What is Valve Flow Coefficient Cv?

Summary

The valve coefficient of a control valve signifies the internal energy of the gas. The internal energy of the gas is the energy source of the gas for adiabatic expansion at the constriction of a valve. The internal energy of the gas is its specific heat at constant volume Cv. Cp/Cv of the gas ensures the Mach no <1, and no choking. The pressure drop at the constriction in the control valve is fixed on the basis of the gas's Cp/Cv ratio.

Helium is a monoatomic gas. Its Cp/Cv = 1.66. It can tolerate more pressure drop without choking at the constriction than CO2 which is a triatomic gas, Cp/Cv= 1.29.

Cp/Cv is the amount of adiabatic work you can extract from a molecule. The more the internal energy, Cv of gas the less Cp/Cv

Detail

Valve Flow Coefficient (Cv) is defined as the flow capability of a control valve at fully open conditions relative to the pressure drop across the valve. It is defined as the volume of water (GPM in the US) at 60°F that will flow through a fully open valve with a pressure differential of 1 psi across the valve.

According to this definition,

Cv = Q x √G/ √ ΔP, Q ∝ √ ΔP

Q = Flow in Gallons per Minute

G = Specific gravity of fluid (estimated as 1 for water systems)

ΔP = Differential pressure over valve (delta P) – stated in psi

What does it mean??For a given flow rate, Q, the Cv is inversely proportional to the square root of the pressure drop, √ ΔP. The relation between pressure and velocity is inversely proportional. Therefore, Cv ∝ √ ΔV.?Cv and ΔV are directly related. At a constriction, this explains, for the velocity to reach Mach = 1 the Cv should be sufficient. In other words, Cv ∝ √ ΔV means that maximum ΔV only occurs when there is sufficient Cv [internal energy].

What is the Mach number? What is choked flow?

The compressibility of fluids is measured by the Mach number. We associate compressibility with sound speed because sound travels at its fastest when fluid becomes compressible. When a choked flow occurs at the maximum compressibility of gas, the gas-particle resistance to sound speed ceases. As a result, the speed of sound becomes a good indicator of fluid compressibility.

When the gas reaches the speed of sound, or Mach number =1, it begins to interact with the surrounding air molecules, attempting to compress airwaves. This is the start of the problems. At a given temperature, the speed of sound remains constant. The sound waves resist the disturbance. Sound waves??"bunch up" in the direction of motion and "stretch out" in the opposite direction. ?When the gas reaches sonic speed (M = 1), it moves at the same rate as the sound waves. In order to prevent the disturbance caused by gas, an infinite number of sound waves "pile up" preventing any further increase in gas velocity or flow rate. This is referred to as choked flow.

On the sound side, the speed of sound increases as the density of fluid reduces because there are a smaller number of particles to cause obstruction in the path of sound waves moving forward.?At the constriction exit thus sound reaches maximum velocity. Therefore, to summarize at the constriction exit the fluid reaches the maximum flow velocity and sound reaches the maximum speed.

Let us go back to the valve coefficient, Cv

Cv is the specific heat at constant volume. Cv is the energy storage of a molecule at constant volume [ no work] Cv is the internal energy of a gas molecule.

?How internal energy of the gas is the valve coefficient for the control valve?

Cv is the source of energy for the adiabatic expansion of gas at the constriction:?Cv, or the internal energy of the fluid, provides the energy for the fluid’ adiabatic 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 Cp/Cv. This means more internal energy. The more the internal energy of a fluid the more it does adiabatic 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.

Explanation: 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.

?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.

Example: Choked flow pressure ratio of gases [The second column stands for gamma]

The table provides the choked flow pressure ratio of gases

Explanation: Look at the above table. Take the case of CO2, y = 1.3, CO2 is a polyatomic gas. Choked flow pressure ratio = 1.83. ?Now look at helium, y = 1.66 Helium is a monoatomic gas and choked flow pressure ratio = 2.05.

?Analysis of the data in the table

CO2 vs He: y [ gamma] for CO2 is lower than He. Cv of CO2 is more than He as a ratio of Cp. The choked pressure ratio of He is more than CO2. In other words, helium can tolerate more pressure ratio at a constriction than CO2 simply because helium does not have sufficient internal energy as a ratio of Cp.

Final point and a summary

The important point to note is that for the gas to expand at the throat when the free energy is equal to the change of enthalpy since the flow is isentropic. The only option for the gas is to draw energy from the internal energy and therefore, Cv is so important for flow choking.

Choked flow always happens at the throat because at the throat the gas draws internal energy for expansion.?

Credit: Google