Gate Valve (G)

Valve with sliding gate as control element

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Description

The Gate Valve (G) block models an orifice with a translating gate (or sluice) as a flow control mechanism. The gate is circular and constrained by the groove of its seat to slide perpendicular to the flow. The seat is annular and its bore, part of the orifice through which the flow must pass, is sized to match the gate. The overlap of the two—the gate and the bore—determines the opening area of the valve.

Gate valve with conical seat

The flow can be laminar or turbulent, and it can reach (up to) sonic speeds. This happens at the vena contracta, a point just past the throat of the valve where the flow is both its narrowest and fastest. The flow then chokes and its velocity saturates, with a drop in downstream pressure no longer sufficing to increase its velocity. Choking occurs when the back-pressure ratio hits a critical value characteristic of the valve. Supersonic flow is not captured by the block.

Gate valves are generally quick-opening. They are most sensitive to gate displacement near the closed position, where a small displacement translates into a disproportionately large change in opening area. Valves of this sort have too high a gain in that region to effectively throttle, or modulate, flow. They more commonly serve as binary on/off switches—often as shutoff and isolation valves—to open and close gas circuits.

Gate Mechanics

In a real valve, the gate connects by a gear mechanism to a handle. When the handle is turned from a fully closed position—by hand, say, or with the aid of an electrical actuator—the gate rises from the bore, progressively opening the valve up to a maximum. Hard stops keep the disk from breaching its minimum and maximum positions.

The block captures the motion of the disk but not the detail of its mechanics. The motion you specify as a normalized displacement at port L. The input (a physical signal) carries the fraction of the instantaneous displacement over its value in the fully open valve. It helps to think of displacements directly as fractions, rather than as lengths to be converted to (and from) fractions.

If the action of the handle and hard stop matter in your model, you can capture these elements separately using other Simscape blocks. A Simscape Mechanical subsystem makes a good source for the gate displacement signal. In many cases, however, it suffices to know what displacement to impart to the disk. You can usually ignore the mechanics of the valve.

Gate Position

The displacement signal allows the block to compute the instantaneous position of the gate, from which the opening of the valve follows. The opening is easily understood as a flow area but, for ease of modeling, it is often best expressed as a flow coefficient or sonic conductance. (The ``ease of modeling'' depends on the data available from the manufacturer.)

The position and displacement variables measure different things. The (instantaneous) position gives the distance of the gate to its resting place on the seat; the displacement gives only that distance to its normal (unactuated) position. The normal position, a fixed coordinate, need not be zero: the gate can be installed so that it is normally off-center with respect to the orifice. (The valve is then partially open even when it is disconnected and therefore idle.)

The normal distance between the gate and its centered position gives the valve lift control offset, specified in the block parameter of the same name. Think of its as the permanent displacement given to the gate while assembling the valve. The variable displacement from port L, on the other hand, captures the motion of the gate during operation of the valve, after it has been assembled and installed. The instantaneous position of the gate is the sum of the two:

$$h(L)=L+h_{\text{0}},$$

where:

• h is the instantaneous position of the gate, normalized against its maximum value. This variable can range from 0 to 1, with 0 giving a maximally closed valve and 1 a fully open valve. If the calculation should return a number outside of this range, that number is set to the nearest bound (0 if the result is negative, 1 otherwise). In other words, the normalized position saturates at 0 and 1.

• L is the variable displacement of the gate, normalized against the maximum position of the same. This variable is obtained from the physical signal at port L. There are no restrictions on its value. You can make it smaller than 0 or greater than 1, for example, to compensate for an equally extreme valve offset.

• h₀ is the fixed offset of the gate relative to its seat in the normal position (when the valve is disconnected and free of inputs). Its value too is normalized against the maximum position of the gate, though there is no requirement that it lie between 0 and 1.

Numerical Smoothing

The normalized position, h, spans three regions. At a sufficiently small displacement, it saturates at 0 and the valve is fully closed. At a sufficiently large displacement, it saturates at 1 and the valve is fully open. In between, it varies linearly between its saturation bounds, giving a valve that is partially open.

