ULTRAPURE MATERIALS — GASES

Establishing a new method for determining valve flow coefficient

William L. Stubbs, Parker Hannifin UHP Products

The most frequently heard question in the engineering departments of valve manufacturers is, "How much will it flow?" Although details such as process gas, inlet pressure, inlet temperature, and pressure drop are often not included in the question, an acceptable flow rate and/or pressure drop prediction must be provided. As O. P. Lovett observed in 1974, "Correct sizing for control valves has been a problem since they were invented."¹ Since control valves were first developed to control fluid transport, proper sizing and flow prediction have been necessary to select the correct valve size and type for a particular application. Incorrect selection can lead to inadequate flow when the valve is undersized, and excessive cost when it is oversized. Many flow prediction methods have been developed over the years, from complicated, theoretically derived equations to simple, one-line plug-and-chug equations.

In the early days of valve manufacturing, fabricators generally avoided flow capacity coefficients and mathematical formulas by presenting graphs or nomographs for liquid and gas flow for various valve sizes. It was then the customer's responsibility to select the chart that best fit the application and size the valve accordingly. This practice was prevalent until mid-1942, when the current valve sizing coefficient, C^_v, was introduced by Ralph A. Rockwell of Maoneilan/Dresser.¹ Even before Rockwell, however, around the middle of the nineteenth century Julius Wiesbach popularized what is now called the "K factor method" by publishing formulas and experimentally determined loss factors for gate valves, stopcocks, butterfly valves, and many other systems components.

There has always been widespread disagreement among valve manufacturers regarding which flow equation should be used. During the 1950s, some users in the process industries began to realize that the gas flow formulas then in use gave incongruous results that could lead to serious sizing errors. The cause of the problem was that valves that had the same C^_v rating but were different shapes could have radically different choked gas flow characteristics. It became apparent that a single experimentally determined coefficient was insufficient to describe gas flow through valves over the full range of pressure drops.

In 1963, H. D. Baumann introduced the experimentally determined critical flow coefficient C^_f to solve this problem. Gas flow was predicted using the standard C^_v, with the gas density determined at the mean pressure. The flow was then limited to a maximum value that was predicted using C^_f. Baumann included experimental data to substantiate his method. J. F. Buresch and C. B. Schuder published a more elegant solution in 1963 in ISA Transactions. They invented a new flow equation that preserved the standard C^_v and introduced a second coefficient, C^_g, which is determined from choked flow of air. Experimental data were included to demonstrate their success at correlating test data with a wide range of valve styles. Their equation was more accurate than Baumann's, and just as easy to use.

The most important work by far was published in 1969 by Les Driskell, who developed a gas formula that led to an Instrument Society of America (ISA) standard that we will discuss in detail later.² In 1971, however, F. E. Sanville invented another empirical gas flow equation, using two experimentally determined coefficients, which he proposed for the periodical Pneumatic Fluid Power Valves. He made no mention of the flow equations previously developed for sizing process control valves, nor did he mention the fact that others before him had used the same equation. His method gives numerical results similar to equations developed by Buresch and Schuder and ISA. It meets the ISO 6358 (1989) requirement for sizing pneumatic fluid power valves; however, the method of presentation is difficult to understand, and the formula is limited to air.

When the modern semiconductor industry emerged in the mid-1970s, high-purity valves were not yet designed, and standard commercial valves from the cryogenic and instrumentation industries were pressed into service. Initially, newly developed high-purity cleaning processes were used to modify these valves. As contamination issues became more prevalent, new valves and components were designed to address specific ultra-high-purity (UHP) issues such as particle entrapment and chemical outgassing.

Eventually, new valves were developed specifically for the UHP industry. These can support submicron semiconductor technology by using specialized materials, design features, and cleaning techniques that are suitable for the various requirements inherent to low-contamination gas distribution systems. Unfortunately, this level of UHP technology has not been applied to the methods used to size valves used in semiconductor fabs.

Valve Sizing Methods

Numerous methods are currently used to size control valves. As mentioned earlier, the need for correct control valve sizing is obvious: when valves are too small, an inadequate supply of gas or liquid is available; when valves are too large, costs increase without any return on investment. Valve sizing can be broken down into two areas for computational purposes: compressible and incompressible fluids. This article examines valve sizing for compressible fluids. It is assumed that the same flow coefficient, C^_v, can be used for incompressible fluids with little difficulty. The most common methods used to size valves in the semiconductor industry are those that have been developed for the instrumentation industry. These techniques are relatively simple to apply, but can lead to significant sizing errors when computing flow rates or pressure drop.

