## 76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX)

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#### Keywords

- 150 bar loop (1)
- Eikonal equation (1)
- Pedestrian FLow (1)
- bursting disk (1)
- dischargeable mass flow rate (1)
- pressure drop (1)
- pressure relief (1)
- rupture disk (1)

#### Faculty / Organisational entity

The Directive 97/23/EC of the European Parliament and of the Council of 29 May 1997 on the approximation of the laws of the Member States concerning pressure equipment (European Commision, 1997) is the basis of the legal framework for protection of pressure equipment within the European Union. Codes and standards are useful to comply with the legal and regulatory responsibilities stipulated in PED Directive regarding the protection of pressure equipment against overpressure, sizing, and selection safety relief devices.
Rupture disk devices are primary relief devices to protect vessels, pipe, and equipment against overpressure. A rupture disk bursts once the so-called burst pressure is reached in the protected system, thereby discharging flow and preventing further increase in pressure. Currently, rupture disks are sized with standards and codes assuming the worst-case scenario at burst pressure. There is however no standardized procedure for sizing rupture disks with two-phase flow and there lacks suited test-facilities, test-sections, and reliable experimental data for model validation. Sizing rupture disk vent-line systems with current characteristic numbers comes with significant uncertainties, especially for high-velocity compressible flows (Schmidt, 2015).
Zero-Emission and Green Safety are current trends for organizations that seek to attain innovative protection concepts beyond regulatory compliance. A procedure to size a rupture disk vent-line should accurately determine the discharge rate and pressure-drop across a rupture disk, from the point of rupture disk activation to the point when the system depressurizes fully. This procedure is critical for further safety considerations, such as for modeling the dispersion of toxic gases released during emergency-relief and calculating the emissions to the environment with time.
Over-dimensioning is one measure taken today to mitigate uncertainties encountered while sizing with current methods. This is not always an option, as over-dimensioning the rupture disk vent-line system leads to unnecessary financial costs. It may also cause malfunction of the collecting systems downstream when the fluids discharged are more than the design limits. Emissions to the environment are thereby potentially higher than necessary, causing excessive harm to the environment. Under-dimensioning, on the other hand, may lead to hazardous incidents with loss of human life and equipment. This work has therefore focused on the investigation of the mass flow rate and pressure-drop through rupture disk devices with compressible gas and two-phase flow.
The experimental focus was in the design, construction, and commissioning of a high-capacity, high-pressure industry-scale test facility for testing small- to large-diameter rupture disks and other fittings with gas flow. The resulting test facility is suited to test safety devices and pipe fittings at near realistic flow conditions at pressures up to 150 bar. This work also presents the design of a pilot plant for testing rupture disks with air/water two-phase flow. These test facilities open-up new frontiers for capacity testing because they have precise and state-of-the-art measurement and instrumentation. Experimental results from these facilities deliver reliable experimental data to validate proposed sizing procedures for rupture disk devices.
The theoretical focus was on the development of a reliable rupture disk sizing procedure for compressible gas and two-phase flow. This required phenomenological studies of flow through rupture disks with both experiments and CFD studies. Better suited rupture disk characteristic numbers and model parameters for determining the mass flow rate and pressure-drop across rupture disks are identified. The proposed sizing procedure with compressible gas and two-phase flow predicts the dischargeable mass flow rate and pressure-drop across a rupture disk within ±4 % of measured value. Experimental validation has been undertaken with different types of rupture disks. The procedure is suited for determine the mass flow rate and pressure-drop through rupture disk seamlessly, from the point of rupture disk activation (worst-case scenario) to the point when the system fully depressurizes beyond regulatory compliance.

The overall goal of the work is to simulate rarefied flows inside geometries with moving boundaries. The behavior of a rarefied flow is characterized through the Knudsen number \(Kn\), which can be very small (\(Kn < 0.01\) continuum flow) or larger (\(Kn > 1\) molecular flow). The transition region (\(0.01 < Kn < 1\)) is referred to as the transition flow regime.
Continuum flows are mainly simulated by using commercial CFD methods, which are used to solve the Euler equations. In the case of molecular flows one uses statistical methods, such as the Direct Simulation Monte Carlo (DSMC) method. In the transition region Euler equations are not adequate to model gas flows. Because of the rapid increase of particle collisions the DSMC method tends to fail, as well
Therefore, we develop a deterministic method, which is suitable to simulate problems of rarefied gases for any Knudsen number and is appropriate to simulate flows inside geometries with moving boundaries. Thus, the method we use is the Finite Pointset Method (FPM), which is a mesh-free numerical method developed at the ITWM Kaiserslautern and is mainly used to solve fluid dynamical problems.
More precisely, we develop a method in the FPM framework to solve the BGK model equation, which is a simplification of the Boltzmann equation. This equation is mainly used to describe rarefied flows.
The FPM based method is implemented for one and two dimensional physical and velocity space and different ranges of the Knudsen number. Numerical examples are shown for problems with moving boundaries. It is seen, that our method is superior to regular grid methods with respect to the implementation of boundary conditions. Furthermore, our results are comparable to reference solutions gained through CFD- and DSMC methods, respectevly.

Pedestrian Flow Models
(2014)

There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries.