for such blades are shown in Figure 3.12 and are seen to differ from those previ­ously considered only by the addition of a mean relative velocity vector, w„o = (w., + w2) to which the blade section is inclined at its angle of attack, a. The mean blade angle is p, with an effec­tive blade angle (blade air angle) between vectors of wand u of P — a.

1st impeller rotation

 V3 »W3 ВГ ~ vu " I ‘
 Figure 3.11 Contra-rotating fan velocity triangles  Figure 3.12 Use of aerofoil section axial flow blades

The static pressure difference across the impeller may be found, since

P = puvu = pt1 — pt2 = p2 + pV2 — (Pi + pV2 )

= p2-p1 + ip(v2-v2)

Static pressure difference, p2-p, = puvu-ip(v2-v12)

= puvu±±pv2

Equ 3.24

Where the negative sign refers to the downstream guide vane impeller, and the positive sign to the upstream guide vane im­peller.

This pressure difference over the impeller swept area may be equated to the axial thrust due to the aerodynamic lift forces L on the blades

Fa = Lcos(p — a) = (p2 — p,) • 2jirdr

If there are z blades, each of chord c,

Zc • dr • CL • rw2 cos (p — a) = pvu(u + ^ vu)2itr • dr

And writing blade spacing, s =2rcr/z and substituting

U + iv’u=wo0cos(p-a)

{-•CLw„ =vu s

Or

-^- = lC,^ Equ3.25

W„ 2 s

The above simplified blade element theory, whilst adequate for exploratory design, ignores the effect of drag. Toconsider more fully the forces on the aerofoils it is necessary to equate the thrust force Fa, which is due to static pressure rise less any pressure loss, to the axial force due to the lift and drag.

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