In a typical application, flow rate through a lateral line is estimated and the pressure drop across several valves and line segments is calculated to determine the desired inlet pressure to the tool. Tool inlet pressure is the most critical factor when determining the time necessary to fill or purge a particular tool volume. The most common method to determine the pressure drop across a valve is to obtain the flow coefficient C^_v from the manufacturer; apply it to the appropriate formula based on flow rate, inlet pressure, and inlet temperature; and then compute pressure drop.

Although various formulas have been published over the years, no standardized procedures or formulas have been set to determine the flow coefficient, pressure drop, or flow rate. Sematech published the earliest semiconductor standards, the venerable Semaspecs.³ Four different methods for calculating flow coefficients are provided here.

Method One

Equation one is published by Sematech as Semaspec 90120394B.³ The flow coefficient C_^v is a constant that is determined from observed flow rate, inlet pressure and temperature, pressure drop, and gas-specific heat.

This method is based on the ISA S75.02 Control Valve Capacity Test Procedure; however, the expansion factor Y, as specified by the ISA procedure, is not used in the Sematech equation. Like the ISA procedure, the Sematech procedure calls for the flow coefficient (C^_v) test to be performed at extremely low pressure ratios, at which the gas can be considered to be incompressible. The formula, however, does not accommodate the change in gas density due to expansion through the valve. Furthermore, the procedure does not specify or limit the test component connection to the test system, which allows the tester to perform "tare" subtractions of the test system tubing from the observed pressure drop measurements. In addition, an erroneous statement is made in the Semaspec document's scope section, which reads, "Since a water-determined flow coefficient can differ from a gas-determined coefficient, the C^_v should be determined using a gas-based test procedure for components utilized in gas service." While this statement offers good advice, it is simply not true. This article will later discuss how the flow coefficient determined from a liquid test will be the same as the flow coefficient determined from a gas test. The overall effect of these simplifications is to determine a single flow coefficient (C^_v) that is higher than the actual value, and to compute flow rates where the error in flow rate prediction increases nonlinearly with increasing pressure drop.

Method Two

Method two is published by a UHP valve manufacturer and is based on the ISA equation. The premise is that if the value of C^_v is known, the flow rate can be predicted under all conditions by using a constant density equation multiplied by an empirical correction factor Y, which varies linearly with the pressure drop ratio x. To simplify the equation, method two assumes the choked flow pressure ratio x^_T = 1/2 for all valves. The flow coefficient C^_v is a constant that is determined from observed flow rate, inlet pressure and temperature, pressure drop, and gas-specific heat. The following equation can be used for normal flow with Mach numbers below choked flow:

Although this equation is easy to use, the assumption of a constant choked flow pressure ratio for various valve geometries causes the accuracy of the flow equation to vary with valve geometry. This is because valves with different flow paths have distinctly different choked flow pressure ratio coefficients. As an example, a ball-type valve has a very low x^_T, well below 1/2, whereas a globe-style valve would have an x^_T significantly higher than 1/2. As shown in Figure 1, valve geometry determines the choked flow pressure ratio. Valves with complicated or circuitous flow paths tend to have numerically high x^_T values; valves with smooth, unobstructed flow paths have very low x^_T values.

Figure 1: Valve geometries and their choked flow coefficients.

To predict choked flow, method two uses another simplification by equating the ISA expansion factor Y to 1/2:

This assumption violates the ISA equation for expansion factor, which empirically found the value of Y never to fall below 2/3; therefore, this equation typically predicts choked flows well in excess of actual values.

Method Three

Method three is also published by a manufacturer of UHP valves, and uses what is known as the mean density equation. The flow coefficient C^_v is a constant that is determined from observed flow rate, inlet pressure and temperature, pressure drop, and gas-specific heat. For normal flow with Mach numbers below choked flow:

For choked flow prediction, method three is truncated at P^₂/P^₁ = 0.528, based on the critical pressure ratio of an ideal nozzle:

For an "ideal nozzle," the maximum flow for a given inlet pressure always occurs when sonic velocity is reached at the throat, where the outlet absolute pressure is 0.528 times the inlet absolute pressure. This is correct for all well-radiused orifices (i.e., ideal nozzles), but incorrect for orifices in general and for valves. This approximation is based on similarities between nozzles (a well-radiused orifice), orifices, and valves. Although many engineers expect valves to behave as nozzles, they do not. Method three has the disadvantage of an increasingly higher error at large values of pressure ratio x.

Method Four

The final method examined here is the Crane TP410. Originally written in the 1930s, Crane TP410 is a compendium of compressible and incompressible fluid properties and flow formulas. For valves and fittings, Crane Co. carefully measured resistance to flow and published a series of K factors and a nomograph to compute C^_v based on pipe diameter using the equation:

Theoretically, the resistance coefficient K is a constant for any given valve geometry, and the flow coefficient C^_v is proportional to the K factor and valve area. Crane's formula for flow rate incorporates an expansion factor Y which accounts for the change in gas density due to expansion of the gas downstream of the vena contracta, or minimum flow area. This approach works well except that the graph for expansion factor Y is a function of the K factor, which for valves must be computed from C^_v data. In practice, the valve expansion factor should be a straight-line function of valve geometry only, not valve size. This simplified approach tends to underestimate actual flow through the valve, and although somewhat conservative, Crane TP410 does an adequate job of predicting choked flow through valves.

Crane published an entire set of flow equations for pressure drop through pipes based on Darcy's formula. These equations work for compressible and incompressible flow through pipes and tubes. For compressible flow, Crane empirically derived an expansion factor Y from measured data which accounts for the change in gas density due to wall friction or Reynolds number loss.

The expansion factor is critical because it enables the Crane formulas to predict the choked flow rate and increases the flow rate prediction as pressure drop increases. For subsonic gas flow, the flow coefficient C^_v is a constant that is determined from an empirically determined K factor:

Comparison of Existing Methods

When the four previously described methods are plotted against actual test data, the results are predictable. Methods one, two, and three predict flow rates that are higher than the actual values for a given pressure drop. Method one continues to predict increasing flow rate with increasing pressure drop, without regard to the changes in gas density that will be caused by expansion, which is obviously an unrealistic condition. Methods two and three both predict choked flow, but the computed values are significantly higher than the actual values. Finally, Crane TP410 is the only method here that uses an expansion factor; however, the predicted flow rates are lower than the actual values, leading to selection of an oversized valve. In Figure 2, these four methods are plotted for purposes of comparison. As the figure demonstrates, methods one and three would undersize the valve for all values of desired pressure drop; method two initially undersizes the valve, and method four consistently leads to an oversized valve.

Figure 2: Comparison of four existing flow rate prediction methods versus actual test data for a 1/2-in. diaphragm valve for air at a 30-psig inlet pressure.

ISA Standard

As mentioned earlier, the gas flow formula that became the ISA standard made its debut in an article by Driskell in Hydrocarbon Processing in July 1969.⁴ This was followed by Driskell's 1970 article in ISA Transactions, and by his 1983 textbook Control Valve Sizing Equations for Compressible Fluids. These works contain the background material necessary to understand the standard. Driskell's contribution was to recognize that the flow through valves was very similar to the flow through thin, sharp-edge flowmeter orifices, and to build his gas flow equation on the existing body of research. There was no need at all to invent another empirical equation.

The thin, sharp-edge orifice meter was introduced in 1910 by J. L. Hodgson of George Kent, Ltd., in England. In 1917 and 1922 papers, he introduced the expansion factor and the present method of computing gas flow through such meters and correlated it with test data. The presently accepted correction for specific heat ratios different from air was published by E. S. Smith in 1929, and further documented by E. Buckingham of the Bureau of Standards in papers published in 1931 and 1932. The accuracy of the flowmetering standards is of great commercial importance and has been carefully investigated. The standards are used for custody transfer of natural gas and other fluids, and even written into contracts. The orifice meter flow equation for gas is the same as for liquid flow, except for using the gas density at the inlet conditions, and correcting for the effects of compressibility by means of the expansion factor Y. The expansion factor is a straight-line function of the pressure drop ratio. The slope of this line is correlated by an empirical equation to the orifice—to—pipe diameter ratio, pressure tap location, and specific heat ratio. The relationship is very well documented with precisely taken experimental data. All modern orifice flowmetering standards are in agreement on the procedure, including ISO 5167, ASME MFC-3M (1989), and the API Manual of Petroleum Measurement Standards (1990).