The transitions between the regions are sharp and their slopes discontinuous. These pose a challenge to variable-step solvers (the sort commonly used with Simscape models). To precisely capture discontinuities, referred to in some contexts as zero crossing events, the solver must reduce its time step, pausing briefly at the time of the crossing in order to recompute its Jacobian matrix (a representation of the dependencies between the state variables of the model and their time derivatives).

This solver strategy is efficient and robust when discontinuities are present. It makes the solver less prone to convergence errors—but it can considerably extend the time needed to finish the simulation run, perhaps excessively so for practical use in real-time simulation. An alternative approach, used here, is to remove the discontinuities altogether.

Normalized position with sharp transitions

To remove the slope discontinuities, the block smoothes them over a small portion of the opening curve. The smoothing, which adds a slight distortion at each transition, ensures that the valve eases into its limiting positions rather than snap (abruptly) into them. The smoothing is optional: you can disable it by setting its time scale to zero. The shape and scale of the smoothing, when applied, derives in part from the cubic polynomials:

$${\lambda }_{\text{L}}=3{\overline{h}}_{\text{L}}^2-2{\overline{h}}_{\text{L}}^3$$

and

$${\lambda }_{\text{R}}=3{\overline{h}}_{\text{R}}^2-2{\overline{h}}_{\text{R}}^3,$$

where

$${\overline{h}}_{\text{L}}=\frac{h}{\Delta h^{*}}$$

and

$${\overline{h}}_{\text{R}}=\frac{h-\left(1-\Delta h^{*}\right)}{\Delta h^{*}}.$$

In the equations:

• ƛ_L is the smoothing expression for the transition from the maximally closed position.

• ƛ_R is the smoothing expression for the transition from the fully open position.

• Δp^* is the (unitless) characteristic width of the smoothing region:

  $$\Delta h^{*}=\frac{1}{2}{f}^{*},$$

  where f^* is a smoothing factor valued between 0 and 1 and obtained from the block parameter of the same name.

  When the smoothing factor is 0, the normalized gate position stays in its original form—no smoothing applied—and its transitions remain abrupt. When it is 1, the smoothing spans the whole of the gate's travel range (with the normalized gate position taking the shape of an S-curve).

  At intermediate values, the smoothing is limited to a fraction of that range. A value of 0.5, for example, will smooth the transitions over a quarter of the travel range on each side (for a total smooth region of half that range).

The smoothing adds two new regions to the normalized gate position function—one for the smooth transition on the left, another for that on the right, giving a total of five regions. These are expressed in the piecewise function:

$$h^{*}=\{\begin{array}{ll}0,\hfill & h\le 0\hfill \\ h{\lambda }_{\text{L}},\hfill & h<\Delta h^{*}\hfill \\ h,\hfill & h\le 1-\Delta h^{*}\hfill \\ h\left(1-{\lambda }_{\text{R}}\right)+{\lambda }_{\text{R}},\hfill & h<1\hfill \\ 1\hfill & h\ge 1\hfill \end{array},$$

where h* is the smoothed valve opening area. The figure shows the effect of smoothing on the sharpness of the transitions.

Opening Area

The opening area of the valve is that of its bore adjusted for the instantaneous overlap of the gate—a function of its displacement—and leakage between its ports:

$$S=\frac{\pi D^2}{4}-S_{\text{C}}+S_{\text{Leak}},$$

where:

• S is the instantaneous valve opening area. This area is later smoothed to remove derivative discontinuities at the limiting valve positions.

• D is the common diameter of the gate and its bore (the two being identical). This value is obtained from the Orifice diameter block parameter.

