.BuildingSystems.Fluid.Movers.BaseClasses.Euler.correlation

Information

This function approximates the following correlation:

where y=η ⁄ η_p (note that η refers to the hydraulic efficiency instead of total efficiency), x=log₁₀(Eu ⁄ Eu_p), with the subscript p denoting the condition where the mover is operating at peak efficiency, and

Z₁=(x-a) ⁄ b

Z₂=(e^c⋅x⋅d⋅x-a) ⁄ b

Z₃=-a ⁄ b

a=-2.732094

b=2.273014

c=0.196344

d=5.267518

The approximation uses two simple polynomials stitched together by a third one of the same order. Care has been taken to ensure that, on the curve constructed by if statements, the differences of dy ⁄ dx evaluated by different groups of coefficients at the connecting points (i.e. at x = - 0.5 and x = + 0.5) are less than 1E-14. This way, the derivative is still continuous to the solver even if the solver requires a precision of 1E-10 when there are nested loops.

The correlation and the approximation have the shape as shown below (plotted by BuildingSystems.Fluid.Movers.BaseClasses.Validation.EulerCurve).

The modified dimensionless Euler number is defined as

Eu=(Δp⋅D⁴) ⁄ (ρ⋅V̇²)

where Δp is the fan pressure rise in Pa, D is the fan wheel outer diameter in m, ρ is the inlet air density in kg/m³, and V̇ is the volumetric flow rate in m³/s. Note that the units in the definition do not matter to this correlation because it is the ratio of the Euler numbers that is used. Since D is constant for the same mover and ρ is approximately constant for common HVAC applications, the Euler number ratio can be simplified to

Eu ⁄ Eu_p=(Δp⋅V̇_p²) ⁄ (Δp_p⋅V̇²)

References

For more information regarding the correlation curve refer to EnergyPlus 9.6.0 Engineering Reference chapter 16.4 equations 16.209 through 16.218. Note that the formula is simplified here from the source document.

Resources

The svg file for the correlation equation was generated on https://viereck.ch/latex-to-svg using this script.

Interface

function correlation
  extends Modelica.Icons.Function;
  input Real x "log10(Eu/Eu_peak)";
  output Real y "eta/eta_peak";
end correlation;

Revisions

• February 2, 2023, by Michael Wetter:
  Rewrote to reduce number of if-then statements and to use smoothMax rather than max.
• December 12, 2022, by Hongxiang Fu and Filip Jorissen:
  First implementation. This is for #2668 and #1645.