For valves, the terms involving orifice area, discharge coefficient, and velocity of approach correction factor are grouped together and replaced by the equivalent C^_v value. The Reynolds number correction is assumed negligible for valves flowing gas. The expansion factor formula requires an experimentally determined coefficient, which for orifice meters is given by an empirical formula, but for valves is determined experimentally and called x^_T.

ISA Test Procedure

Since the ISA test method for liquid is similar to the test method for gas, this article only examines the gas method. There are two procedures. The first involves finding the maximum flow rate directly, and the second, which is the alternative test procedure, finds the choked flow pressure ratio through extrapolation of test data.

In the first procedure, flow is measured at a low pressure ratio (<0.02), which is assumed to approach incompressible flow. A separate test is required to determine the maximum flow rate, which is defined 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 is difficult to perform because of the large volume of gas required and is nearly impossible for high-flow valves, such as ball valves, because of the choking effect of the test apparatus relative to the valve itself.

This high-flow difficulty is eliminated from the alternative test method. This procedure requires a minimum of five widely spaced pressure differentials, measured at a constant upstream pressure. From these data points, values of YC^_v are calculated using the equation:

The test points are plotted on linear coordinates as YC^_v versus x and a linear curve fitted to the data. The value of C^_v for the test specimen is taken from the curve at x = 0, Y = 1 (the y intercept of the linear regression curve fit). The value of x^_T for the test specimen is taken from the curve at YC^_v = 0.667_*C^_v(0). This method has the advantage of determining the choked flow pressure ratio without having to actually attain the choked flow rate. An interesting caveat regarding the ISA procedure should be noted here. The choked flow pressure drop ratio, x^_T, is not the pressure ratio at which the choked flow rate occurs, but the choked pressure ratio value that yields the actual choked flow rate in the ISA equation. Although this seems somewhat confusing, it is important to note the value of x^_T may differ from the observed pressure ratio when the choked flow rate is achieved. Figure 3 shows a typical YC^_v versus x^_T plot for a 1-in., full-flow valve. Extensive tests conducted by ISA prove that when plotting a linear regression curve of the observed data points, the choked flow pressure ratio is the value of x when the product YC^_v = 0.667.

Figure 3: The ISA flow coefficient alternative test method plots expansion factor times flow coefficient YC^_v versus pressure drop ratio.

Flow Rate Prediction Using ISA Method

The most common method of sizing valves is to predict flow rate and compare the calculated flow rate to the requirement for the particular application. In this regard, the ISA method of flow rate prediction provides excellent correlation with actual performance, when using the ISA-generated flow coefficients, C^_v and x^_T. The ISA equation for flow rate is the function of pressure ratio, inlet pressure and temperature, published flow coefficient C^_v, published choked flow pressure ratio x^_T, and gas-specific heat and expansion factor Y, which itself is a function of pressure drop and published choked flow pressure ratio x^_T:

In Figure 4, a plot is generated to predict flow rate based on the flow coefficients calculated by the ISA method and compared to the actual flow rates. This and many other examples demonstrate that flow rates can be predicted with excellent results. The ISA method has proved to be extremely accurate when comparing predicted flow rate to the observed data points.

Figure 4: ISA flow rate prediction for a 2-in., full-flow valve for air at a 60-psig inlet pressure.

Pressure Drop Prediction

Another goal of using any flow coefficient calculation is to predict pressure drop. Predicting pressure drop is more difficult than predicting flow rate because of the iterative nature of the solution. Fortunately, modern computer programs make this once difficult calculation simple. The ISA equation for pressure drop is a function of observed flow rate, inlet pressure and temperature, pressure drop, published flow coefficient, gas-specific heat, and expansion factor, which itself is a function of pressure drop and published choked flow coefficient:

This method of pressure drop prediction using the ISA-derived flow coefficients, C^_v and x^_T, has proven to be very accurate for a wide range of gas properties, inlet pressure conditions, and valve geometries, as shown in Figure 5. The ISA equation for pressure drop prediction is an iterative solution; however, it is just as accurate as the ISA flow rate prediction method when compared to observed data.

Figure 5: ISA pressure drop prediction for a 1/2-in. diaphragm valve for air at a 60-psig inlet pressure.

