FI CdComp Elements

FI CdComp General Description

The FI CdComp element is typically used to model flow through a circular hole in turbomachinery applications. This element is similar to a conventional orifice except that the Cd (Compressible Loss Coefficient) is calculated based on the geometry and flow. “CdComp” refers to the Cd being computed and “FI” is used to identify this element from a previous version of the CdComp element that is no longer recommended.

The calculation of the Cd is based on the method summarized in the paper by Idris et al. (ref 1.). A typical application would be for flow going through a hole in a compressor or turbine disk of a turbomachine. The incidence angle effect (#6 above) will account for the difference in the air and hole rotational speed. This element can be used for a wide range of applications where the circular hole Cd is not known. However, for an orifice plate in a nonrotating pipe user should consider the “Orifice Plate” element.

The element is typically used for compressible flow but Flow Simulator can use this element with incompressible or compressible fluid.

Quick Guide for CdComp Creation in the GUI

FI CdComp can be found under the “Compressible Gas Elements” or “Incompressible Liquid Elements” - “Orifices and Generic Losses” section

The correct “Part Surface Angle” and “Element Orientation” are critical for an accurate flow incidence angle. One thing to remember is that if the hole centerline is perpendicular to the part surface the “Part Surface Angle” and “Element Orientation” are the same. The “Part Surface Angle” and “Element Orientation” will only be different if the hole is at an angle to the part surface. Examples of “Part Surface Angle” and “Element Orientation” are shown below.

Figure 1.

FI CDComp Inputs

Table of the inputs for the Cd comp element. See the image in the user interface to help understand geometric variables.

FI CdComp Theory

Flow Rate Calculation

The compressible flowrate is calculated using the following “flow function” equation (ref 5):

The mach number (MN) is related to the pressure ratio using

(aka TTS)

(aka PTS)

(aka PSEB)

Other references (1,2) also have the flow function equation in different forms but they all represent the same thing.

The incompressible flowrate is calculated using the following:

Cd Calculation

The calculation of the Cd is based on the method summarized in the paper by Idris et al. (ref 1.). The method steps though several characteristics of the orifice and the flow to calculate the final Cd. The details of each step are shown below.

Reynolds Number Effect

The first step is the related to the hole Reynolds number (Re). The Reynolds number is based on the “ideal” velocity through the hole.

For an orifice:

Figure 2.

For a nozzle:

$C{d}_{Re}=.9981-\frac{4.73}{\sqrt{Re}}$

The Cd is not very sensitive to the Re. The minimum Re used in this equation is 10,000. The Cd used as the basis for the next step is always around 0.6 for an orifice.

Hole Corner Fillet or Chamfer

A fillet or chamfer can significantly reduce the losses in the hole (a higher Cd).

For a hole with a circular fillet (McGreehan Fillet, ref 1):

Figure 3.

Figure 4.

The example shows streamlines entering a hole with a sharp corner and a hole with a large fillet. The hole with a sharp corner shows flow separation while the hole with a radius does not. The plot demonstrates the radius effect and may not represent the final Cd.

For a hole with a chamfer (Idris Chamfer, ref 1):

Figure 5.

Figure 6.

The plot demonstrates the radius effect and may not represent the final Cd.

Orifice Length over Diameter (L/D)

The length of the orifice affects the reattachment of the orifice flow. If the flow can reattach inside the orifice length the losses can be reduced (Cd increased). The program adjusts the length due to a fillet or chamfer so the user should enter the orifice length as if there was no fillet or chamfer. The program does not adjust the length for the hole angle. The user should enter a hole length that includes the angle effect. Equation taken from Huning, ref. 2.

Figure 7.

Figure 8.

The plot demonstrates the length effect and may not represent the final Cd. Note that the maximum Cd occurs between L/D = 1.5 and 2. Friction losses increase as the orifice gets longer so Cd decreases for L/D > 2

Ratio of Pressure Upstream and Downstream of the Hole

The pressure ratio effect accounts for the compressibility of the gas. It will have no affect for liquid (incompressible) fluids. The maximum pressure ratio used in this equation is 5.0. As the pressure ratio increases the vena contracta increases to cause the higher Cd. Equation taken from Idris, ref. 1.

Figure 9.

The plot demonstrates the pressure ratio effect and may not represent the final Cd. Note that the Cd increases after the choking pressure ratio (~1.9). So, the orifice flowrate may increase after the PR exceeds the choking PR.

Pumping to Account for Hole Angle and Rotation

Idris (ref 1) describes a pumping effect that accounts for the fact that the Cd for Incidence angle = 0^o is higher for a rotating hole (which must be angled to achieve zero incidence) that it is for a stationary hole. The physics behind this are not clear. The effect is included to match the Idris data, ref. 1.

Figure 10.

The plot demonstrates the pumping effect and may not represent the final Cd.

Incidence Angle of the Flow Approaching the Hole

The final Cd is very sensitive to the incidence angle effect. If the upstream flow approaches a hole at a steep angle a large separation region can fill a significant portion of the hole. This will decrease the Cd and flowrate. The incidence angle affects are taken from ref 1. The hole corner treatment and the angle of the hole influence the incidence angle effect.