• S_C is the area of overlap between the gate and bore, computed as a function of the gate position, h (which in turn depends on the gate displacement signal, L):

  $$S_C=\frac{D^2}{2}\text{acos}\left(h\right)-\frac{hD}{2}\sqrt{D^2-h^2{D}^2},$$

• S_Leak is the residual area that remains open after the valve has closed to its maximum. This area can be due to bore tolerances, surface defects, or an imperfect seal between the gate and its seat. This area is obtained from the Leakage area block parameter.

The figure shows a front view of the valve maximally closed (left), partially open (middle), and fully open (right). The parameters and variables used in the opening area calculation are shown.

Sonic Conductance

As the opening area varies during simulation, so does the mass flow rate through the valve. The relationship between the two variables, however, is indirect. The mass flow rate is defined in terms of the valve's sonic conductance and it is this quantity that the opening area truly determines.

Sonic conductance, if you are unfamiliar with it, describes the ease with which a gas will flow when it is choked—when its velocity is at its theoretical maximum (the local speed of sound). Its measurement and calculation are covered in detail in the ISO 6358 standard (on which this block is based).

Only one value is commonly reported in valve data sheets: one taken at steady state in the fully open position. This is the same specified in the block dialog box (when the Valve parameterization setting is Sonic conductance). For values across the opening range of the valve, this maximum is scaled by the (normalized) valve opening area:

$$C(S)=\frac{S}{S_{\text{Max}}}{C}_{\text{Max}},$$

where C is sonic conductance and the subscript Max denotes the specified (manufacturer's) value. The sonic conductance varies linearly between C_Max in the fully open position and $$S_{\text{Leak}}÷S_{\text{Max}}\times C_{\text{Max}}$$ in the maximally closed position—a value close to zero and due only to internal leakage between the ports.

Other Parameterizations

Because sonic conductance may not be available (or the most convenient choice for your model), the block provides several equivalent parameterizations. Use the Valve parameterization drop-down list to select the best for the data at hand. The parameterizations are:

• Compute from geometry

• Sonic conductance

• Cv coefficient (USCS)

• Kv coefficient (SI)

The parameterizations differ only in the data that they require of you. Their mass flow rate calculations are still based on sonic conductance. If you select a parameterization other than Sonic conductance, then the block converts the alternate data—the (computed) opening area or a (specified) flow coefficient—into an equivalent sonic conductance.

Flow Coefficients

The flow coefficients measure what is, at bottom, the same quantity—the flow rate through the valve at some agreed-upon temperature and pressure differential. They differ only in the standard conditions used in their definition and in the physical units used in their expression:

• C_v is measured at a generally accepted temperature of 60 ℉ and pressure drop of 1 PSI; it is expressed in imperial units of US gpm. This is the flow coefficient used in the model when the Valve parameterization block parameter is set to Cv coefficient (USCS).

• K_v is measured at a generally accepted temperature of 15 ℃ and pressure drop of 1 bar; it is expressed in metric units of m3/h. This is the flow coefficient used in the model when the Valve parameterization block parameter is set to Kv coefficient (SI).

Sonic Conductance Conversions

If the valve parameterization is set to Cv Coefficient (USCS), the sonic conductance is computed at the maximally closed and fully open valve positions from the Cv coefficient (SI) at maximum flow and Cv coefficient (SI) at leakage flow block parameters:

$$C=\left(4\times {10}^{-8}{C}_{\text{v}}\right)m^3/(sPa),$$

where C_v is the flow coefficient value at maximum or leakage flow. The subsonic index, m, is set to 0.5 and the critical pressure ratio, b_cr, is set to 0.3. (These are used in the mass flow rate calculations given in the Momentum Balance section.)

If the Kv coefficient (SI) parameterization is used instead, the sonic conductance is computed at the same valve positions (maximally closed and fully open) from the Kv coefficient (USCS) at maximum flow and Kv coefficient (USCS) at leakage flow block parameters:

$$C=\left(4.758\times {10}^{-8}{K}_{\text{v}}\right)m^3/(sPa),$$

where K_v is the flow coefficient value at maximum or leakage flow. The subsonic index, m, is set to 0.5 and the critical pressure ratio, b_cr, is set to 0.3.

