Axial fan design

Variation of Pressure and Velocity of Flow through a Propeller disc.[1]

In the figure, the thickness of the propeller disc is assumed to be negligible. The boundary between the fluid in motion and fluid at rest is shown. Therefore, the flow is assumed to be taking place in an imaginary converging duct[1] [2] where:

• D = Diameter of the Propeller Disc.
• D_s = Diameter at the Exit.

In the figure, across the propeller disc, velocities (C₁ and C₂) cannot change abruptly across the propeller disc as that will create a shockwave but the fan creates the pressure difference across the propeller disc.[1]

${\displaystyle C_{\rm {1}}=C_{\rm {2}}=C}$ and ${\displaystyle P_{\rm {1}}\neq P_{\rm {2}}}$

• The area of the propeller disc of diameter D is:

${\displaystyle A={\frac {\pi D^{2}}{4}}}$

• The mass flow rate across the propeller is:

${\displaystyle {\dot {m}}={\rho AC}}$

• Since thrust is change in mass multiplied by the velocity of the mass flow i.e., change in momentum, the axial thrust on the propeller disc due to change in momentum of air, which is:[1]

${\displaystyle F_{\rm {x}}={\dot {m}}{(C_{\rm {s}}-C_{\rm {u}})}={\rho AC}{(C_{\rm {s}}-C_{\rm {u}})}}$

• Applying Bernoulli's principle upstream and downstream:

${\displaystyle {\begin{aligned}P_{a}+{\frac {1}{2}}{\rho C_{u}^{2}}&=P_{1}+{\frac {1}{2}}{\rho C^{2}}\\P_{a}+{\frac {1}{2}}{\rho C_{s}^{2}}&=P_{2}+{\frac {1}{2}}{\rho C^{2}}\end{aligned}}}$

On subtracting the above equations:[1]

${\displaystyle P_{2}-P_{1}={\frac {1}{2}}\rho (C_{s}^{2}-C_{u}^{2})}$

• Thrust difference due to pressure difference is projected area multiplied by the pressure difference. Axial thrust due to pressure difference comes out to be:

${\displaystyle F_{x}=A(P_{2}-P_{1})={\frac {1}{2}}\rho A\left(C_{s}^{2}-C_{u}^{2}\right)}$

Comparing this thrust with the axial thrust due to change in momentum of air flow, it is found that:[1]

${\displaystyle C={\frac {C_{s}+C_{u}}{2}}}$

A parameter 'a' is defined such that[1] -

${\displaystyle C=(1+a)C_{u}}$ where ${\displaystyle a={\frac {C}{C_{u}}}-1}$

Using the previous equation and "a", an expression for C_s comes out to be:

${\displaystyle C_{s}=(1+2a)C_{u}}$

• Calculating the change in specific stagnation enthalpy across disc:[1]

${\displaystyle \Delta h_{o}=h_{od}-h_{ou}=\left(h_{d}+{\frac {1}{2}}C_{s}^{2}\right)-\left(h_{u}+{\frac {1}{2}}C_{u}^{2}\right)={\frac {1}{2}}\left(C_{s}^{2}-C_{u}^{2}\right)}$

Now, Ideal Value of Power supplied to the Propeller = Mass flow rate * Change in Stagnation enthalpy;[1]

${\displaystyle P_{i}={\dot {m}}{\Delta h_{o}}}$ where ${\displaystyle {\dot {m}}=\rho AC}$

If propeller was employed to propel an aircraft at speed = C_u; then Useful Power = Axial Thrust * Speed of Aircraft;[1]

${\displaystyle P=F_{x}C_{u}}$

• Hence the expression for efficiency comes out to be:[1]

${\displaystyle \eta _{p}={\frac {{\text{Actual Power}}(P)}{{\text{Ideal Power}}(P_{i})}}={\frac {F_{x}C_{u}}{{\frac {1}{2}}\rho AC\left(C_{s}^{2}-C_{u}^{2}\right)}}={\frac {C_{u}}{C}}={\frac {1}{1+a}}}$

• Let D_s be the diameter of the imaginary outlet cylinder. By Continuity Equation;

  ${\displaystyle {\begin{aligned}C{\frac {\pi D^{2}}{4}}&=C_{s}{\frac {\pi D_{s}^{2}}{4}}\\\Rightarrow D_{s}^{2}&={\frac {C}{C_{s}}}D^{2}\end{aligned}}}$

• From the above equations it is known that -

  ${\displaystyle C_{s}={\frac {1+2a}{1+a}}C}$

Therefore;

${\displaystyle D_{s}^{2}=\left({\frac {1+a}{1+2a}}\right)D^{2}}$

Hence the flow can be modeled where the air flows through an imaginary diverging duct, where diameter of propeller disc and diameter of the outlet are related.[1]

Long Blade of a Propeller Fan with varying Blade Section.[1]

In this theory, a small element (dr) is taken at a distance r from the root of the blade and all the forces acting on the element are analysed to get a solution. It is assumed that the flow through each section of small radial thickness dr is assumed to be independent of the flow through other elements.[1][3]

