By William Rahmeyer,1 A. M. ASCE and Les Driskell2
(Reviewed by the Pipeline Division)
ABSTRACT: The flow coefficient for a control valve is an important hydraulic parameter that is used to select and design the controls for a closed conduit system. The parameter is used to determine flow capacities, valve positioning, and energy losses of the system. The coefficient is also used for the analysis of the flow system for other phenomena such as cavitation and transients. It is important to understand the definition of the flow coefficient, how it is used, and how it is determined for a control valve. At present there are two sets of testing standards to determine the flow coefficient for a valve. The standards are different in the definition of how the coefficient is determined and can result in two different sets of flow coefficient data for a valve. Confusion has resulted in identifying which set of standards was used to generate the valve coefficient information, and the importance of designing a system using the same definition of the coefficient.
INTRODUCTION
One of the main components for the design, regulation, and operation of a closed conduit system is the control valve, and one of the most useful sources of information about a valve is its flow coefficient. Complex procedures (2,6,8-10) have been developed for the selection and design of valves, and for considering the effects of different flow phenomena such as cavitation, torque, and transients on valves. The basic parameter that all design procedures mus t rely upon is the flow coefficient.
A popular and widely used form of the flow coefficient is the coefficient, Cv . It is the relation of the valve discharge in gallons per minute to the pressure drop across the valve in pounds differential per square inch. Each control valve has a unique numerical value of Cv that varies with the opening or position of the valve. It is very important to understand the definition of the flow coefficient, how it is determined in laboratory testing, and the units of the different flow variables that are used in calculating and applying the flow coefficient.
As simple as the flow coefficient appears as a mathematical parameter, there are problems in using the correct units of the flow variables, and in considering the flow coefficient for different sizes of geometrically similar valves. There is also a very serious problem in that there are two different testing standards and procedures for determining the flow coefficient in laboratory tests. This conflict can result in two different sets of values of Cv for the same valve which may vary as much as 50%. The purpose of this paper is to present cautions for using CB, to em-
'Asst. Research Prof., Dept. of Civ. Engrg., Colorado State Univ., Fort Collins, Colo.
Consulting Engr., Pittsburgh, Pa. Note.—Discussion open until December 1, 1985. To extend the closing date
phasize the conflict in testing procedures, and recommend standard definitions.
BACKGROUND
The flow coefficient, C„, is a mathematical expression, Eq. 1, used to relate the flow through a valve to the pressure drop or energy loss caused by the valve. It can be derived directly from the theorem
/ G \ V 2
C " = Q U) (1)
of the conservation of energy into its present form which is widely used (10,14). The flow, Q, is in gallons per minute; Gf = the specific gravity of the liquid; and dP is in psid (pounds per square inch differential). However, the coefficient is not dimensionless, and it is not independent of the size of the system or valve. Cv is a function of the opening of the valve, the flow rate in gallons per minute, and the energy loss caused by the valve to the piping system. The dP term is not the pressure drop across the valve, a definition which is often times mistakenly used. It is actually the local energy loss associated with the addition of the valve to a closed conduit system. The pressure differential across a valve includes the effect of changing kinetic energy in the nonuniform flow immediately downstream of the valve. A pressure differential can exist where the change in pressure is due solely to a change in pipe velocity without any loss of energy.
Therefore, dP is the loss of energy due to the valve, and is measured in units of psid from pressure taps located upstream and downstream of the valve in regions of uniform, fully developed pipe flow where the kinetic energy or the average pipe velocity are the same. The energy loss (4) refers only to the effect or local loss of the valve and not to the additional friction loss of piping between the valve and pressure taps.