For the Restriction area parameterization, the sonic conductance is computed (at the same valve positions) from the Maximum opening area, and Leakage area block parameters:

$$C=\left(0.128\times 4S_{\text{R}}/\pi \right)L/(sbar),$$

where S_R is the opening area at maximum or leakage flow. The subsonic index, m, is set to 0.5 while the critical pressure ratio, b_cr is computed from the expression:

$$0.41+0.272{\left(\frac{S_{\text{R}}}{S_P}\right)}^{0.25},$$

where the subscript P refers to the inlet of the connecting pipe.

Momentum Balance

The causes of those pressure losses incurred in the passages of the valve are ignored in the block. Whatever their natures—sudden area changes, flow passage contortions—only their cumulative effect is considered during simulation. This effect is assumed to reflect entirely in the sonic conductance of the valve (or in the data of the alternate valve parameterizations).

Mass Flow Rate

When the flow is choked, the mass flow rate is a function of the sonic conductance of the valve and of the thermodynamic conditions (pressure and temperature) established at the inlet. The function is linear with respect to pressure:

$${\dot{m}}_{\text{ch}}=C{\rho }_{\text{0}}{p}_{\text{in}}\sqrt{\frac{T_{\text{0}}}{T_{\text{in}}}},$$

where:

• C is the sonic conductance inside the valve. Its value is obtained from the block parameter of the same name or by conversion of other block parameters (the exact source depending on the Valve parameterization setting).

• ρ is the gas density, here at standard conditions (subscript 0), obtained from the Reference density block parameter.

• p is the absolute gas pressure, here corresponding to the inlet (in).

• T is the gas temperature at the inlet (in) or at standard conditions (0), the latter obtained from the Reference temperature block parameter.

When the flow is subsonic, and therefore no longer choked, the mass flow rate becomes a nonlinear function of pressure—both that at the inlet as well as the reduced value at the outlet. In the turbulent flow regime (with the outlet pressure contained in the back-pressure ratio of the valve), the mass flow rate expression is:

$${\dot{m}}_{\text{tur}}=C{\rho }_{\text{0}}{p}_{\text{in}}\sqrt{\frac{T_{\text{0}}}{T_{\text{in}}}}{\left[1-{\left(\frac{p_{\text{r}}-b_{\text{cr}}}{1-b_{\text{cr}}}\right)}^2\right]}^m,$$

where:

• p_r is the back-pressure ratio, or that between the outlet pressure (p_out) and the inlet pressure (p_in):

  $$P_{\text{r}}=\frac{p_{\text{out}}}{p_{\text{in}}}$$

• b_cr is the critical pressure ratio at which the flow becomes choked. Its value is obtained from the block parameter of the same name or by conversion of other block parameters (the exact source depending on the Valve parameterization setting).

• m is the subsonic index, an empirical coefficient used to more accurately characterize the behavior of subsonic flows. Its value is obtained from the block parameter of the same name or by conversion of other block parameters (the exact source depending on the Valve parameterization setting).

When the flow is laminar (and still subsonic), the mass flow rate expression changes to:

$${\dot{m}}_{\text{lam}}=C{\rho }_{\text{0}}{p}_{\text{in}}\left[\frac{1-p_{\text{r}}}{1-b_{\text{lam}}}\right]\sqrt{\frac{T_{\text{0}}}{T_{\text{in}}}}{\left[1-{\left(\frac{b_{\text{lam}}-b_{\text{cr}}}{1-b_{\text{cr}}}\right)}^2\right]}^m$$

where b_lam is the critical pressure ratio at which the flow transitions between laminar and turbulent regimes (obtained from the Laminar flow pressure ratio block parameter). Combining the mass flow rate expressions into a single (piecewise) function, gives:

$$\dot{m}=\{\begin{array}{ll}{\dot{m}}_{\text{lam}},\hfill & b_{\text{lam}}\le p_{\text{r}}<1\hfill \\ {\dot{m}}_{\text{tur}},\hfill & b_{\text{cr}}\le p_{\text{r}}<p_{\text{lam}}\hfill \\ {\dot{m}}_{\text{ch}},\hfill & p_{\text{r}}<b_{\text{Cr}}\hfill \end{array},$$

with the top row corresponding to subsonic and laminar flow, the middle row to subsonic and turbulent flow, and the bottom row to choked (and therefore sonic) flow.

