Improper valve sizing can be both expensive and inconvenient. A valve that is too small will not pass the required flow, and the process will be starved. An oversized valve will be more expensive, and it may lead to instability and other problems. The days of selecting a valve based upon the size of the pipeline are gone. Selecting the correct valve size for a given application requires a knowledge of process conditions that the valve will actually see in service. The technique for using this information to size the valve is based upon a combination of theory and experimentation.
Sizing For Liquid Service
Using the principle of conservation of energy, Daniel Bernoulli found that as a liquid flows through an orifice, the square of the fluid velocity is directly proportional to the pressure differential across the orifice and inversely proportional to the specific gravity of the fluid. The greater the pressure differential, the higher the velocity; the greater the density, the lower the velocity. The volume flow rate for liquids can be calculated by multiplying the fluid velocity times the flow area.By taking into account units of measurement, the proportionality relationship previously mentioned, energy losses due to friction and turbulence, and varying discharge coefficients for various types of orifices (or valve bodies), a basic liquid sizing equation can be written as follows:
Thus, Cv is numerically equal to the number of U.S. gallons of water at 60°F that will flow through the valve in one minute when the pressure differential across the valve is one pound per square inch. Cv varies with both size and style of valve, but provides an index for comparing liquid capacities of different valves under a standard set of conditions.
To aid in establishing uniform measurement of liquid flow capacity coefficients (Cv) among valve manufacturers, the Fluid Controls Institute (FCI) developed a standard test piping arrangement, shown in Figure 1.
Figure 1. Standard FCI Test Piping for C Measurement
Using such a piping arrangement, most valve manufacturers develop and publish Cv information for their products, making it relatively easy to compare capacities of competitive products.
To calculate the expected Cv for a valve controlling water or other liquids that behave like water, the basic liquid sizing equation above can be re-written as follows:
Flashing and Cavitation
The occurrence of flashing or cavitation within a valve can have a significant effect on the valve sizing procedure. These two related physical phenomena can limit flow through the valve in many applications and must be taken into account in order to accurately size a valve. Structural damage to the valve and adjacent piping may also result. Knowledge of what is actually happening within the valve may permit selection of a size or style of valve which can reduce, or compensate for, the undesirable effects of flashing or cavitation. The "physical phenomena" label is used to describe flashing and cavitation because these conditions represent actual changes in the form of the fluid media. The change is from the liquid state to the vapor state and results from the increase in fluid velocity at or just down-stream of the greatest flow restriction, normally the valve port. As liquid flow passes through the restriction, there is a necking down, or contraction, of the flow stream. The minimum cross-sectional area of the flow stream occurs just downstream of the actual physical restriction at a point called the vena contracta, as shown in Figure 3.
Figure 3. Vena Contracta
Figure 4. Comparison of Pressure Profiles for High and Low Recovery Valves
To maintain a steady flow of liquid through the valve, the velocity must be greatest at the vena contracta, where cross sectional area is the least. The increase in velocity (or kinetic energy) is accompanied by a substantial decrease in pressure (or potential energy) at the vena contracta. Further downstream, as the fluid stream expands into a larger area, velocity decreases and pressure increases. But, of course, downstream pressure never recovers completely to equal the pressure that existed upstream of the valve. The pressure differential (ΔP) that exists across the valve is a measure of the amount of energy that was dissipated in the valve. Figure 4 provides a pressure profile explaining the differing performance of a streamlined high recovery valve, such as a ball valve and a valve with lower recovery capabilities due to greater internal turbulence and dissipation of energy.
Regardless of the recovery characteristics of the valve, the pressure differential of interest pertaining to flashing and cavitation is the differential between the valve inlet and the vena contracta. If pressure at the vena contracta should drop below the vapor pressure of the fluid (due to increased fluid velocity at this point) bubbles will form in the flow stream. Formation of bubbles will increase greatly as vena contracta pressure drops further below the vapor pressure of the liquid. At this stage, there is no difference between flashing and cavitation, but the potential for structural damage to the valve definitely exists.
If pressure at the valve outlet remains below the vapor pressure of the liquid, the bubbles will remain in the downstream system and the process is said to have "flashed." Flashing can produce serious erosion damage to the valve trim parts and is characterized by a smooth, polished appearance of the eroded surface. Flashing damage is normally greatest at the point of highest velocity, which is usually at or near the seat line of the valve plug and seat ring.
However, if downstream pressure recovery is sufficient to raise the outlet pressure above the vapor pressure of the liquid, the bubbles will collapse, or implode, producing cavitation. Collapsing of the vapor bubbles releases energy and produces a noise similar to what one would expect if gravel were flowing through the valve. If the bubbles collapse in close proximity to solid surfaces, the energy released gradually wears the material leaving a rough, cinderlike surface. Cavitation damage may extend to the downstream pipeline, if that is where pressure recovery occurs and the bubbles collapse. Obviously, "high recovery" valves tend to be more subject to cavitation, since the downstream pressure is more likely to rise above the liquid's vapor pressure.
