4.33. VALVE¶

4.33.1. Valve (class)¶

../_images/image787.png — Fig. 4.33.1 Control valve, block valve¶

Fall type

type label	description	active
Valve	Control or block valve with choice out of four predefined Deltares standard head loss characteristics or user specified characteristics; several initial settings can be used for flow or pressure balancing	Yes

4.33.1.1. Mathematical model¶

The discharge characteristic indicates the relation between the flow Q through the valve and the pressure ∆p across the valve as a function of the valve position. This relation is expressed in a discharge coefficient K_v or C_v and a loss coefficient ξ.

The coefficients are derived from the general equation for a Newtonian flow through a restriction in a pipeline under cavitation free circumstances:

(4.33.1)¶\[\Delta p=\xi \frac{\rho v^{2}}{2}\]

in which:

∆p	=	net pressure difference across the valve	[N/m²]
ξ	=	loss coefficient of the valve	[-]
ρ	=	density of the fluid	[kg/m³]
v	=	velocity	[m/s]

or

(4.33.2)¶\[\Delta H=\xi \frac{v^{2}}{2 g}\]

in which:

∆H	=	pressure head difference across the valve	[m]
ξ	=	loss coefficient of the valve	[-]
v	=	velocity	[m/s]
g	=	gravitational acceleration	[m/s²]

In practice the discharge coefficients K_v is used.

(4.33.3)¶\[K_{v}=\frac{Q}{\sqrt{\Delta p}}\]

with:

K_v	=	discharge coefficient	[m³/h/√bar]
Q	=	flow	[m³/h]
∆p	=	net pressure difference across the valve	[bar]

In words: the discharge coefficient K_v denotes the flow in m³/h which flows through a valve at a pressure difference of 1 bar.

Apart from Kv, Cv is also defined as a discharge coefficients for American units.

(4.33.4)¶\[C_{v}=\frac{Q}{\sqrt{\Delta p}}\]

with

C_v	=	discharge coefficient	[USGPM/√psi]
Q	=	flow	[US gallons/min]
∆p	=	net pressure difference across the valve	[psi]

The relation between ξ, Kv and Cv is as follows:

(4.33.5)¶\[\xi=1.6 \cdot 10^{9} \frac{1}{K_{v}^{2}} \cdot D^{4}\]

(4.33.6)¶\[K_{v}=0.865 C_{v}\]

A valve is characterised by ξ = f (θ) or K_v = f (θ) or C_v = f (θ).

θ denotes the dimensionless valve opening. θ ranges from 0 to 1 (in SI-units, or 0 = 100 % in percentage annotation)

θ = 0, valve is closed.
θ = 1, valve is completely open.

The different discharge characteristic are always translated to the following equation:

(4.33.7)¶\[\Delta H=a \xi Q_{1}\left|Q_{1}\right|\]

in which:

∆H	=	H₁ ‑ H₂ in positive flow direction	[m]
H₁	=	upstream head	[m]
H₂	=	downstream head	[m]
a	=	1 / (2 g A_f²)	[s²/m⁵]
ξ	=	loss coefficient	[‑]
Q₁	=	discharge upstream	[m³/s]
A_f	=	discharge area valve	[m²]

The discharge characteristic may be defined by one of Deltares’ standard characteristics (See Hydraulic specifications) or by a user-defined discharge characteristic. If the valve position does not coincide with a tabulated position, interpolation must be performed to obtain the discharge coefficient for intermediate valve positions. The standard characteristics and the user-defined ξ characteristic are interpolated logarithmically according to the following equation:

(4.33.8)¶\[{\theta(z) = z \cdot \theta_{1} + \left( 1 - z \right) \cdot \theta_{2} }{\xi\left( \theta(z) \right) = \xi_{1}^{z} \cdot \xi_{2}^{1 - z}}\]

The user-defined K_v and C_v characteristics are interpolated such that K_v or C_v values are interpolated linearly:

(4.33.9)¶\[\frac{1}{\sqrt{\xi(\theta(z))}}=\frac{z}{\sqrt{\xi_{1}}}+\frac{1-z}{\sqrt{\xi_{2}}}\]

in which:

θ1, θ2	=	tabulated valve positions	[-]
z	=	fraction, which defines intermediate valve position (0 < z < 1)	[-]
θ(z)	=	intermediate valve position	[-]
ξ	=	loss coefficient at θ1, θ2	[‑]
ξ ( θ(z))	=	interpolated loss coefficient	[‑]

If the valve is fully closed the governing equations is:

(4.33.10)¶\[Q_{1}=0\]

Cavitation

Cavitation depends on the pressure conditions around the valve. Usually these pressure conditions are defined by a pressure relation. Several different definitions are use in the industrial standards.

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

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

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 depends of the valve opening: X_f = f (θ)

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

If the cavitation characteristic is specified, the program calculates the pressure ratio in the system and warns the user if it exceeds the available value as defined in the characteristic.