In laboratory conditions, the flow coefficient of a valve is determined from a measured flow rate and a calculated local head loss. The local head loss due to the valve cannot be measured directly due to the nonuniform flow immediately downstream of the valve. It is calculated from a pressure differential measured upstream and well downstream of the valve in regions of constant velocity and uniform pipe flow. Since the measured pressure differential (4) then includes both the local head loss and friction loss due to the length of pipe between pressure taps, the friction loss must be subtracted from the measured pressure differential. Actually the friction loss is not used, but the energy loss that exists between the pressure taps without the installation of the valve is used. That way the local loss calculated by subtracting the friction or pipe energy loss is the head loss due only to the effect or addition of the valve. The Instrument Society of America testing standards, ISA S39.01 and ISA S39.02 (7,8), describe the foregoing testing procedures and definition.
The recent standards, ISA S75.01 and S75.02 (5,6), include the friction of the pipe between the pressure taps in determining the pressure drop or local energy loss of the control valve. The difference between ISA
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S39.02 and ISA S75.2 standards has created problems in other standards and testing procedures (3,11) which refer to and include both ISA standards. The difference between the two definitions of testing procedures can result in confusion and the existence of two different sets of test data for a control valve. Control valves are being tested and evaluated by one set of standards, and control systems incorporating the valves are being evaluated and designed from another set of standards. This paper will document the difference in C„ values.
Another discrepancy in test procedures for determining the flow coefficient is the recommended locations of the downstream and upstream pressure taps of the valve. Both ISA S39.2 and ISA S75.02 recommend taps at equivalent pipe lengths of 2 diam upstream and 6 pipe diam downstream of the test valve. These recommendations are based on the assumption that the nonuniform flow and velocities immediately downstream of the valve will become uniform or the pressure will recover within the 6 diam distance. It is possible that valves with small pressure losses and high discharges will create flow disturbances extending downstream past the pressure tap locations. Testing at Colorado State University with large, low loss valves confirms this discrepancy.
MATHEMATICS
The flow coefficient, Cv, can also be derived from the Euler Number (15). The dimensionless Euler number is formulated from dimensional analysis to relate the ratio of inertial forces to the forces from a pressure differential. It is used to describe the distribution of velocity and pressure associated with a flow disturbance such as a valve where a local pressure loss or energy loss occurs.
The variables of the Euler number (Eq. 2) are density, pressure differential, and flow velocity. The variable of
dP
V
E = —> (2)
2
viscosity of the fluid is not considered in the Euler number or C„ . Therefore, for viscous flows where the viscosity affects the velocity and pressure distribution, C„ will not be constant. Both the Euler number and C„ will become constant at flows with large Reynolds numbers, Eq. 3, where the effect of viscosity
VD R = v • (3)
is minor compared to the influence of the inertial forces. Typical pipe flows above a pipe Reynolds number of 100,000 will not be influenced by viscosity and have constant E and C„.
As mentioned, C„ is a relationship of the inertia force represented by the flow, Q, and the pressure differential in psid. It can be related to the Euler number by Eq. 4.
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890.603 , c . 4 (4)
in which D = diameter in inches; and Gf = specific gravity. The Euler number is dimensionless, but C„ is not dimensionless with units of gal in./min lb1/2. Also the Euler number is independent of the size of the valve, i.e., it is constant for different sizes of geometrically identical sizes of valves. By dividing C„ by the diameter squared to form another flow coefficient, the coefficient, Cv/D2, is relatively independent of size. The use of this coefficient is preferred over Q, because it avoids the confusion and need to modify C„ for each size of similar valve.
Because there are two different definitions of the testing procedures to determine C„, the coefficient can be derived from the Conservation of Energy or Bernoulli equation in two different forms. C„39 per ISA S39.2 Standards, is defined and derived as the local energy loss exclusive of the friction loss of the test piping. Cv75, per ISA S75.02, is defined and derived as the combination of the local loss of the valve and the friction loss of the test piping. Eqs. 5 and 6 show the relationship of the two derivations of C„ from the energy equation.
2 -1-1/2 0,39
D2
O75
D2
0,75
Q 3 9
D2
- 0.008986 Gff
- 2
+ 0.008986 Gff -1/2
(5)
(6)
in which / = the friction factor of the pipe per the Darcy Wiesbach equation for friction loss.