Mass Balance

The volume of fluid inside the valve, and therefore the mass of the same, is assumed to be very small and it is, for modeling purposes, ignored. As a result, no amount of gas can accumulate there. By the principle of conservation of mass, the mass flow rate into the valve through one port must therefore equal that out of the valve through the other port:

$${\dot{m}}_A+{\dot{m}}_B=0,$$

where $$\dot{m}$$ is defined as the mass flow rate into the valve through port A or B. Note that in this block the flow can reach but not exceed sonic speeds.

Energy Balance

The valve is modeled as an adiabatic component. No heat exchange can occur between the gas and the wall that surrounds it. No work is done on or by the gas as it traverses from inlet to outlet. With these assumptions, energy can flow by advection only, through ports A and B. By the principle of conservation of energy, the sum of the port energy flows must then always equal zero:

$${\varphi }_{\text{A}}+{\varphi }_{\text{B}}=0,$$

where ϕ is defined as the energy flow rate into the valve through one of the ports (A or B).

Ports

Input

L — Valve control signal, unitless

physical signal

Input port through which to specify the relative displacement of the gate (as a fraction of its maximum). The block uses this input to compute the position of the gate—and from it, the mass flow rate through the valve.

The gate position is the sum of this signal and the valve lift control offset (specified in the block parameter of the same name). This position should range between 0 for a maximally closed valve to 1 for a fully open valve.

The control signal range is enforced by saturating the calculated gate position at these limits. If no control signal is provided, the gate position is fixed to the specified valve lift control offset.

Conserving

A — Valve entrance

gas

Opening through which the working fluid can enter or exit the valve. The direction of flow depends on the pressure differential established across the valve. Both forward and backward directions are allowed.

B — Valve entrance

gas

Opening through which the working fluid can enter or exit the valve. The direction of flow depends on the pressure differential established across the valve. Both forward and backward directions are allowed.

Parameters

Orifice diameter — Diameter of the bore of the valve

0.01 m (default) | positive scalar in units of length

Diameter of the bore of the valve and of the gate that controls the opening area. The orifice is assumed to be constant in cross section throughout its length (from one port to the other).

Gate displacement fraction offset — Distance of the gate from its centered position

0 (default) | unitless scalar

Displacement of the gate from the maximally closed position, expressed as a fraction of the travel range for the same. This displacement is the position of the gate in the normal valve position, when no control input is provided or when that input is zero. The instantaneous gate position is calculated during simulation as the sum of this offset and the control signal specified at port L. The valve is partially open in its normal position when the offset is a fraction between 0 and 1.

Valve parameterization — Method by which to characterize the opening of the valve

Sonic conductance (default) | Cv coefficient (USCS) | Kv coefficient (SI) | Restriction area

Choice of ISO method to use in the calculation of mass flow rate. All calculations are based on the Sonic conductance parameterization; if a different option is selected, the data specified in converted into equivalent sonic conductance, critical pressure ratio, and subsonic index. See the block description for more information on the conversion.

This parameter determines which measures of valve opening you must specify—and therefore which of those measures appear as parameters in the block dialog box.

Sonic conductance at maximum flow — Measure of maximum flow rate at reference upstream conditions

1.6 l/s/bar (default) | positive scalar in units of volume/time/pressure

Equivalent measure of the maximum flow rate allowed through the valve at some reference inlet conditions, generally those outlined in ISO 8778. The flow is at a maximum when the valve is fully open and the flow velocity is choked (it being saturated at the local speed of sound). This is the value generally reported by manufacturers in technical data sheets.

