Proper Valve Size Helps Determine Flow
Sizing valves for vapor applications based on liquidsizing calculations can lead to poor results, noise, and excessive cost.
T he most common practice used in industry to determine the pressure drop or flow capacity of a valve is to obtain the valve flow coefficient C _{v} from the valve manufacturer and apply it to an appropriate formula. Today, most valve manufacturers publish flow coefficients, together with equations to predict flow versus pressure drop. Most manufacturers' equations for incompressible fluid (liquid) flow are correct.
However, it is amazing how many different compressible fluid (gas) flow formulas are used; and worse still, they do not all provide the same answers. Incorrect valve selection can lead to inadequate flow or noise when undersized, and excessive cost when oversized.
Correct sizing for control valves has been a problem since they were invented. Originally, valve manufacturers avoided the use of valve flow coefficients and mathematical formulas, by presenting graphs or nomographs for water, air and steam flow, for each valve size. This system was prevalent until mid1942, when the current valve flow coefficient C _{v} , was introduced. The coefficient C _{v} is used in computations for both compressible and incompressible flow.
During the 1950's and 1960's, there was widespread disagreement between valve manufacturers as to which compressible fluid flow equation should be used. Some users in the process industries began to realize the compressible fluid flow formulas then in use gave results that were in disagreement, and could lead to serious sizing errors. The cause of the problem was that valves with the same C _{v} rating and different shapes could have radically different gas flow characteristics. It became apparent that a single experimentally determined valve flow coefficient C _{v} was insufficient to describe both liquid and gas flow through valves over the full range of pressure drops.
The compressible fluid flow formula that became the ISA (Research Triangle Park, N.C.) standard appeared in an article by Les Driskell in Hydrocarbon Processing , Jul. '69, p.131 . This was followed by Driskell's article in ISA Transactions, Vol. 9, No. 4, '70, and his textbook Control Valve Selection and Sizing. Driskell recognized flow through valves was very similar to flow through thin, sharpedge flowmeter orifices. His work rested on a solid foundation of many years of research conducted on sharpedged orifices for precise flow measurement.
Pressure drop across an orifice or valve causes a reduction in density because the gas reacts to the pressure drop by expanding. This is not the case with liquids because density can not change significantly. Mass flow rate ( density x velocity x flow area ) does not change along a flow path in steady flow, so an expanding gas must accelerate to higher velocities to maintain the mass flow rate.
The orifice meter flow equation for compressible fluids is the same as for incompressible fluid flow, except for using the gas density at the inlet conditions, and correcting for the effects of compressibility by means of the Y 'expansion factor.' The expansion factor is the ratio of flow coefficient for a gas to that of a liquid.
For orifice meters, the expansion factor is given by an empirical formula. For valves, the expansion factor accounts for the change in density of a fluid as it passes from the valve inlet to the vena contracta and for the change in area of the vena contracta as the pressure drop is varied. The expansion factor formula for valves requires the experimentally determined critical pressure drop ratio factor x _{T} .
Existing valve sizing methods
Beginning about 1995, it became apparent more and more valve manufacturers were publishing simplified ISA compressible fluid formulas and not using the x _{T} coefficient. Today, most formulas in widespread use will not predict compressible fluid flow accurately for all types of valves over the full range of pressure drops.
Of three valve manufacturer's formulas reviewed, the first simplifies the ISA equation by assuming x _{T} = 0.50 for every valve. The second uses the 'downstream density' equationan equation reinvented many times since the early 19thcentury, which approximates the textbookderived equation for the flow of a compressible fluid through an ideal nozzle. Another manufacturer uses the 'mean density' equation, but truncates it at p _{2} / p _{1} = 0.528 (based on the critical pressure drop ratio of an ideal nozzle). All of these formulas have the potential to predict compressible fluid flows in excess of actual values.
ISA standard S75.01 Flow Equations for Sizing Control Valves provides excellent valve sizing equations for both compressible and incompressible fluids. When the equations are applied, reliable results over the full range of pressure drops are obtained, regardless of media or valve type.
The calculation example in the first illustration demonstrates the accuracy of the ISA compressible fluid formula to actual flow, and the significant error that can occur when using a simplified form of the ISA equation.
The ISA equation for compressible fluid pressure drop prediction versus a typical simplified equation. 
