Forget the Omega Method and Master VdP Integration Instead

Reliable flow estimates are essential for the sizing and selection of process equipment including but not limited to relief devices, process piping, and depressuring systems. In addition, reliable flow estimates from loss of containment scenarios can significantly influence the quality of consequence, risk analysis, and facility siting studies as well.

Existing methods for the calculation of flow rates range from those for simple, non-reacting, single phase, steady state flow to methods for dynamic, multiphase, reacting flows. Ideal nozzle flow calculation methods are heavily used in relief systems and risk analysis studies and are detailed in numerous standards and industry guidelines including the International Organization for Standardization (ISO), the American Society of Mechanical Engineers (ASME), the American Institute of Chemical Engineers (AIChE) Center for Chemical Process Safety (CCPS) and Design Institute for Emergency Relief Systems (DIERS), and the American Petroleum Institute (API).


Relief systems studies often include different types of flows such as non-equilibrium, subcooled, liquid, vapor, two phase, supercritical, and retrograde and phase change (RPC) flows. Flow estimates can be influenced by vapor quality, the presence of solids, slip, viscosity, chemical reactions, piping and fittings losses and geometry, chemical composition, temperature, pressure, etc. In addition to flow rates, reliable flow methods are expected to yield reliable estimates for reaction forces, location of choke points, sound power levels, exit temperatures and pressures, exit compositions and vapor quality, two phase flow regimes, etc.

We explore in this document the origins of ideal nozzle flow methods for single and multiphase flow including Δh, direct vdP integration ( vdP), simple reduced analytical models, and complex reduced analytical models. We also explore the advantages and disadvantages of those methods for simple and complex flow systems.

Several case studies with practical examples are included using Process Safety Office® SuperChems™ software.

The Thermodynamics of Nozzle Flow

The origin of all ideal nozzle flow methods can be traced back to the first law of thermodynamics or conservation of energy. Let’s consider a vessel containing a multiphase mixture that is exchanging mass and energy with its surroundings. If we define our thermodynamic system to include all the vessel contents, we can write the first law of thermodynamics as follows:


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