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Restriction

This function shall be used within the restricted limits according to the referenced literature.

• Reynolds number (for vena contraction) Re > 1e3 [Idelchik 2006, p. 222, diag. 4-15]
• Relative length of vena contraction (L/d_hyd_0) > 0.015 [Idelchik 2006, p. 222, diag. 4-15]
• Darcy friction factor lambda_FRI = 0.02 [Idelchik 2006, p. 222, sec. 4-15]

Geometry

Calculation

The pressure loss dp for a thick edged orifice is determined by:

    dp = zeta_TOT * (rho/2) * (velocity_1)^2

with

rho	as density of fluid [kg/m3],
velocity_1	as mean velocity in large cross sectional area [m/s],
zeta_TOT	as pressure loss coefficient [-].

The pressure loss coefficient zeta_TOT of a thick edged orifice can be calculated for different cross sectional areas A_0 and relative length of orifice l_bar =L/d_hyd_0 by:

    zeta_TOT = (0.5*(1 - A_0/A_1)^0.75 + tau*(1 - A_0/A_1)^1.375 + (1 - A_0/A_1)^2 + lambda_FRI*l_bar)*(A_1/A_0)^2 [Idelchik 2006, p. 222, diag. 4-15] 

with

A_0	cross sectional area of vena contraction [m2],
A_1	large cross sectional area of orifice [m2],
d_hyd_0	hydraulic diameter of vena contraction [m],
lambda_FRI	as constant Darcy friction factor [-],
l_bar	relative length of orifice [-],
L	length of vena contraction [m],
tau	geometry parameter [-].

The geometry factor tau is determined by [Idelchik 2006, p. 219, diag. 4-12]:

    tau = (2.4 - l_bar)*10^(-phi)
    phi = 0.25 + 0.535*l_bar^8 / (0.05 + l_bar^8) .

Verification

The pressure loss coefficient zeta_TOT of a thick edged orifice in dependence of a relative length (l_bar = L /d_hyd) with different ratios of cross sectional areas A_0/A_1 is shown in the figure below.

Incompressible case [Pressure loss = f(m_flow)]:

The pressure loss DP of an thick edged orifice in dependence of the mass flow rate m_flow of water for different ratios A_0/A_1 (where A_0 = 0.001 m^2) is shown in the figure below.

And for the compressible case [Mass flow rate = f(dp)]:

References

Elmqvist,H., M.Otter and S.E. Cellier:

Inline integration: A new mixed symbolic / numeric approach for solving differential-algebraic equation systems.. In Proceedings of European Simulation MultiConference, Praque, 1995.

Idelchik,I.E.:

Handbook of hydraulic resistance. Jaico Publishing House,Mumbai, 3rd edition, 2006.