The failure probability of a pipeline is a quantification of the likelihood of an accident occurring in the pipeline, which is an indispensable part of the pipeline risk assessment process. To solve the problems of strong subjectivity, low feasibility, and low accuracy in the existing pipeline failure probability calculation methods, a three-layer Bayesian network topology model of “pipeline failure–failure cause–influencing factor” is proposed, with the pipeline failure as the subnode, the type of pipeline failure as the intermediate node, and the factors affecting the pipeline failure as the parent node of the network. Based on data fitting and fuzzy theory analysis methods, the functional relationship between the impact factor and the failure frequency of various pipelines is quantified. Using the mean value theorems for definite integrals and the analytic hierarchy process, the conditional probability of the directed edge in the network is calculated. The proposed function relationship provides a method to calculate the prior probability according to the parameters of the pipeline and its surroundings and a new idea to train the network model even without sufficient data.

In this paper, the effect of nozzle height on the combustion dynamics of jet fires in rotating flow fields (JFRFFs) is systematically investigated through experimental and numerical simulations. As the nozzle height increases, the JFRFF flame state transitions from stable rotation (SR) to unstable rotation (USR), and eventually to non-rotation (NR), indicating a weakening interaction between the vortex flow and the jet flame. The radial distribution of tangential velocity gradually deviates from the Burgers vortex model as the nozzle height increases, providing a criterion for distinguishing different flame states. Both vortex intensity and flame length are found to decrease with increasing nozzle height, whereas the maximum flame diameter increases. The relative position of the maximum flame diameter to the whole flame length firstly increases and then decreases to match that of the free jet fires, as the flame evolves from SR to USR and NR. In addition, the air entrainment near the nozzle exit decreases with increasing nozzle height, as evidenced by the gradual rise in lift-off height. These findings establish a theoretical basis for the fire performance design of flares in pipeline retrofitting and process industries.