What does Cv stand for and its role in control valve sizing
Predicting flow choking across the nozzle or orifice is the most important factor in sizing a control valve or PSV. When flow is choked upstream, it becomes unpredictable, and any device with a nozzle or orifice becomes functionally erratic. Cv, or the internal energy of the fluid, provides the energy for fluid expansion at the nozzle/orifice, and thus Cv is important in determining flow across a nozzle/orifice and choked pressure drop. There is a wonderful relationship between Cv and choked flow. The more the Cv, the less is Cp/Cv. This means more internal energy. The more the internal energy of a fluid the more it does work or expand and therefore more prone to choking. A triatomic gas with gamma [Cp/Cv] = 1.33 has more chance to choke than a diatomic gas with gamma = 1.4. The most interesting point is while a bigger Cv would give you more flow but the flow is proportional to the square root of the choked pressure drop.
Therefore, the engineers, I am sure, consider this point and suitably adjust the pressure drop to avoid flow choking. This has been explained in more detail later.
This post will concentrate on the role of Cv in control valve sizing. The post would not go into control valve sizing because API 520/521 has some equations to determine the minimum orifice size you need, as well as good advice and factors to put into their equations if you don't yet have information from the valve manufacturer. There are numerous programmes and spreadsheets available for sizing the orifice, so find out which ones suit you.
Cv and choked flow
Cv stands for specific heat at constant volume. Cv is a measure of the internal energy of a fluid passing through an orifice in a pipe. When a fluid flows through an orifice and it expands at the vena contracta. In the case of liquids vaporization of the liquid occurs when the static pressure within the valve drops below the vapor pressure of the liquid. We call it choked flow. When the flow is choked no increase in flow takes place.
Pertaining to gases/steam – Critical flow is a choked flow condition caused by the increased gas velocity at the vena contracta. When velocity at the vena contracta reaches sonic velocity, additional increases in dP by reducing downstream pressure produce no increase in flow. Choked flow in short is a phenomenon that limits the mass flow rate of a compressible fluid flowing through nozzles, orifices, and sudden expansions. Generally speaking, it is the mass flux after which a further reduction in downstream pressure will not result in an increase in mass flow rate.?Fundamentally, at initially subsonic upstream conditions, the conservation of energy principle requires the fluid velocity to increase as it flows through the smaller cross-sectional area of the constriction. At the same time, the venturi effect causes the static pressure, and therefore the density, to decrease at the constriction and expansion of the fluid.
Cv provides energy
Cv provides the energy to the fluid for its behavioral change at the orifice.?The fluid uses its own internal energy. In addition, the process is so fast that the process is approximated as an adiabatic and internally reversible isentropic process.?
Cv and choked flow are related
Please see the equations in the figure
The basic liquid sizing equation, shown in the upper left corner of the figure states that the flow rate of liquid through a control valve is proportional to the square root of the pressure drop. The green portion of the graph in the figure depicts this simple relationship graphically. (Note that the horizontal axis scale is the square root of the pressure drop.) This linear relationship is not always valid. The flow reaches a point where it no longer increases as the pressure drop is increased by lowering the downstream pressure. Once this occurs, further increases in pressure drop across the valve result in no additional flow, and the flow is said to be choked. This is referred to as limiting or choking in this context.
What causes flow choking?
The presence and extent of flow choking depends on many process conditions, including the physical properties of the fluid involved, flow rates, upstream and downstream pressures, process temperature, and inlet and outlet piping configurations—as well as a number of details associated with the control valve itself. Special parameters, such as pressure drop ratio, pressure recovery factor, and cavitation index, help predict exactly when cavitation or choking will occur, and how much flow a valve will pass
Role of Cp/Cv in choked flow
When a gas expands adiabatically it uses its internal energy to expand. The ability of a gas to expand comes from how much internal energy it has to supply for expansion. The more gas has internal energy the more it is compressible and therefore, the more it is prone to choked flow.
Detail
When you push a high-pressure gas through a constriction it expands adiabatically following the polytropic equation PV ^n = C. n = Cp/Cv = y = 1.4 for a diatomic gas. Cp/Cv depends on the atomicity of a gas.
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?Y [Gamma] = Cp/Cv = 1 + 2/[DOF], DOF is degrees of freedom, the way a molecule can store energy
Cp/Cv ratio for monoatomic, diatomic, and triatomic is 1.67,1.4,1.33 respectively.
dH / dT =Cp, H is enthalpy
dU / dT = Cv, U is the internal energy
Cp/Cv = dH / dU
Cp/Cv is simply the amount of adiabatic work you can extract from a molecule. The more internal energy the less Cp/Cv. The more adiabatic work you get. The change in potential energy results in a change in Internal Energy, even though Kinetic energy (or temperature) doesn't change. In thermodynamics, this is why work being done on a system is positive.
The choked flow pressure ratio of gases [The second column stands for gamma]
Look at this table. Take the case of CO2, y = 1.3, Choked flow pressure ratio = 1.83 and helium, y = 1.66 and choked flow pressure ratio = 2.05.
?y [ gamma] for CO2 is the lowest in the list because it is a polyatomic polar molecule with a larger DOF [ degrees of freedom to store energy]. Its two oxygen atoms with two lone pairs of electrons each make CO2 more polar than any other molecules in the list of gas. It has the highest internal energy stored in the molecule for the above reasons for adiabatic work so it is the most compressible gas in this list of gases and has the maximum ability for choked flow.
He [ helium] on the other hand has the highest Cp/Cv in the list, it has the lowest internal energy to do adiabatic work because it is an inert small molecule. It has no polarity. It has very few van der Waal forces. So, it is the least compressible Z >1, almost like an ideal gas compared to CO2 compressibility factor Z < 1 with the minimum choked flow potential.
But nothing is absolute. Everything is on a comparative scale
All gases are compressible. But within the gases, some real gases are more compressible than others. Like CO2 is more compressible than He [ comparatively]. The reason is CO2 has z < 1. In CO2 because it is a big polar molecule the attractive forces dominate. On the other side helium, is a small nonpolar molecule. It has dominating repulsion forces acting between the molecules therefore it has z > 1 [ z is the compressibility factor]. This makes on a relative scale CO2 more compressible than He
Finally, it is the atomicity or degrees of freedom (DOF) which relates to y (gamma) Cp / Cv = 1 + 2/ DOF that takes the final call. The more degrees of freedom the less is CP/ Cv and this is a fixed number. For all diatomic molecules, it is 1.4. So, all diatomic molecules have identical choked flow potential. For triatomic molecules like CO2 or H2O, because there are more degrees of freedom, y = 1.33.
Therefore, a triatomic molecule has more chances to choke than a diatomic molecule
The bottom line is if you reach a mass flow rate that generates sonic velocity at the point of constriction, MAC = 1 regardless of what the gas is there will be choked flow.
Credit: Google