Eqs. 5 and 6 show that the relationships are dependent upon the variables of friction factor, and specific weight of the fluid, and relatively independent of size. Also, by comparing the two equations it can be shown that O39 will always be greater than O75 • The difference between the coefficients will increase with an increase in friction factor and specific weight. Applying the theoretical equations for a standard steel pipe and water as the flow medium, predicts a possible difference of 30-50% between the two coefficients.
RESULTS
Testing at Colorado State University has included evaluating the hydraulic performance of most types of control valves in sizes from 1 in.-36 in. Each valve tested has shown that there is a difference in C„ values, and 0 3 9 is always greater than Q 7 5 . For example, a 6-in. gate valve at full open and a pipe friction factor of 0.0161 had a Cv39/D
2 of 82.5. This value was almost 42% larger than the Cv75 of 56.78 determined in the same tests.
Data for tests of a symmetrical 6-in. butterfly valve are shown in Fig. 1 as a plot of Cv/D
2 versus valve opening. The friction factor for the 6 in. pipe was measured as 0.018. The difference between Cv39 and Cv75 at full open was about 26%. The results from testing both the gate valve
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1 1 1 1 1 1 1 r—
MEASURED C v „ I MEASURED Cv,,, A~ THEORETICAL Cv38 , EQUATION 5 /
FIG. 1.—Relationship of Flow Coefficients and Valve Opening
and butterfly show that derivation from theory and the corresponding Eqs. 5 and 6 are valid and accurately predict the difference between the two C„ values.
Another result that the tests of the two valves show is that there is a 10% difference between Cv39/D
2 and Cv75/D2 at a value of 40. The results
also show that the difference in the reported C„ depends on the friction factor of the actual pipe used in the test. The discrepancy becomes greater with increasing values of pipe friction factor. The discrepancy also increases as the size of the specimen decreases, because the friction factor becomes larger with the lower Reynolds numbers that result when smaller valves are tested at the prescribed pressure differentials, and with the greater relative roughness of the pipe wall.
Fig. 2 shows the relationship of valve size and the percent difference between the coefficients, Cv39/D
2 and Cv75/D2. Relationships for constant
values of Cv39/D2 were plotted for a constant pressure drop of 5 psid
and a constant pipe wall roughness of 0.0002 ft. The friction factors of the pipe varied with the different Reynolds numbers due to the change in valve size with a constant pressure drop.
Another objective that was considered in the tests of the butterfly and gate valve was the location of the pressure taps used to measure dP. The latest ISA standard requires that the taps shall be located 2 nominal pipe diam upstream and 6 pipe diam downstream of the test valve. The actual tests used pressure taps at 1 and 2 diam upstream, and at 6, 8, and 10 diam downstream. Test results show that only a minimum of 1 diam upstream, and a minimum of at least 8-10 diam downstream were needed. Pressure differentials measured at 6 diam downstream were found to be 4%-8% less than the recovered pressure should have been.
Considerable testing has been done at Colorado State University on
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60.0
50.0
40.0 Q
> o 30.0
20 .0 -
10.0
r\ n
O a A
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80.0 CONSTANT 5 PS ID CONSTANT e = 0.0002 ft
0 0 l 1 1 1 1 1 1 1 1 0 5 10 15 20 25 30 35 40
DIAMETER (inches)
FIG. 2.—Relationship of Pipe Diameter and Percent Difference of Flow Coefficients
many different types of valves ranging in size from 1 in.-36 in. It has been found that tap locations at 1 diam upstream and 10 diam downstream were adequate. The error produced using taps at 6 diam was found to increase with an increase in the C„ of the valve and a decrease in the friction factor of the pipe.