Choked Flow
Aside from the possibility of physical equipment damage due to flashing or cavitation, formation of vapor bubbles in the liquid flowstream causes a crowding condition at the vena contracta which tends to limit flow through the valve. So, while the basic liquid sizing equation implies that there is no limit to the amount of flow through a valve as long as the differential pressure across the valve increases, the realities of flashing and cavitation prove otherwise. If valve pressure drop is increased slightly beyond the point where bubbles begin to form, a choked flow condition is reached. With constant upstream pressure, further increases in pressure drop (by reducing downstream pressure) will not produce increased flow. The limiting pressure differential is designated DP allow and the valve recovery coefficient (Km) is experimentally determined for each valve, in order to relate choked flow for that particular valve to the basic liquid sizing equation. Km is normally published with other valve capacity coefficients. Figures 4 and 5 show these flow vs. pressure drop relationships.
Figure 5. Flow Curve Showing Cv and Km
Figure 6. Relationship Between Actual ΔP and ΔP Allowable
Use the following equation to determine maximum allowable pressure drop that is effective in producing flow. Keep in mind, however, that the limitation on the sizing pressure drop,ΔP allow, does not imply a maximum pressure drop that may be controlled by the valve.
Figure 7. Critical Pressure Ratios for Water
Figure 8. Critical Ratios for Liquid Other than Water
After calculating ΔPallow, substitute it into the basic liquid sizing equation Q = Cv to determine either Q or Cv √——ΔP/G.. If the actual ΔP is less the ΔP allow , then the actual ΔP should be used in the equation.
The equation used to determine ΔPallow should also be used to calculate the valve body differential pressure at which significant cavitation can occur. Minor cavitation will occur at a slightly lower pressure differential than that predicted by the equation, but should produce negligible damage in most globe-style control valves.
Consequently, it can be seen that initial cavitation and choked flow occur nearly simultaneously in globe-style or low-recovery valves.
However, in high-recovery valves such as ball or butterfly valves, significant cavitation can occur at pressure drops below that which produces choked flow. So while ΔPallow and Km are useful in predicting choked flow capacity, a separate cavitation index (Kc) is needed to determine the pressure drop at which cavitation damage will begin (ΔPc) in high-recovery valves.
The equation can be expressed:
This equation can be used anytime outlet pressure is greater than the vapor pressure of the liquid.
Addition of anti-cavitation trim tends to increase the value of Km. In other words, choked flow and insipient cavitation will occur at substantially higher pressure drops than was the case without the anti-cavitation accessory.
Liquid Sizing Summary
The most common use of the basic liquid sizing equation is to determine the proper valve size for a given set of service conditions. The first step is to calculate the required Cv by using the sizing equation. The ΔP used in the equation must be the actual valve pressure drop or DPallow, whichever is smaller. The second step is to select a valve, from the manufacturer's literature, with a Cv equal to or greater than the calculated value.Accurate valve sizing for liquids requires use of the dual coefficients of Cv and Km. A single coefficient is not sufficient to describe both the capacity and the recovery characteristics of the valve. Also, use of the additional cavitation index factor Kc is appropriate in sizing high recovery valves, which may develop damaging cavitation at pressure drops well below the level of the choked flow.
Liquid Sizing Equation Application
1. 
Basic liquid sizing equation. Use to determine proper valve size for a given set of service contitions. (Remember that viscosity effects and valve recovery capabilities are not considered in this basic equation.)
2. 
Use to calculate expected Cv for valve controlling water or other liquids that behave like water.
3. 
Use to find actual required Cv for equation (2) after including viscosity correction factor.
4. 
Use to find maximum flow rate assuming no viscosity correction is necessary.
5. 
Use to predict actual flow rate based on equation (4) and viscosity factor correction.
6. 
Use to calculate corrected sizing coefficient for use in equation (7).
7. 
Use to predict pressure drop for viscous liquids.
8. 
Use to determine maximum allowable pressure drop that is effective in producing flow.
9. 
Use to predict pressure drop at which cavitation will begin in a valve with high recovery characteristics.
Sizing for Gas or Steam Service
A sizing procedure for gases can be established based on adaptions of the basic liquid sizing equation. By introducing conversion factors to change flow units from gallons per minute to cubic feet per hour and to relate specific gravity in meaningful terms of pressure, an equation can be derived for the flow of air at 60°F. Since 60°F corresponds to 520° on the Rankine absolute temperature scale, and since the specific gravity of air at 60°F is 1.0, an additional factor can be included to compare air at 60°F with specific gravity (G) and absolute temperature (T) of any other gas. The resulting equation can be written:
The equation shown above, while valid at very low pressure drop ratios, has been found to be very misleading when the ratio of pressure drop (ΔP) to inlet pressure (P1) exceeds 0.02. The deviation of actual flow capacity from the calculated flow capacity is indicated in Figure 8 and results from compressibility effects and critical flow limitations at increased pressure drops.
