6.15. Valve

Fig. 6.15.1 Schematic of a Heat valve.

Fall type

Type label	Description	Active
Heat 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

6.15.1. Mathematical model

6.15.1.1. Pressure loss coefficients

The discharge characteristic indicates the relation between the flow \(Q\) through the valve and the pressure loss \(\Delta p\) 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 \(\xi\).

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

(6.15.1)\[\Delta p = \xi\frac{\rho_{1}v_{1}^{2}}{2}\]

with:

Variable	Description	Units
\(\Delta p\)	Pressure difference over the valve	Pa
\(\xi\)	Loss coefficient of the valve	-
\(\rho_1\)	Density of the fluid (upstream)	kg/m3
\(v_1\)	Velocity of the fluid (upstream)	m/s

In practice the discharge coefficients \(K_v\) is often used:

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

with:

Variable	Description	Units
\(K_v\)	Discharge coefficient	m3 /h/\(\sqrt{\text{bar}}\)
\(Q\)	Flowrate	m3/h
\(\Delta p\)	Pressure difference over 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 \(K_v\), \(C_v\) is also defined as a discharge coefficients for American units.

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

with

Variable	Description	Units
\(C_v\)	Discharge coefficient	USGM/ \(\sqrt{\text{psi}}\)
\(Q\)	Flowrate	USGM
\(\Delta p\)	Pressure difference over the valve	psi

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

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

(6.15.5)\[K_{v}=0.865 C_{v}\]

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:

(6.15.6)\[\Delta H = a\xi Q_{1}|Q_{1}|\]

with:

Variable	Description	Units
\(\Delta H\)	\(H_1 - H_2\) in the positive flow direction	m
\(H_1 / H_2\)	Upstream and downstream head	m
\(a\)	\(1 / \left(2 g A_f^2 \right)\)	s2/m5
\(\xi\)	Loss coefficient	-
\(Q_1\)	Upstream discharge	m3/s

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:

(6.15.7)\[\theta (z) = z \theta_1 + \left( 1 - z \right) \theta_2\]

(6.15.8)\[\xi\left( \theta (z) \right) = \xi_1^z \xi_2^{1 - z}\]

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

(6.15.9)\[\frac{1}{\sqrt{\xi\left( \theta (z) \right)}} = \frac{z}{\sqrt{\xi_{1}}} + \frac{1 - z}{\sqrt{\xi_{2}}}\]

with:

Variable	Description	Units
\(\theta_1, \theta_2\)	Tabulated valve opening positions	-
\(z\)	Fraction defining intermediate valve position (\(0 < z < 1\))	-
\(\theta (z)\)	Intermediate valve position loss coefficient	-
\(\xi ( \theta (z))\)	Interpolated loss coefficient	-

If the valve is closed the governing equations is:

(6.15.10)\[Q_{1} = 0\]

To calculate the temperature the following equation is used:

(6.15.11)\[\dot{m}c_{p, 1}T_{1} - \dot{m}c_{p, 2}T_{2} + fr Q_{\text{gen}} = 0\]

with:

Variable	Description	Units
\(\dot{m}\)	Mass flow rate	kg/s
\(c_{p, i}\)	Specific heat at connection point \(i\)	J/kg/K
\(T_{i}\)	Temperature at connection point \(i\)	K
\(fr\)	Fraction generated heat supplied to fluid	-
\(Q_{gen}\)	Heat generated by friction	W

The generated heat is given by:

(6.15.12)\[Q_{gen} = \xi \frac{\dot{m}^3}{2 \rho^2 A^2}\]

Variable	Description	Units
\(A\)	Area of the valve	m2

6.15.1.2. Cavitation

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

In WANDA the factor \(X_f\) is used, according the German VDMA standard.

(6.15.13)\[X_{f} = \frac{\Delta p}{p_1 - p_v}\]

with:

Variable	Description	Units
\(X_f\)	Pressure ratio	-
\(\Delta p\)	Pressure difference over valve	Pa
\(p_1\)	Absolute pressure upstream of the valve	Pa abs
\(p_v\)	Vapour pressure of the fluid	Pa

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:

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

In some standards (e.g. ISA) the Thoma number (\(\sigma\)) is used.

(6.15.15)\[\sigma_{} = \frac{p_1 - p_v}{\Delta p}\]

with:

Variable	Description	Units
\(X_f\)	Pressure ratio
\(\Delta p\)	Pressure difference over valve	Pa
\(p_1\)	Absolute pressure upstream of the valve	Pa abs
\(p_v\)	Vapour pressure of the fluid	Pa

The relationship between Xf and σ is:

(6.15.16)\[\sigma_{} = \frac{1}{X_{f}}\]

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

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

where \(\sigma_{} = 1 + \sigma_{2}\) and \(\sigma_{2} = \frac{1}{X_f} - 1\)

6.15.2. Valve properties

6.15.2.1. Hydraulic specifications

Description	input	SI-units	Remarks
Characteristic type	Standard Kv Cv Xi
Standard type	Buttrfly Ball Gate Gate_sqr		if char.type = Standard, default = Buttrfly
Kv characteristic	table	[m3/h/√bar]	if char.type = Kv
Cv characteristic	table	[USGM/√psi]	if char.type = Cv
Xi characteristic	table	[-]	if char.type = Xi
Inner diameter	real	[m]
Initial setting	Position P_upstream P_downstr Mass flow T_downstream
Initial position (open)	real	[-]	0 = closed 1= open If init_set = Position
Initial upstream pressure	real	[N/m²]	If init_set = P_upstream
Initial downstream pressure	real	[N/m²]	If init_set = P_downstr
Initial mass flow	real	[kg/s]	If init_set = Mass flow
Check cavitation	Yes/No
Cavitation table	table		If check cavitation=Yes
Fraction gen. heat to fluid	real	[-]	Default = 1 (100%)

Deltares standard characteristics

Butterfly valve

\(\theta\)	\(\xi\)
0.000	1.0E+10
0.010	10000000
0.025	1700000
0.050	140000
0.075	23000
0.100	6000
0.125	2400
0.150	1150
0.200	440
0.250	195
0.300	97.5
0.400	31.0
0.500	13.8
0.600	5.80
0.700	2.40
0.800	1.00
0.900	.420
1.000	0.150

Ball valve

\(\theta\)	\(\xi\)
0.000	1.0E+10
0.015	900000
0.025	350000
0.050	40000
0.075	9500
0.100	2750
0.150	650
0.200	270
0.300	79.5
0.400	30.0
0.500	13.8
0.600	6.1
0.700	2.7
0.800	1.03
0.900	0.14
1.000	0.01

Gate valve

\(\theta\)	\(\xi\)
0.000	1.0E+10
0.0025	270000
0.025	2850
0.050	625
0.075	270
0.100	140
0.150	58
0.200	31
0.300	11.5
0.400	5.35
0.500	2.55
0.600	1.27
0.700	0.67
0.800	0.355
0.900	0.188
1.000	0.100

Square gate valve

\(\theta\)	\(\xi\)
0.000	1.0E+10
0.0025	249000
0.050	850
0.075	370
0.100	195
0.150	82
0.200	45
0.300	17.8
0.400	8.2
0.500	4.0
0.600	2.1
0.700	0.95
0.800	0.39
0.900	0.09
1.000	0.001

6.15.2.2. Component specific output

• Valve position (open) [-]

• Pressure ratio X_f system [-]

• Generated Heat flux [W]

6.15.2.3. Actions

An activated valve can be opened or closed. How the valve opens or closes is arranged via a θ-time relation in tabular form.

An example:

Valve action table

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.

6.15.2.4. Component messages

Message	Type	Explanation
starts in open phase	Info
starts in closed phase	Info
Opens	Info
Closes	Info
Initial valve position below minimum position; minimum table value used	Warning
Initial valve position above maximum position; maximum table value used	Warning
Valve position truncated to input range	Warning
Indifferent valve position (zero discharge and zero dH) max. open position taken	Warning
Valve characteristic table entries not in between [0-1]	Error	The input table is not correct
Valve resistance in max. open position too large to obtain prescribed state	Error	The desired flow rate is too large for this valve with the calculated pressure drop.
Valve resistance in min. open position too small to obtain prescribed state	Error
Initial setting not physically realistic Discharge opposite to pressure drop	Error	Check your model or chose different settings.
Cavitation in steady state not allowed’	Error
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.