Sonic conductance is defined as the ratio of the mass flow rate through the valve to the product of the pressure and density upstream of the valve inlet. This parameter is often referred to in the literature as the C-value.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Critical pressure ratio — Back-pressure ratio, at reference upstream conditions, at which the flow rate is a maximum

0.3 (default) | positive unitless scalar

Ratio of downstream to upstream absolute pressures at which the flow becomes choked (and its velocity becomes saturated at the local speed of sound). This parameter is often referred to in the literature as the b-value. Enter a number greater than or equal to zero and smaller than the Laminar flow pressure ratio block parameter.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Subsonic index — Exponent used to more accurately characterize flow in the subsonic regime

0.5 (default) | positive unitless scalar

Empirical exponent used to more accurately calculate the mass flow rate through the valve when the flow is subsonic. This parameter is sometimes referred to as the m-index. Its value is approximately 0.5 for valves (and other components) whose flow paths are fixed.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Cv coefficient (USCS) at maximum flow — Flow coefficient of the fully open valve expressed in US customary units

0.4 (default) | positive scalar in units of ft^3/min

Flow coefficient of the fully open valve, expressed in the US customary units of ft3/min (as described in NFPA T3.21.3). This parameter measures the relative ease with which the gas will traverse the valve when driven by a given pressure differential. This is the value generally reported by manufacturers in technical data sheets.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Cv coefficient (USCS).

Kv coefficient (SI) at maximum flow — Flow coefficient of the fully open valve expressed in SI units

0.3 (default) | positive scalar in units of L/min

Flow coefficient of the fully open valve, expressed in the SI units of L/min. This parameter measures the relative ease with which the gas will traverse the valve when driven by a given pressure differential. This is the value generally reported by manufacturers in technical data sheets.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Kv coefficient (SI).

Leakage area — Opening area in the maximally closed position due to sealing imperfections

1e-12 m^2 (default) | positive scalar in units of area

Opening area of the valve in the maximally closed position, when only internal leakage between the ports remains. This parameter serves primarily to ensure that closure of the valve does not cause portions of the gas network to become isolated (a condition known to cause problems in simulation). The exact value specified here is less important that its being a (very small) number greater than zero.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Opening area.

Cross-sectional area at ports A and B — Area normal to the flow path at the valve ports

0.01 m^2 (default) | positive scalar in units of area

Area normal to the flow path at the valve ports. The ports are assumed to be the same in size. The flow area specified here should ideally match those of the inlets of adjoining components.

Laminar flow pressure ratio — Pressure ratio at which the flow transitions between laminar and turbulent

0.999 (default) | positive unitless scalar

Pressure ratio at which the flow transitions between laminar and turbulent flow regimes. The pressure ratio is the fraction of the absolute pressure downstream of the valve over that just upstream of it. The flow is laminar when the actual pressure ratio is above the threshold specified here and turbulent when it is below. Typical values range from 0.995 to 0.999.

Reference temperature — Inlet temperature used in the measurement of sonic conductance

293.15 K (default) | positive scalar in units of temperature

Absolute temperature used at the inlet in the measurement of sonic conductance (as defined in ISO 8778).

Reference density — Inlet density used in the measurement of sonic conductance

1.185 (default) | positive scalar in units of density

Gas density established at the inlet in the measurement of sonic conductance (as defined in ISO 8778).

Smoothing factor — Amount of smoothing to apply to the valve opening area function

0 (default)

Amount of smoothing to apply to the opening area function of the valve. This parameter determines the widths of the regions to be smoothed—one located at the fully open position, the other at the fully closed position.

The smoothing superposes on each region of the opening area function a nonlinear segment (a third-order polynomial function, from which the smoothing arises). The greater the value specified here, the greater the smoothing is, and the broader the nonlinear segments become.

At the default value of 0, no smoothing is applied. The transitions to the maximally closed and fully open positions then introduce discontinuities (associated with zero-crossings), which tend to slow down the rate of simulation.

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