For holes that are perpendicular to the part surface:

For a hole with a sharp corner (Idris):

Figure 11.

The plot demonstrates the incidence effect and may not represent the final Cd. The absolute value of the incidence angle is used in this equation. The maximum incidence angle allowed is 45^o.

For a hole with a circular fillet (Idris):

Figure 12.

The plot demonstrates the incidence effect and may not represent the final Cd. The absolute value of the incidence angle is used in this equation. The maximum incidence angle allowed is 45^o. The maximum r/d allowed is 0.25 .

For a hole with a chamfer (Idris):

Figure 13.

The plot demonstrates the incidence effect and may not represent the final Cd. The absolute value of the incidence angle is used in this equation. The maximum incidence angle allowed is 30^o. The maximum c/d allowed is 0.20.

There is also an equation from Huning (ref 2) for holes that are perpendicular to the surface. The same equation is used for all corner types (sharp, fillet, and chamfer).

$C{d}_i=C{d}_{previous}*Hunin{g}_{mult}$

$Hunin{g}_{mult}=-.00044513*i^2-.0029937*i+1.0$

Figure 14.

For holes that are angled to the part surface:

There is no data available for the incidence angle effect on an angled hole with a fillet or chamfer. The sharp corner Cd will be used even if the hole has a fillet or chamfer. The effect is different for holes with positive and negative incidence.

Figure 15.

For a hole with a sharp corner (Positive incidence) :

For a hole with a sharp corner (Negative incidence) :

The following plots show the effect of incidence angle on holes that have a 15^o, 30^o, and 45^o angle with the part surface. The plots demonstrate the incidence effect and may not represent the final Cd.

Figure 16.

Figure 17.

Figure 18.

Weisbach Adjustment for Angled Hole

The Weisbach adjustment for an angled hole is valid when the upstream velocity is 0, in other words, a plenum chamber is upstream of the Cdcomp element. It is based on the following equation from Weisbach, but can also be found in Ref 9.

$K=0.5+0.3\cos \left(i\right)+0.2{\text{cos}}^2\left(i\right)$

The K is converted into a Cd using Cd=1/sqrt(1+K) to get an adjustment to the previous Cd.

$C{d}_i=C{d}_{previous}+Weisbach Cd adjustment$

Figure 19.

Difference between Element and Test Upstream Pressure

The final effect is not from the Idris paper and is specific to Flow Simulator. The empirical based Cd equations and Flow Simulator may use different upstream pressures (P_tup, aka PTS) in the flow function calculation. The test typically used a total pressure based on the relative velocity but Flow Simulator typically uses a pressure less than this total pressure. The pressure that Flow Simulator uses is based on the upstream chamber velocity vector that is aligned with the centerline of the hole, and the entrance loss setting (Portion of Ustrm Cham. Dyn. Head Lost).

FI CDComp Outputs

Outputs in file with RES extension. Output units controlled by user setting in “Output Control” panel.

FI CDComp Validation

The FI CDComp results were compared to Cd and flow results from various tests as shown below.

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Figure 24.

Figure 25.

CDComp References

1. A. Idris, K. R. Pullen, “Correlations for the discharge coefficient of rotating orifices based on the incidence angle”, Proceedings of IMechE, Part A: J. Power and Energy, Vol-219, Year-2005.
2. Marcus Hüning, “Comparison of Discharge Coefficient Measurements and Correlations for Orifices with Cross-Flow and Rotation”, Journal of Turbo machinery, Vol-132, p031017-1-10, Year-2010.
3. A. Alexiou, N.J. Hills, C.A. Long, A.B. Turner, L.-S. Wong, J.A. Millward, “Discharge coefficients for flow through holes normal to a rotating shaft”, International Journal of Heat and Fluid Flow, Vol-21, p701-709, Year-2000.
4. W. F. McGreehan, M. J. Schotsch, “Flow Characteristics of Long Orifices With Rotation and Corner Radiusing”, Journal of Turbo machinery, Vol-110, p213-217, Year-1988.
5. Miller, D.S., “Internal Flow Systems,” Miller Innovations, 2^nd Edition, 1990.
6. Rhode, J. E., Richards, H. T., Metger, G. W., Discharge Coefficients for Thick Plate Orifices With Approach Flow Perpendicular and Inclined to the Orifice Axis, 1969, Technical Report, NASA-TN-D-5467.
7. Dittmann, M., Dullenkopf, K., Witting, S., Discharge coefficients of rotating short orifices with radiused and chamfered inlets, J. Turbomach.,2004, 126, 803-808.
8. Wittig, S., Kim, S., Jakoby, R., WeiBert, I., Experimental and Numerical Study of Orifice Discharge Coefficients in High-Speed Rotating Disks, Transactions of the ASME, 1996, 118, 400-407.
9. Blevins, R. D., Applied Fluid Dynamics Handbook, Krieger Publications, 2003.