Critical flow limitation is the more significant of the two problems mentioned. Critical flow is a choked flow condition caused by increased gas velocity at the vena contracta. When velocity at the vena contracta reaches sonic velocity, additional increases in ΔP by reducing downstream pressure produce no increase in flow. So, after critical flow condition is reached (whether at a pressure drop/inlet pressure ratio of about 0.5 for glove valves or at much lower ratios for high recovery valves) the equation above becomes completely useless. If applied, the Cv equation gives a much higher indicated capacity than actually will exist. And in the case of a high recovery valve which reaches critical flow at a low pressure drop ratio (as indicated in Figure 8), the critical flow capacity of the valve may be over-estimated by as much as 300 percent.
The problems in predicting critical flow with a Cv -based equation led to a separate gas sizing coefficient based on air flow tests. The coefficient (Cg) was developed experimentally for each type and size of valve to relate critical flow to absolute inlet pressure. By including the correction factor used in the previous equation to compare air at 60°F with other gases at other absolute temperatures, the critical flow equation can be written:
Figure 9. Critical Flow for High and Low Recovery Valves with Equal Cv
Universal Gas Sizing Equation
To account for differences in flow geometry among valves, equations (A) and (B) were consolidated by the introduction of an additional factor (C1). C1 is defined as the ratio of the gas sizing coefficient and the liquid sizing coefficient and provides a numerical indicator of the valve's recovery capabilities. In general, C1 values can range from about 16 to 37, based on the individual valve's recovery characteristics. As shown in the example, two valves with identical flow areas and identical critical flow (Cg) capacities can have widely differing C1 values dependent on the effect internal flow geometry has on liquid flow capacity through each valve. Example:
So we see that two sizing coefficients are needed to accurately size valves for gas flow Cg to predict flow based on physical size or flow area, and C1 to account for differences in valve recovery characteristics. A blending equation, called the Universal Gas Sizing Equation, combines equations (A) and (B) by means of a sinusoidal function, and is based on the "perfect gas" laws. It can be expressed in either of the following manners:
In either form, the equation indicates critical flow when the sine function of the angle designated within the brackets equals unity. The pressure drop ratio at which critical flow occurs is known as the critical pressure drop ratio. It occurs when the sine angle reaches π/2 radians in equation (C) or 90 degrees in equation (D). As pressure drop across the valve increases, the sine angle increases from zero up to π/2 radians (90°). If the angle were allowed to increase further, the equations would predict a decrease in flow. Since this is not a realistic situation, the angle must be limited to 90 degrees maximum.
Although "perfect gases," as such, do not exist in nature, there are a great many applications where the Universal Gas Sizing Equation, (C) or (D), provides a very useful and usable approximation.
Gas and Steam Sizing Summary
The Universal Gas Sizing Equation can be used to determine the flow of gas through any style of valve. Absolute units of temperature and pressure must be used in the equation. When the critical pressure drop ratio causes the sine angle to be 90 degrees, the equation will predict the value of the critical flow. For service conditions that would result in an angle of greater than 90 degrees, the equation must be limited to 90 degrees in order to accurately determine the critical flow.Most commonly, the Universal Gas Sizing Equation is used to determine proper valve size for a given set of service conditions. The first step is to calculate the required Cg by using the Universal Gas Sizing Equation. The second step is to select a valve from the manufacturer's literature. The valve selected should have a Cg which equals or exceeds the calculated value. Be certain that the assumed C1 value for the valve is selected from the literature.
It is apparent that accurate valve sizing for gases that requires use of the dual coefficient is not sufficient to describe both the capacity and the recovery characteristics of the valve.
Proper selection of a control valve for gas service is a highly technical problem with many factors to be considered. Leading valve manufacturers provide technical information, test data, sizing catalogs, nomographs, sizing slide rules, and computer or calculator programs that make valve sizing a simple and accurate procedure.
Gas and Steam Sizing Equation Application
A. 
Use only at very low pressure drop (DP/P1) ratrios of 0.02 or less.
B.
Use only to determine critical flow capacity at a given inlet pressure.
C. 
D. 
Universal Gas Sizing Equation.
Use to predict flow for either high or low recovery valves, for any gas adhering to the perfect gas lows, and under any service conditions.
E. 
Use to predict flow for perfect or non-perfect gas sizing applications, for any vapor including steam, at any service condition when fluid density is known.
F. 
Use only to determine steam flow when inlet pressure is 1000 psig or less.