CONCLUSIONS
Figs. 1 and 2 clearly show the difference between the flow coefficient, Q3 9 , defined by ISA Standards S39.2, and the coefficient, Cv75, defined by ISA Standards S75.02. Difference of 20%-40% are possible even with the large-size valves tested. Fig. 2 indicates that the discrepancy will exceed 40% for valves less than the 10-in. size when Cv3g/D
2 is 80 or greater. It is very important to recognize that the values of the two coefficients can be different, and to be able to identify by which standards the data base of CB for a control valve was generated.
In the design of a closed conduit system, most engineers use the modified Bernoulli equation to calculate the head losses of the system and the local losses of the individual components. The local loss calculated for a control valve is usually the head loss due only to the valve and does not include any pipe friction loss. For this reason, it is necessary to use Q39 for calculating head losses and flow rates. The use of C„75 could produce calculations that would be drastically different. Eqs. 5 and 6 presented in this paper can be used to convert C„ data from one definition to another. Tests with many different types and sizes of valves verify the equations.
Another important consideration for the difference in definitions of C„ is for determining the opening or position of a control valve. It is pos-
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sible to calculate differences in valve position as large as 20%. Following design procedures for determining cavitation (13,14), torque (1), maxim u m possible flow (9), and transients (16) can result in errors and differences as large as 150-200%.
Therefore, it is recommended that the testing of control valves should carefully consider the effect of pipe friction between the test pressure taps; and pipe friction should be accounted for in the testing of control valves that have Cv/D
2 values greater than 20. Also, it is recommended that tests for the flow coefficient should use pressure taps located at a minimum of 1 diam upstream and at 10 diam downstream of the valve.
APPENDIX.—REFERENCES
1. American Water Works Association, "Method for Calculating Torque Required to Operate Butterfly Valves," Appendix A, ANSI/AWWA C504-80 Standards, American Water Works Assoc, Denver, Colo., Jan., 1980, pp. 13-16.
2. Driskell, L. R., Control Valve Selection and Sizing, Instrument Society of America, Research Triangle Park, N.C., 1983.
3. Fluid Controls Institute, "American National Standards for Control Valve Seat Leakage," ANSI/FCT 70-2-1976, 1982.
4. Harthun, F. P., "Valve Characteristics for Pipeline Applications," ASME, Paper No. 69-PET-23, Sept., 1969.
5. Instrument Society of America, "Control Valve Sizing Equations," ANSI/ISA S75.01 Standard, Research Triangle Park, N.C., 1981.
6. Instrument Society of America, "Control Valve Capacity Test Procedures," ISA-S75.02 Standards, Research Triangle Park, N.C., 1981.
7. Instrument Society of America, "Control Valves Sizing Equations for Incompressible Flow," ISA-S39.1 Standards, Research Triangle Park, N.C., 1972.
8. Instrument Society of America, "Control Valve Capacity Test Procedures for Incompressible Flow," ISA-S39.2 Standards, Research Triangle Park, N.C., 1972.
9. Instrument Society of America, "Handbook of Control Valves," J. W. Hutchison, ed., 2nd Ed., Research Triangle Park, N.C., 1976.
10. Lyon, J. L., "Lyon's Valve Designer's Handbook," Van Nostrand Reinhold, New York, N.Y., 1982, pp. 667-698.
11. Military Specifications, "Valve, Butterfly, Wafer and Lug Style," MIL-V-24624, Nov., 1983.
12. Moore, R. W., "Allocating Pressure Drop to Control Valves," Instrument Technology, Oct., 1970, pp. 102-105.
13. Rahmeyer, W. J., "Cavitation Testing of Control Valves," Instrument Society of America, Paper No, C.1.83-R931, Oct., 1983.
14. Rahmeyer, W. J., "Cavitation Limits for Valves," American Water Works Association Journal, Nov., 1981.
15. Rouse, H., "Elementary Mechanics of Fluids," Dover Publishing, New York, N.Y., 1978, pp. 61-63.
16. Wylie, E. B„ and Streeter, V. L., "Fluid Transients," McGraw-Hill Publishing Co., New York, N.Y., 1978, pp. 37-70.
