Theoretical analysis about model simplifications and boundary conditions on two-phase flow instability of parallel heated tube systems

Flow instability may happen in both subcritical two-phase flow boiling systems (such as U tube natural circulation steam generator of Pressurized Water Reactor, once through steam generator of High-Temperature GasCooled Reactor, etc) and supercritical evaporation systems. In previous theoretical researches, the model of a single heated tube (SHT) with constant pressure drop boundary condition (ConPD) was always used to investigate the system stability. However, in real applications, there are usually multiple parallel heated tubes (MHT) and unheated connection regions before the branch heated tubes. The system is usually driven by a pump, and there is always a long unheated main pipe and throttling between the heated tubes (boiler or steam generator) and the pump. However, there lacks rigorous analysis about the effects of these model simplifications and boundary conditions. Take superheated two-phase flow boiling systems as examples, the theoretical models with different flow-driven boundary conditions (ConPD, constant mass flow rate (ConMF), and pump driven (PumpD)), different numbers of heat tubes (SHT, MHT), with or ignoring the unheated region (UHR), with or ignoring the bypass of a pump, with identical or un-identical geometry parameters and operation parameters are established. Linear ordinary differential equations describing the instability are obtained. Eigenvalues of the coefficient matrix of linear ordinary differential equations, and the property of the block matrix and the similarity matrix are used to analyze the effects of the above boundary conditions and model simplifications on the system stability. Two new dimensionless numbers, dimensionless pump number (Npump) and dimensionless bypass number (Nbypass), are proposed to describe the effect of pump and its bypass on stability quantitatively. The effect of the pump bypass on the stability can be attributed to the decrease of Npump. No matter ConPD or PumpD is adopted, SHT considering UHR is proved to be more stable than SHT ignoring UHR. No matter UHR is considered or ignored, SHT with PumpD is proved to be more stable than SHT with ConPD. No matter UHR of MHT at the main pipe is considered or ignored, the stability of MHT with ConPD or ConMF or PumpD is proved to be the same as that of SHT with ConPD ignoring UHR. When ConPD ignoring UHR is adopted, the stability of MHT with un-identical geometry and operation parameters is proved to be the same as that of its most unstable heated tube. The above conclusions about the effect of model simplification and boundary condition are obtained from superheated two-phase flow boiling systems, they also hold for saturated two-phase flow boiling systems and supercritical evaporation systems.