1. Numerical Simulation in Real Time Operation Purpose of the Numerical Simulation unit Simulate the whole system - Provide the Control Input for the Small Sector Rigs Simulation Provide Control Input Small Sector Rigs Whole System

1-(1) Modeling for the Numerical Simulation using d’Alembert Solution & Derived Heat Source Model Model of the System Separate Models for the regions Inside & Outside of the Heat Source Heat Source Model Equation Sets Sample Result 1 (No Mean flow, Traveling  Standing Wave) Initial Condition: Only (+) going Traveling Wave Sample Result 2 (Mean flow, Standing  Travelling Wave) Initial Condition: Standing Wave with Mean Flow Current Set-Up of Real Time System (Ongoing)

Model of the System Separate Models for the regions Inside & Outside of the Heat Source Model of Heat Source Linear Acoustic Equation d’Alembert Solution Region of Heat Source - 1-D Toroidal Cavity Assume 1-D (Neglect Geometrical Effect, Only Physical Effects) - The Heat Sources are located in the small zone, not in the entire domain The Zone is assumed as an Acoustically Compact Zone - The Outside the Heat Source Region No heat Addition  Eventually, Isentropic Process (from Energy Relation) And the Wave Solutions

Model of the System Model of Heat Source + 3. Reaction Depends on Injector 1. Heat Source affects Waves 2. Waves & Reaction affect Heat source - Perturbation in the Cavity - The Change in Energy Balance -> Heat Source perturbed + 3. Reaction Depends on - Chemical Energy in the Cavity - Pressure & Velocity -> Heat Source affected

Model of the System Final Equations Set Where

Numerical Simulation Numerical Simulation on the Simple Case No Heat Source Compact Heat Source Region Closed Circular Tube - 1-D - Closed Circular Tube - 1 Heat Source - In the Region of No Heat Source - d’Alembert’s Solutions works - In the Compact Heat Source Region - Use Previous Equation Set or the Combined Equation

Result 1 (No Mean flow, Traveling  Standing Wave) Initial Condition: Only (+) going Traveling Wave imin imax iheat i i+1 i-1 Pressure Transition from Travelling Wave to Standing Wave Animation

Result 1 (No Mean flow, Traveling  Standing Wave) Transition from Travelling Wave to Standing Wave t=10 t=0.25 t=10.25 t=0.5 t=10.5 t=0.75 t=10.75 t=1 t=11 t=1.25 t=11.25

Result 2 (Mean flow, Standing  Travelling Wave) Initial Condition: Standing Wave with Mean Flow imin imax iheat i i+1 i-1 Mean Flow Pressure Transition from Standing Waves to Travelling Waves Animation

Result 2 (Mean flow, Standing  Travelling Wave) Transition from Standing Waves to Travelling Waves t = ~0.25 t = ~16.25 t = ~0.5 t = ~16.5 t = ~0.75 t = ~16.75 t = ~1 t = ~17 T = ~1.25 T = ~17.25 Standing Wave propagates at Mean Velocity Traveling Wave propagates at Mean Velocity + Speed of Sound

Current Set-Up of Real Time System (Ongoing) Achieved Fast Operation Heat Source Eqn: Linear Acoustic Eqn: d’Alembert Sol. DS1103 Board Numerical Simulation ADC DAC Control Signal Input Signal 4 channel inputs 4 channel outputs Simulation Time ~ 6~7 micro sec ~ 2 micro sec ~ 2 micro sec Sampling Rate ~ 11.2 micro sec