4.17. ORIFICE¶

4.17.1. ORIFICE (class)¶

../_images/image591.png — Fig. 4.17.1 Orifice¶

Fall type

type label	Description	active
Orifice	Orifice plate complying with ISO 5167-2:2003	No

4.17.1.1. Mathematical model¶

The resistance in a weakly compressible liquid of an orifice plate that complies with the international standard ISO 5167-2:2003 is described by the following equations:

(4.17.1)¶\[Q=\frac{C}{\sqrt{1-\beta^{4}}} \frac{\pi}{4} d^{2} \sqrt{\frac{2 \cdot \Delta p}{\rho}}=\frac{C}{\sqrt{1-\beta^{4}}} \frac{\pi}{4} \beta^{2} D^{2} \sqrt{2 g \cdot\left(H_{1}-H_{v c}\right)}\]

(4.17.2)¶\[H_{1}-H_{2}=\frac{\sqrt{1-\beta^{4}\left(1-C^{2}\right)}-C \beta^{2}}{\sqrt{1-\beta^{4}\left(1-C^{2}\right)}+C \beta^{2}}\left(H_{1}-H_{v c}\right)\]

in which:

H₁	=	Upstream head	[m]
H₂	=	Downstream head	[m]
H_vc	=	Head in the vena contracta	[m]
P_vc	=	Pressure in the vena contracta	[Pa]
z_i	=	Elevation of connecting up- (1) and downstream (2) pipe	[m]
Q	=	discharge through orifice	[m³/s]
C	=	Discharge coefficient	[-]
d	=	Throat diameter	[m]
D	=	Pipe diameter	[m]
ẞ	=	Diameter ratio d / D	[-]

Equation (2) represents the pressure recovery by expansion of the jet downstream of the orifice plate. Pressure recovery is completed approximately 6D downstream of the orifice plate.

Equations (1) and (2) can be rewritten to a general head loss equation:

(4.17.3)¶\[H_{1}-H_{2}=\xi \frac{v|v|}{2 g}=\xi \frac{Q|Q|}{2 g A^{2}}\]

(4.17.4)¶\[\xi=\left(\frac{\sqrt{1-\beta^{4}\left(1-C^{2}\right)}}{C \cdot \beta^{2}}-1\right)^{2}\]

Where ξ, denoted as K in ISO 5167, is the dimensionless head loss coefficient.

ISO 5167-2:2003 proposes the Reader-Harris/Gallagher (RHG) equation to model the discharge coefficient, C, as a function of the location of the pressure tappings. The RHG-equation is:

(4.17.5)¶\[\begin{split}C=0.5961+0.0261 \beta^{2}-0.216 \beta^{8}+5.21 \cdot 10^{-4}\left(\frac{10^{6} \beta}{\operatorname{Re}_{D}}\right)^{0.7}+ \\ (0.0188+0.0063 A) \beta^{3.5}\left(\frac{10^{6}}{\operatorname{Re}_{D}}\right)^{0.3}+ \\ \left(0.043+0.08 e^{-10 L_{1}}-0.123 e^{-7 L_{1}}\right)(1-0.11 A) \frac{\beta^{4}}{1-\beta^{4}}- \\ 0.031\left(M_{2}-0.8 M_{2}^{1.1}\right) \beta^{1.3}\end{split}\]

Where D < 71.12 mm (2.8 in.), the following term shall be added to equation (5):

(4.17.6)¶\[+0.011(0.75-\beta)\left(2.8-\frac{D}{0.0254}\right)\]

In these equations

Re_D	=	Reynolds number calculated with respect to D	[-]
L₁	=	quotient of distance of upstream tapping to the orifice and the pipe diameter; l₁ / D	[-]
L₂	=	quotient of distance of downstream tapping to the orifice and the pipe diameter; l₂ / D	[-]
M₂	=	\(\frac{2 L_{2}}{1-\beta}\)	[-]
A	=	\(\left(\frac{19000 \beta}{\mathrm{Re}_{D}}\right)^{0.8}\)	[-]

The model assumes that the vena contracta occurs D/2 downstream of the orifice plate, such that the RHG equation for D and D/2 tappings is applicable; hence L₁ = 1 and L₂ = 0.47.

