Orifice Flow Calculation Basis
Inglenook received an e-mail the other day from a long-time client asking what references we used when specifying restriction orifices for several emergency blowdown systems we had designed a few years earlier. The primary concern was making sure that our calculations would match up with the vendor calculations, especially with respect to the orifice flow discharge coefficient which can have a significant impact on the end results.
This is one of those questions that seems like it should be easy and straightforward, but like most things in life, it depends…
- It depends on the type of fluid, is it compressible, incompressible or two phase?
- It depends on the pressure drop across the orifice, is critical (choked) flow established?
- And lastly, it depends on how much engineering effort is warranted for the calculation?
It depends on so many parameters because flow through an orifice behaves differently or, at the very least, the models are simplified differently depending on the above conditions.
Design equations for sub-critical, or unchoked, compressible and incompressible flow are well established in industry references such as Perry’s Chemical Engineers Handbook, ASME PTC 19.5 (Flow Measurement) and ISO 5167-2 (Measurement of Fluid Flow by Means of Pressure Differential Devices Inserted in Circular Cross-section conduits running full: Part 2 Orifice Plates). These references also include formulas for calculating the orifice’s discharge coefficient, usually as a function of the beta ratio and Reynolds number, but vendor specific data is always recommended, if available.
Design equations for critical flow in compressible fluids are where things start to get interesting and where significantly less work has been completed. The reason for this is the bulk of the restriction orifices in plants are used for the measurement of flow where low pressure drops (sub critical flow) are desired, not the reduction of flow where high pressure losses are often the goal. Critical flow presents unique challenges because an orifice does not act like a well formed nozzle. As the pressure downstream of a sharp edged orifice is reduced below the fluid’s choking pressure, the location of the vena contracta gradually moves downstream and its diameter increases. The result is an increase in flow or what has become known as supercritical flow.
Perry’s Handbook provides a simplified approach to sizing critical flow across a thin, sharp-edged orifice by using the ideal nozzle as a mathematical basis for the flow calculation, just as for subcritical flow , which is then derated by the orifice’s discharge coefficient. There is some evidence to suggest this is a good approximation for relatively small orifices (beta ratio < 0.2), but as the orifice diameter increases, so does the error. For a quick calculation, this approach may be acceptable as there is always a tradeoff of engineering effort versus accuracy, but in many cases a more rigorous approach is required. Such an approach is offered in the following article, which is referenced in Perry’s Handbook as well:
Cunningham, R.G., “Orifice Meters with Supercritical Compressible Flow”, Transactions of the ASME, July 1951 pg 625-638. (Available through the Linda Hall Library in Kansas City, MO)
The final fluid or flow type to contend with is the two-phase or flashing flow scenario. Flashing and two phase flow have many unique characteristics and are among the hardest to accurately characterize. Changing fluid compressibility and the potential for mechanical (slip) or thermodynamic non-equilibrium between the liquid and vapor phases can greatly influence the flow of fluid through an orifice (or any flow device for that matter). Changes in compressibility and fluid flow then influence the discharge coefficient, further complicating matters. Ralf Diener and Jürgen Schmidt with BASF have published one method for characterizing two-phase orifice flow:
Ralf Diener and Jürgen Schmidt, Sizing of Throttling Device for Gas/Liquid Two-Phase Flow Part 2: Control Valves, Orifices and Nozzles, Process Safety Progress, Vol. 24, No. 1, March 2005, pgs 29-37.[Updated 08 November 2017] The reference above provides the basis for the two-phase nozzle sizing in ISO 4126-10; however, we’ve encountered some problems with implementing the calculations in this reference for orifice flow – see Orifice Flow for Subcooled Flashing Liquids and Orifice Flow for Two-Phase Fluids – Part 1.
Whatever the type of fluid, flow regime, or even calculation type is, the rule for calculating that flow is the same: it depends. It depends on the fluid and how that fluid will behave in the flow system in question. By understanding the intended application, limitations, and assumptions for a given engineering calculation an Engineer will be better prepared to ensure the right calculation is being applied to the right situation.