Control valve choked flow and Cv [specific heat at constant volume]

Is choked flow a game of entropy? Yes, it is [explained below.]

[In?layman's?terms,?sound?waves?slap?gas?molecules?when?they?try?to?outrun?the?speed?of?sound. Sound?waves?slap?the?gas?molecules?so?hard?that?they?can no?longer?increase?their?velocity. Flow?becomes?obstructed and choked. In the meantime, the?sound?waves?acquire?the?maximum?speed?at?that?temperature after?removing?the?resistance?from?their?path. Flow?choking?is?caused?by?the?speed?of?sound?overpowering?the?velocity?of?gas?molecules?at?the?constriction.]

The important point to note is that the gas expands at the throat because the gas has constant free energy as the flow is isentropic and reversible. At constant free energy, the only option for the gas is to draw energy when expanding adiabatically 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. There is no other location for flow choking.

Is flow choking a game of entropy? Yes, it is.

At the throat, the flow is isentropic and therefore, reversible. This is primarily due to back pressure by sound waves at the throat. The temperature and pressure of both air and gas stay at equilibrium. the gas molecules cannot exceed the speed of sound.

The post is divided into two parts, [1] Explanation of choked flow and [2] Control valve choked flow.?I have explained how Cv in the conventional definition stands for internal energy.

What is choked flow?

Critical or choked flow is important in many practical applications, such as power generation and chemical process industries, where system safety or performance can be jeopardized without a precise understanding of critical flow behavior. In general, a choking flow occurs around a minimum area location [constriction] or an abrupt area change location in a flow system, such as orifices and nozzles. When there is a critical flow through a nozzle or orifice, throat pressure provides the maximum back pressure for a given upstream condition, such as a choked pressure. Choked flow occurs when the downstream static pressure falls below 0.487 to 0.587 times the absolute pressure in the stagnant upstream source vessel, depending on the gas.

Details:

To put it simply, choked flow is a compressible flow effect. Because the distance between atoms/molecules is so great, gases are the most compressible. When the velocity of a gas increases, or when gas molecules move away from one another, the pressure decreases, and the volume of the gas increases. Gas density decreases as it expands. The Mach number is an important concept. The Mach number (M) is defined as the ratio of an object's [gas] (or a flow's) speed to the speed of sound.?When gas flows at a low velocity or low Mach number, the Mach number has no significance. Only gases with a Mach number greater than 0.3 are considered compressible. As a result, gases with Mach numbers less than 0.3 are considered incompressible because the density change in the fluid is insignificant. When the speed of the gas approaches the speed of sound, the Mach number becomes critical. The gas expands at the constriction adiabatically. ?Cv, the specific heat of the gas or the internal energy is the energy source for the expansion of gases. More Cv for gas means more prone to choking.

Explanation:?

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, 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.

Explanation: As a gas accelerates from subsonic toward supersonic speed in the air, different types of wave phenomena occur. To illustrate these changes, at a stationary point (M = 0) sound waves are symmetric.?The speed of sound is the same in all directions in a uniform fluid, so these waves are simply concentric spheres. As the sound-generating point begins to accelerate, the 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 travels at the same speed as the sound waves. At this stage, an infinite number of sound waves "pile up" ahead of the gas preventing any further increase in gas velocity or flow rate.

Risk of flow choking in a control valve: 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

Choked flow is defined as the point at which increasing the pressure drop (?P), while maintaining a constant inlet pressure, yields no further increase in flow rate through a fixed area. Predicting flow choking across the nozzle or orifice is the most important factor in sizing a control valve or PSV because it can jeopardize the performance and safety of a control valve. When flow is choked upstream, it becomes unpredictable, and any device with a nozzle or orifice becomes functionally erratic.

The conventional definition of Cv : Valve Flow Coefficient (Cv) is 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].

?How internal energy relates conventional definition of Cv?

Cv is the source of energy for the expansion of gas at the constriction:?Cv, or the internal energy of the fluid, provides the energy for 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

It is interesting to note that CO2 is more prone to choking than Helium.

Why?

It is all the magic of internal energy. Helium just does not have sufficient internal energy, Cv to cause its expansion at the valve throat and choke.?

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. 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.

The final point

The important point to note is that the gas expands at the throat because the gas has constant free energy since the flow is isentropic and reversible. At constant free energy, the only option for the gas is to draw energy from the internal energy in an adiabatic process, 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. There is no other location for flow choking.

Is flow choking a game of entropy? Yes, it is.

At the throat, the flow is isentropic and therefore, reversible. This is primarily due to back pressure by sound waves at the throat. The temperature and pressure of both air and gas stay at equilibrium. the gas molecules cannot exceed the speed of sound.