The location of the vena contracta determines whether local cavitation and choking flow will occur. If local cavitation occurs the above equations are not valid anymore. WANDA will generate a warning. The velocity in the vena contracta is approximated, based on Bernoulli, by:

(4.17.7)¶\[\begin{split}P_{v c}=\left(H_{v c}-z_{2}\right) \rho g \\ v_{v c}=\sqrt{2 g\left(H_{1}-H_{v c}\right)+v_{D}^{2}} \quad, Q>0\end{split}\]

(4.17.8)¶\[\begin{split}P_{v c}=\left(H_{v c}-z_{1}\right) \rho g \\ v_{v c}=\sqrt{2 g\left(H_{2}-H_{v c}\right)+v_{D}^{2}} \quad, Q \leq 0\end{split}\]

If the flow reverses, the same loss coefficient is applied, which is physically correct if the orifice plate is not bevelled. If the orifice plate is bevelled, the numerical result for negative flow rates are unreliable.

Cavitation depends on the pressure conditions around the orifice. Usually these pressure conditions are defined by a pressure ratio, for which various specific numbers are used.

In WANDA the factor Xf is used, according the German VDMA standard.

(4.17.9)¶\[X_{f}=\frac{\Delta p}{p_{1}-p_{v}}\]

in which:

X_f	=	pressure ratio	[-]
∆p	=	net pressure difference across the valve	[N/m²]
p₁	=	absolute pressure upstream of the valve	[N/m²]
p_v	=	vapour pressure of the fluid	[N/m²]

The pressure ratio X_f is only calculated for positive flow; for negative flow X_f = 0.

4.17.2. Orifice plate¶

4.17.2.1. Hydraulic specifications¶

Description	input	unit	range	default	remarks
Pipe diameter	real	[m]	(0-5]
Initial setting	throat d. / discharge			throat d.	if the user specifies the initial discharge, then the required throat diameter is calculated
Throat diameter	real	[m]	(0-4]		if initial setting = throat d.
Initial discharge	real	[m³/s]	(0-100]		if initial setting = discharge

See also “Mathematical model” on page 327.

4.17.2.2. Component specific output¶

Pressure vena contracta	[Pa]
Head vena contracta	[m]
Velocity vena contracta	[m/s]
Loss coefficient (xi)	[-]
Pressure ratio Xf	[-]
Throat diameter	[m]
Beta (d/D)	[-]

4.17.2.3. H-actions¶

None

4.17.2.4. Component messages¶

Message	Type	Explanation
Throat diameter is out of scope of standard	Warning	Standard requires d ≥ 12.5 mm. Same equations are applied but uncertainty of standard is not guaranteed.
Pipe diameter is smaller than standard scope	Warning	Standard requires D ≥ 50 mm. Same equations are applied but uncertainty of standard is not guaranteed.
Pipe diameter is greater than standard scope	Warning	Standard requires D ≤ 1000 mm. Same equations are applied but uncertainty of standard is not guaranteed.
Diameter ratio is smaller than standard scope	Warning	Standard requires β ≥ 0.1. Same equations are applied but uncertainty of standard is not guaranteed.
Diameter ratio is greater than standard scope	Warning	Standard requires β ≤ 0.75. Same equations are applied but uncertainty of standard is not guaranteed.
Reynolds number drops out of scope	Warning	The actual Reynolds number has dropped below the minimum Re-number in the standard. The applicable Re-number is fixed to the minimum Re-number. The uncertainty may increase.
Reynolds number returns into scope	Info	The actual Reynolds number returns into the scope of the standard.
Degassing may occur	Warning	This message is displayed if the pressure in the vena contracta drops to vapour pressure + 0.5 bar
Vapour pressure reached. Choking flow	Warning	Vapour pressure reached in the vena contracta. Choking flow is not supported at this stage. Results become unreliable.
Reverse flow	Warning	If orifice is bevelled, then results are unreliable.