So why doesn't everyone use the ISA standard compressible fluid formula? First, the testing and data reduction required to obtain the experimentally determined capacity factors is substantially more complex than traditional procedures. Related higher costs dissuade manufacturers from obtaining this data. As a result, most valve manufacturers simply do not know their products critical pressure drop ratio x _{T} factors. Secondly, valve users do not realize how approximate traditional compressible fluid valve sizing methods are because suitable educational material is rarely published.
Determining C _{v} and x _{T} ISA standard S75.02 Control Valve Capacity Test Procedure provides two methods for determining the valve flow coefficient C _{v} and critical pressure drop ratio factor x _{T} . The first involves finding the maximum flow rate q _{max} (referred to as choked or sonic flow) of the valve, and the second, or alternative test procedure, obtains the information through linear curve fitting of test data.
With the first test procedure, C _{v} is determined at pressure drop ratios ( x = D p / p _{1} ) less than 0.02. A separate test to determine x _{T} requires the valve be tested at its maximum flow rate, which is defined by ISA as 'the flow rate at which, for a given upstream pressure, a decrease in downstream pressure will not produce an increase in flow rate.' This test requires a large volume of gas, and becomes a significant problem for larger high flow valves, such as ball and butterfly valves. Employing the ISA alternative test procedure eliminates this high flow test difficulty.
It has been determined the expansion factor Y is a linear function of the pressure drop ratio x . At a fixed stem travel C _{v} is constant, but the expansion factor Y changes with x . As x approaches zero (no expansion), Y approaches 1.0 in value. With the ISA alternate test procedure, both C _{v} and x _{T} are determined by testing the valve at a minimum of five widely spaced pressure differentials, measured at a constant upstream pressure. From these data, values of YC _{v} are calculated using the equation:
The test points are then plotted as YC _{v} versus x , and a linear curve is fitted to the data as shown. The value of C _{v} for the test specimen is taken from the curve at x = 0 and Y = 1. Since Y is linear with x , it can be shown the flow rate reaches a maximum when Y = 2/3. Therefore, the value of x _{T} for the test specimen is taken from the curve at YC _{v} = 0.66 7C _{v} . This method has the advantage of determining the critical pressure drop ratio factor without having to achieve choked flow.
This provides an example of the ISA Flow Coefficient alternative test method plot of expansion factor Y times flow coefficient Cv versus pressure drop ratio x. 
ISA Equation
The ISA equation for compressible fluid flow rate is a function of the pressure drop ratio x , the inlet pressure p , temperature T , flow coefficient Cv , critical pressure drop ratio factor x _{T} , gas specific gravity G _{g} , and expansion factor Y :
The expansion factor Y also includes the effect of the specific heat ratio F _{k} of the compressible fluid on the flow rate. For air, both F _{k} and G _{g} are equal to 1.0 at moderate temperatures and pressures.
The flow predicted by the above equation, assuming a constant inlet pressure and temperature, rises to a maximum until x = x _{T} at which point Y = 2/3. Flow conditions where the value of x exceeds x _{T} are known as choked flow. Choking occurs when the jet stream at the valve's vena contracta attains its maximum crosssectional area at sonic velocity. No further increase in flow rate can occur regardless of decreases in downstream pressures after this point.
When choked flow occurs, the above ISA equation is modified to:
Flows impact on x _{T} Valves with different flow paths have distinctly different critical pressure drop ratio factors. Valve geometries with complicated or circuitous flow paths tend to have numerically higher x _{T} values than valves with smooth, unobstructed flow paths. Therefore, two valves with identical flow coefficients C _{v} but with different x _{T} values will produce remarkably different flow rates.
For example, compare a full open ball valve with a C _{v} of 1.0 and an x _{T} of 0.14 with a needle globe valve with the same C _{v} but having an x _{T} of 0.84. Assume the upstream pressure is maintained at 100 psia (6.9 bar) and 70 °F (21 °C). Flow rates will be the same at low values of x . As x increases, the flows become quite different. The ball valve will reach maximum flow at x = 0.14, while the needle valve will reach maximum flow at x = 0.84. The needle valve will have a flow rate at choked flow of almost twice that of the ball valve!
Eliminating sources of variability in processes is more important than ever and key to variability elimination is ensuring valves are properly sized for the operational range of the application. A new era is dawning for compressible fluid valve selection; one where the need to guess or use obsolete flow capacity data is unnecessary. An era where accurate valve sizing will be virtually effortless.
Solving the testing problem 

Representative x _{T} examples 