The figure shows the Velocities and Blade Forces for the Flow across the element dr, where w is the Mean Velocity at a direction β from Axial direction. ΔL = Lift Force (Perpendicular to w) and ΔD = Drag Force (Parallel to 'w'). The Axial and Tangential forces are ΔFx and ΔFy respectively and the Resultant force ΔFr is at an angle Φ to the Lift.[1]

Resolving Forces in the figure[1] -

${\displaystyle \Delta F_{x}=\Delta L\sin(\beta )-\Delta D\cos(\beta )}$

${\displaystyle \Delta F_{y}=\Delta L\cos(\beta )+\Delta D\sin(\beta )}$

Lift Coefficient (C_L) and Drag Coefficient (C_D) are given as -

${\displaystyle \mathrm {Lift} (\Delta L)={\frac {1}{2}}C_{L}\rho w^{2}(ldr)}$

${\displaystyle \mathrm {Drag} (\Delta D)={\frac {1}{2}}C_{D}\rho w^{2}(ldr)}$

Also from the figure [1]-

${\displaystyle \tan(\phi )={\frac {\Delta D}{\Delta L}}={\frac {C_{D}}{C_{L}}}}$

Now,

${\displaystyle \Delta F_{x}=\Delta L(\sin \beta -{\frac {\Delta D}{\Delta L}}\cos \beta )=\Delta L(\sin \beta -\tan \phi \cos \beta )={\frac {1}{2}}C_{L}\rho w^{2}ldr{\frac {\sin(\beta -\phi )}{\cos \phi }}}$

No. of Blades (z) and Spacing (s) are related as,[1] ${\displaystyle s={\frac {2\pi r}{z}}}$ and the total thrust for the elemental section of the propeller is zΔF_x.

Therefore,[1]

${\displaystyle \Delta p(2\pi rdr)=z\Delta F_{x}}$

${\displaystyle \Rightarrow \Delta p={\frac {1}{2}}C_{L}\rho w^{2}({\frac {l}{s}}){\frac {\sin(\beta -\phi )}{\cos \phi }}={\frac {1}{2}}C_{D}\rho w^{2}({\frac {l}{s}}){\frac {\sin(\beta -\phi )}{\sin \phi }}}$

Similarly, solving for ΔF_y, ΔF_y is found out to be[1] -

${\displaystyle \Delta F_{y}={\frac {1}{2}}C_{L}\rho w^{2}ldr{\frac {\cos(\beta -\phi )}{\cos \phi }}}$

and ${\displaystyle (\mathrm {Torque} )\Delta Q=r\Delta F_{y}}$

Finally, thrust and torque can be found out for an elemental section as they are proportional to F_x and F_y respectively.[1]

The cause for this is the separation of the flow from the blade surfaces. This effect can be explained by the flow over an air foil. When the angle of incidence increases (during the low velocity flow) at the entrance of the air foil, flow pattern changes and separation occurs. This is the first stage of stalling and through this separation point the flow separates leading to the formation of vortices, back flow in the separated region. For a further the explanation of stall and rotating stall, refer to compressor surge. The stall zone for the single axial fan and axial fans operated in parallel are shown in the figure.[4]

The Figure shows the Stall Prone Areas differently for One fan and Two fans in parallel.[4]

The following can be inferred from the graph :

• For the Fans operated in parallel, the performance is less when compared to the individual fans.
• The fans should be operated in safe operation zone to avoid the stalling effects.

Many Axial fan failures have happened after controlled blade axial fans were locked in a fixed position and Variable Frequency Drives (VFDs) were installed. The VFDs are not practical for some Axial fans. Axial fans with severe instability regions should not be operated at blades angles, rotational speeds, mass flow rates, and pressures that expose the fan to stall conditions.[5]

Surging should not be confused with stalling. Stalling occurs only if there is insufficient air entering into the fan blades causing separation of flow on the blade surface. Surging or the Unstable flow causing complete breakdown in fans is mainly contributed by the three factors

• System surge
• Fan surge
• Paralleling

This situation occurs when the system resistance curve and static pressure curve of the fan intersect have similar slope or parallel to each other. Rather than intersecting at a definite point the curves intersect over certain region reporting system surge. These characteristics are not observed in axial fans.

This unstable operation results from the development of pressure gradients in the opposite direction of the flow. Maximum pressure is observed at the discharge of the impeller blade and minimum pressure on the side opposite to the discharge side. When the impeller blades are not rotating these adverse pressure gradients pump the flow in the direction opposite to the direction of the fan. The result is the oscillation of the fan blades creating vibrations and hence noise.[6]

This effect is seen only in case of multiple fans. The air flow capacities of the fans are compared and connected in same outlet or same inlet conditions. This causes noise, specifically referred to as Beating in case of fans in parallel. To avoid beating use is made of differing inlet conditions, differences in rotational speeds of the fans, etc.