The New SEMI Standard

The SEMI standards committee decided in 1995 to form a task force to find the best method for UHP valve manufacturers and users to determine flow coefficients for gas shutoff valves. The task force included most of the current UHP valve manufacturers and several valve end-users. The group considered both current and out-of-date flow coefficient test methods, including the ISA method. It was determined that the ISA method, if simplified, would meet the needs of both manufacturers and users. The SEMI standard (document No. 2755) Test Method for Determination of Flow Coefficient for High-Purity Shutoff Valves is based on the ISA standard but has been modified to accommodate the unique features of UHP valves.⁵ The document is written in two parts. The first describes the flow coefficient test method; the second describes how to predict flow rate and pressure drop based on supplied flow coefficients.

An example of how the SEMI standard simplified the ISA method is best illustrated by the SEMI definition of test specimen. The ISA standard, which differentiates between the connection flow coefficient and valve flow coefficient, requires the determination of separate coefficients for end connections and fittings. In contrast, the SEMI standard considers the unique tube and fitting end connections of the UHP valve to be part of the test specimen (see Figure 6). This greatly simplifies the process of testing by including end connections in determination of both flow coefficients, C^_v and x^_T, and the formulas for predicting flow rate and pressure drop. In addition, the document provides formulas in both metric and standard versions.

Figure 6: (a) Test valve with face seal connections; (b) test valve with tube stub connections.

Before the task force approved the final version, each member participated in a round-robin test using the new standard to determine the flow coefficients of a special test specimen developed by the group. Results of this validation testing proved that the new SEMI standard works very well with UHP valves, and can be used to predict flow rates and pressure drops with far greater accuracy than previous methods. Figure 7 uses this new method to predict flow rates of a large, full-flow valve. Excellent correlation as shown in the figure is important when sizing large, expensive bulk gas distribution valves. When compared to actual or measured flow rates, this method allows selection of expensive valves to be determined with good accuracy.

Figure 7: Comparison of flow rate predicted by the SEMI flow coefficient test method versus observed test data for a large, 6-in. full-flow valve with air at a 60-psig inlet pressure.

Conclusion

The most important effect of the SEMI standard is that it will cause valve manufacturers to provide end-users with flow coefficients determined by the new method. Certainly this will be a paradigm shift for both valve users and producers because, in many cases, the C^_v values that have been published by the manufacturer will change when valves are tested by the new SEMI method. Most important, however, is the requirement that both flow coefficients, C^_v and x^_T, must be provided by the valve manufacturer, in contrast to the standard industry practice of publishing only C^_v data. There is at least one UHP valve manufacturer that is currently publishing both C^_v and x^_T data for its entire product line. As the pressure increases to reduce the cost of semiconductor fab ownership, these data will become increasingly important to valve users attempting to downsize lines, valves, and components by computing flow and pressure requirements more accurately using the new methods in the SEMI standard. These methods are very accurate for predicting flow rates and pressure drops when the manufacturer follows the test method and provides the proper flow coefficients to the end-user. It is up to the end-user to require that valve suppliers provide the necessary test data for the products they specify.

References

1. Lovett OP, "History of Control Valve Sizing," in Flow, Its Measurement and Control in Science and Industry, Triangle Park, NC, Instrument Society of America, vol. 1, p1053, 1974.

2. Kenyon RL, "Calculating Flow through Orifice Flowmeter," internal report, Irvine, CA, Parker Hannifin, 1991.

3. Semaspec 90120394B, Test Method for Determination of Valve Flow Coefficient for Gas Distribution Systems Components, Austin, TX, Sematech, 1993.

4. Kenyon RL, "Using the ISA Gas Flow Equation," internal report, Irvine, CA, Parker Hannifin, 1997.

5. Test Method for Determination of Flow Coefficient for High-Purity Shutoff Valves, Mountain View, CA, SEMI, to be published in 1998.

William L. Stubbs is the chief engineer for Parker Hannifin UHP products division (San Luis Obispo, CA), where he has worked for three years. He is a registered professional engineer with more than 17 years' experience as a design engineer and 10 years' experience in engineering management. He holds a BS in mechanical engineering from California Polytechnic State University (San Luis Obispo, CA) and an MS in software engineering from National University (Irvine, CA). He is the leader of a SEMI committee task force and a member of ASME, ISA, and the Cal Poly Industrial Advisory Council. (Stubbs can be reached at 805/786-1159 or bill_stubbs@saes-group.com.)

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