Sliding Motion and Stability of a Class of Discontinuous Dynamical Systems

Abstract

Abstract. A closed container with gas and liquid inflows and one outlet for outflow, described by a nonlinear hydraulic model, is shown to reach a state of sliding mode. Through a Lyapunov function, the sliding motion is shown to be stable for all positive exponents in the flow model. This property is independent of factors, such as, valve opening, valve coefficients, friction factor and initial states, making this device a suitable one to study two phase flow. The sliding solution of this system with square root hydraulic model is found using Filippov's equivalent dynamic approach. The discontinuous system is also solved using an accurate and efficient method that accounts for persistent discontinuity sticking. These two solutions are shown to be in exact agreement.Keywords: Discontinuous System, Sliding Mode, Lyapunov Function, Self Regulating System, Slug Flow, Two Phase Flow. Nomenclature:Di -Region in (MG, ML) phase plane; FG -Gas feed rate, mol/s; FL -Liquid feed rate, mol/s; kG -Valve coefficient for liquid flow; kL -Valve coefficient for gas flow; L -Liquid outflow rate, mol/s and subscript to indicate liquid; G -Gas outflow rate, mol/s and subscript to indicate gas; MG -Gas holdup in the system, mol; M * G -Gas holdup at the beginning of sliding motion, mol; MG,ss -Gas holdup at steady state, mol; M L G -Limiting value of gas holdup defined for sliding motion, mol; ML -Liquid holdup in the system, mol; P -System pressure, atm; PssSteady state pressure, atm; Pout -Outlet pressure, atm; R -Gas constant; SiSwitching/Sliding surface; T -Temperature, K; t -Time, s; t * -Time when sliding begins, s; V -Volume of the reactor, lit; V d -Volume of the reactor below the dip tube, lit; x -Extent of valve opening; α -Fraction time of applicability of liquid model while sliding; ϕ -Switching function; ϕ1 -Switching function restricted to MG > M L G ; ρL -Density of liquid in mol/lit.

Hybrid model of an industrial HDPE reactor

“…It is assumed that the liquid and the gas are ideal. The total accumulation of each component in the reactor is the sum of it in the two phases: this is expressed by (7) and (8). Finally, (9) and (10) express the fact that the liquid and the gas phases are in equilibrium.…”

“…It is well known that there is usually quite a bit of measurement error in experiments involving two phase flow. Through Lyapunov function, it has been shown that the state of sliding is stable so long as the outflow and pressure drop have a monotonic relationship: that is, both of them increase or decrease simultaneously [8]. This stability property is independent of factors, such as, valve opening, valve coefficient, friction factor and starting points.…”