Note:

In other standards (ISA, BS, IEC) the pressure ratio \(X_{T}\) is used:

(4.33.12)¶\[X_{T}=\frac{p_{1}-p_{2}}{p_{1}}=\frac{\Delta p}{p_{1}}\]

In some standards (e.g. ISA) the Thoma number σ is used. \(\sigma=\frac{p_{1}-p_{v}}{\Delta p}\)

in which:

σ	=	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 relationship between Xf and σ is: \(\sigma=\frac{1}{X_{f}}\)

Another definition for the Thoma number is based on the downstream pressure p2:

(4.33.13)¶\[\sigma_{2}=\frac{p_{2}-p_{v}}{\Delta p}\]

where \(\sigma=1+\sigma_{2}\) and \(\sigma_{2}=\frac{1}{X f}-1\)

4.33.2. Valve properties¶

4.33.2.1. Hydraulic specifications¶

Description	input	Unit	range	default	remarks
Characteristic type	Standard/ Kv/ Cv/ Xi
Standard type	Buttrfly/ Ball/ Gate/ Gate_sqr			Buttrfly	if char.type = Standard
Kv characteristic	Table				if char.type = Kv
Cv characteristic	Table				if char.type = Cv
Xi characteristic	Table				if char.type = Xi
Inner diameter	Real	[m]	(0-1000]
Initial setting	Position/ H_upstream H_downstr Discharge P_upstream P_downstr
Initial position (open)	Real	[-]	[0-1]		0 = closed 1= open If init_set = Position
Initial upstream head	Real	[m]	[-1000 – 1000]		If init_set = H_upstream
Initial downstream head	Real	[m]	[-1000 – 1000]		If init_set = H_downstr
Initial discharge	Real	[m³/s]	(0, 10)		If init_set = Discharge
Initial upstream pressure	Real	[N/m²]	[-10⁵ – 10⁷]		If init_set = P_upstream
Initial downstream pressure	Real	[N/m²]	[-10⁵ – 10⁷]		If init_set = P_downstr
Check cavitation	Yes/No
Cavitation table	Table				If check cavitation=Yes

Deltares standard characteristics

Butterfly valve		Ball valve
θ	ξ		θ	ξ
0.000	1.0E+10		0.000	1.0E+10
0.010	10000000		0.015	900000
0.025	1700000		0.025	350000
0.050	140000		0.050	40000
0.075	23000		0.075	9500
0.100	6000		0.100	2750
0.125	2400		0.150	650
0.150	1150		0.200	270
0.200	440		0.300	79.5
0.250	195		0.400	30.0
0.300	97.5		0.500	13.8
0.400	31.0		0.600	6.1
0.500	13.8		0.700	2.7
0.600	5.80		0.800	1.03
0.700	2.40		0.900	0.14
0.800	1.00		1.000	0.01
0.900	.420
1.000	0.150

Gate valve		Square gate valve
θ	ξ		θ	ξ
0.000	1.0E+10		0.000	1.0E+10
0.0025	270000		0.0025	249000
0.025	2850		0.050	850
0.050	625		0.075	370
0.075	270		0.100	195
0.100	140		0.150	82
0.150	58		0.200	45
0.200	31		0.300	17.8
0.300	11.5		0.400	8.2
0.400	5.35		0.500	4.0
0.500	2.55		0.600	2.1
0.600	1.27		0.700	0.95
0.700	0.67		0.800	0.39
0.800	0.355		0.900	0.09
0.900	0.188		1.000	0.001
1.000	0.100

4.33.2.2. Component specific output¶

Valve position (open) [-]

Pressure ratio X_f system [-]

4.33.2.3. H-actions¶

A valve can be opened or closed. To do that the valve must be activated. How the valve opens or closes is arranged via a θ-time relation in tabular form (menu ‘actions’ in model).

An example:

Input of table valve type VALVE ACTION
Time (s)	Position (-)
0
0.5
10.0
12.0
14.0	0.5
18.0	0.5

Note: the position unit depends on the setting made in menu Units

The valve closes linearly in 0.5 s. It remains closed until 10 s. Then the valve opens again in 2 s and starts to close directly until theta = 0.5 at 14.0 s. From then on the valve remains in that position.

4.33.2.4. Component messages¶

Message	Type	Explanation
starts in open phase	Info
Opens	Info
starts in closed phase	Info
Closes	Info
Xf valve exceeded, valve may be choking	Info	Only if “Check Cavitation” = “Yes” Current pressure ratio exceeds the table value
Inconsistent valve position valve position between steady state results and action table; modify input	Error	The action table input for the valve position does not match with the Initial setting. The user should verify that the required Initial setting corresponds with the valve position of the action table. The values in the action table are allowed to deviate slightly from the Initial setting.