Formulation

In this section, the methodology of the fan capabilities of STAR-CCM+ is presented in detail.

Fan Momentum Sources

To understand the forces imparted by the fan blades to the fluid, consider the velocity vector diagram below (the top view of a single fan blade). Air approaches the fan with velocity , which is assumed to be uniform and to have no tangential component . The fan rotates in the -direction with the angular velocity . The fan blades are assumed to have no curvature.

The air velocity leaving the fan, , is the vector sum of the relative velocity, (assumed to be parallel with the blade angle) and the rotational velocity, , imparted to the air by the fan blades.

Continuity implies that . Conservation of momentum states that the sum of the forces acting on a volume is equal to the net flow of momentum across the surfaces of that volume. Applying this principle and assuming the momentum flux contributions apply only to the volume force (surface forces are not affected), the equation for the -direction becomes:

(4)

where is the tangential force, is the fluid density, and A is the surface area. From the previous vector diagram, it can be shown that:

(5)

Recall that was taken to be zero so that Eqn. 4 can be written as:

(6)

This is the tangential force of the fan blades on the fluid. If the total force is assumed to be normal to the blade face, then the axial direction force can be written as:

(7)

If the fan lies in the x-y plane, then . To convert the tangential force into Cartesian components, consider an arbitrary point P in the fan with global coordinates , as shown in the figure below (the front view of a fan). Let the center point C have global coordinates .

The radial distance between point C and P is given by:

(8)

so the components of the force in the x- and y-directions at point P are given by:

(9)

(10)

STAR-CCM+ requires that the momentum sources be specified in terms of force per unit volume. To get this, Eqns. (7), (9) and (10) may be divided by the fan volume.

Adjusting the Momentum Source Scale Factor

When STAR-CCM+ adjusts the momentum source scale factor, we have two points on the load curve, corresponding to two different momentum source scale factors. These two points my be used to calculate an equation for the load curve, which is assumed to have the form:

(11)

where is the fan pressure rise, is the volumetric flow rate, and and are the constants determined using the two points on the load curve. The intersection point of the load curve, given by Eqn. 11, and the user-supplied fan curve is then determined. This is the desired operating point. The scale factor is then adjusted to try to reach this operating point.

Fan Laws

(12)

(13)

where is the fan speed (rpm) and T is the air temperature (K).