What is the relationship between the Flow Coefficient and the Discharge Coefficient?

In our previous article, the difference between the Resistance Coefficient (K) and Flow Coefficient (Cv) was evaluated and a relationship between the two was derived. The Flow Coefficient (Cv in US units, Kv in SI units) is typically associated with the hydraulic performance of a control valve, but other devices such as safety relief valves are characterized by the Discharge Coefficient (Cd, sometimes designated by Kd), which is also associated with orifices and nozzles. They are not numerically equivalent, so what is the relationship between the two?

There are various standards in the U.S. and internationally that are used to size and select control valves and relief valves, most notably the ANSI/ISA-75.01.01 Flow Equations for Sizing Control Valves (IEC 60534-2-1 equivalent) and the API Standard 520 Part 1, Sizing, Selection, and Installation of Pressure-relieving Devices in Refineries. These two standards can be used to derive the relationship between the Flow Coefficient (Cv) and the Discharge Coefficient (Cd) for relief valves. There are minor differences in the nomenclature used in each standard, so for the purpose of this article, the nomenclature will be defined for the equations below along with the engineering units being used.

Good communication between various groups involved with fluid piping systems is critical for the proper design, operation, and determination of cost for many piping systems in residential, commercial, and industrial applications. It is crucial that the engineer understand and apply equations correctly to prevent costly mistakes in the sizing and selection of equipment, operating within safety limits, and avoiding unnecessary modifications later in plant life. One potential area for costly miscommunication is the use of coefficients for devices that have a fluid flowing through them. Manufacturers of various equipment use different coefficients to characterize the hydraulic performance of their devices and these differences must be understood when applying them to calculations involving piping systems.