Check Valve (G)
Valve for limiting flow to a single (forward) direction
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Simscape / Fluids / Gas / Valves & Orifices / Directional Control Valves
Description
The Check Valve (G) block models an orifice with a unidirectional opening mechanism to prevent unwanted backflow. The opening mechanism, often spring-loaded by design, responds to pressure, (typically) opening the orifice when the pressure gradient across it falls from inlet (port A) to outlet (port B), but forcing it shut otherwise. Check valves protect components upstream against pressure surges, temperature spikes, and (in real systems) chemical contamination stemming from points downstream.
The valve opens by degrees, beginning at its cracking pressure, and continuing to the end of its pressure regulation range. The cracking pressure gives the initial resistance, due to friction or spring forces, that the valve must overcome to open by a sliver (or to crack open). Below this threshold, the valve is closed and only leakage flow can pass. Past the end of the pressure regulation range, the valve is fully open and the flow at a maximum (determined by the instantaneous pressure conditions).
The cracking pressure assumes an important role in check valves installed upside down. There, the weight of the opening element—such as a ball or piston—and the elevation head of the fluid can act to open the valve. (The elevation head can arise in a model from a pipe upstream of the inlet when it is vertical or given a slant.) A sufficient cracking pressure keeps the valve from opening inadvertently even if placed at a disadvantageous angle.
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.
Control and Other Pressures
The pressure to which the valve responds is its control pressure. In a typical check valve (and by default in this block), that pressure is the drop from inlet to outlet. This setting ensures that the valve in fact closes if the direction of flow should reverse.
For special cases, an alternative control pressure is provided: the gauge pressure at the inlet. Use it if you know that the inlet will always be at a higher pressure than the outlet (for example, when the inlet connects to a pressure source, such as a pump).
You can select an appropriate control pressure for your model—either
Pressure differential or Pressure at port A
(gauge)
—using the Pressure control
specification drop-down list.
Pressure Differential
When the Pressure control specification parameter is set
to Pressure differential
, the control pressure is
computed as:
where p is instantaneous pressure. The
subscript Ctl
denotes the control value and the subscripts
A and B the inlet and outlet,
respectively. The port pressures are instantaneous values determined (against
absolute zero) during simulation. The cracking pressure is likewise defined as:
where P is a constant pressure parameter.
The subscript Crk
denotes the cracking value (here a
differential). The term in parentheses is obtained as a constant from the
Cracking pressure differential block parameter.
Similarly for the maximum pressure of the valve (at which the valve is fully open):
where the subscript Max
denotes the
maximum value of the valve. Here too the term in parentheses is obtained as a
constant, from the Maximum opening pressure differential
block parameter.
Pressure at port A
When the Pressure control specification parameter is set to Pressure at port A, the control pressure is computed as:
The port pressure is an instantaneous value determined (against absolute zero) during simulation. For the cracking pressure:
where the subscript A,Crk
denotes the
cracking value, specified as a gauge pressure at port A.
This value is obtained as a constant from the Cracking pressure
(gauge) block parameter. The subscript Atm
denotes the atmospheric value (specified in the Gas Properties (G) block
of the model). The maximum pressure of the valve is:
where the subscript A,Max
denotes the
maximum value, specified as a gauge pressure at port A.
This value is obtained as a constant from the Maximum opening pressure
(gauge) block parameter.
Control Pressure Overshoot
The degree to which the control pressure exceeds the cracking pressure determines how much the valve will open. The pressure overshoot is expressed here as a fraction of the (width of the) pressure regulation range:
The control pressure (pCtl), cracking
pressure (pSet), and maximum opening
pressure (PMax) correspond to the control
pressure specification chosen (Pressure differential
or
Pressure at port A
).
The fraction—technically, the overshoot normalized—is valued
at 0
in the fully closed valve and 1
in the
fully open valve. If the calculation should return a value outside of these bounds,
the nearest of the two is used instead. (In other words, the fraction
saturates at 0
and
1
.)
The normalized control pressure, p, spans three pressure
regions. Below the cracking pressure of the valve, its value is a constant zero.
Above the maximum pressure of the same, it is 1
. In between,
it varies, as a linear function of the (effective) control pressure,
pCtl.
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 pressure overshoot with sharp transitions
The block removes the discontinuities by smoothing them over a specified time scale. The smoothing, which adds a slight distortion to the normalized inlet pressure, 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:
and
where
and
In the equations:
λL is the smoothing expression for the transition from the maximally closed position.
λR is the smoothing expression for the transition from the fully open position.
Δp* is the (unitless) characteristic width of the pressure smoothing region:
where f* is a smoothing factor valued between
0
and1
and obtained from the block parameter of the same name.When the smoothing factor is
0
, the normalized inlet pressure stays in its original form—no smoothing applied—and its transitions remain abrupt. When it is1
, the smoothing spans the whole of the pressure regulation range (with the normalized inlet pressure 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 pressure regulation range on each side (for a total smooth region of half the regulation range).
The smoothing adds two new regions to the normalized pressure overshoot—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:
where the asterisk denotes a smoothed variable (the normalized control pressure overshoot). The figure shows the effect of smoothing on the sharpness of the transitions.
Sonic Conductance
As the normalized control pressure 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 normalized inlet pressure 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 Sonic
conductance at maximum flow parameter when the Valve
parameterization setting is Sonic
conductance
. For values across the opening range of the valve,
this maximum is scaled by the normalized pressure overshoot:
where C is sonic conductance and the
subscripts Max
and Min
denote its values in
the fully open and fully closed valve.
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:
Restriction area
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.
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:
Cv is measured at a generally accepted temperature of
60 ℉
and pressure drop of1 PSI
; it is expressed in imperial units ofUS gpm
. This is the flow coefficient used in the model when the Valve parameterization block parameter is set toCv coefficient (USCS)
.Kv is measured at a generally accepted temperature of
15 ℃
and pressure drop of1 bar
; it is expressed in metric units ofm3/h
. This is the flow coefficient used in the model when the Valve parameterization block parameter is set toKv coefficient (SI)
.
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:
where Cv is the
flow coefficient value at maximum or leakage flow. The subsonic index,
m, is set to 0.5
and the critical
pressure ratio, bcr, 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:
where Kv is the
flow coefficient value at maximum or leakage flow. The subsonic index,
m, is set to 0.5
and the critical
pressure ratio, bcr, 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:
where S is the opening area at maximum or
leakage flow. The subsonic index, m, is set to
0.5
while the critical pressure ratio,
bcr is computed from the expression:
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).
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:
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:
where:
pr is the back-pressure ratio, or that between the outlet pressure (pout) and the inlet pressure (pin):
bcr 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:
where blam 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:
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:
where 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:
where ϕ is defined as the energy flow rate into the valve through one of the ports (A or B).