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Application of the Root-Locus Method to the Design and Sensitivity Analysis of Closed-Loop Thermoacoustic Engines C Mark Johnson

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**Overview The Design Challenge Frequency Domain Analysis**

Laplace Domain Analysis Root Locus Approach to Instability Design Parametric Optimisation Addition of Heat Transfer Parametric Output Power Response Use of Tuning Stubs Conclusions

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The Design Challenge Travelling wave loop with alternator mid-loop and 2 side branches Simplified equivalent Loop must oscillate: added power = losses + alternator Steady-state frequency domain analysis Pressure and volume velocity must match around the loop

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Power Balance For travelling wave loop, available acoustic power is a fraction of the loop power: Carnot efficiency = max. possible: Alternator absorbs acoustic power: For steady oscillation: Depends on ratio of volume velocities, alternator parameters & engine conditions

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**Pressure and Vol. Velocity**

Simultaneous solution requires 4 independent variables (e.g. q1, q2, Ae, ra) Power balance equality must be satisfied

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**Effect of Alternator Position**

Normalised pressure, volume velocity and impedance at the alternator as a function of alternator position (q1) Position of alternator must satisfy condition for velocity ratio from both ends of the loop Determines the minimum thermoacoustic gain (kTA) for oscillation under varying acoustic engine conditions (Ae)

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**Conditions for Oscillation**

Minimum achievable velocity ratios (determined from AHX and HHX) and corresponding onset temperature as a function of (real) pressure ratio Ae Possible to establish boundaries to design space but… Difficult to gain further insight into optimisation of system parameters

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**Laplace-Domain Approach**

Laplace-domain analysis can be applied to study the time-domain (transient) behaviour as well as steady state Solution of the characteristic equation for the loop gives the poles of the transfer function Opens up new possibilities: Parametric stability analysis of a system, for example by plotting root loci as functions of temperature, load resistance, feedback pipe parameters Design optimisation in which geometrical and physical parameters may be varied to achieve specific targets, for example: Maximising the real part of dominant pole pair Minimising the temperature at which the dominant pole pair lies on the imaginary axis Note that the pressure and velocity are not solved for explicitly: reduces the number of equations from 4 to 2

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Two-Port Networks Analysis based on the assumption that all thermoacoustic loop elements may be represented by a 2-port equivalent Each element is based on a modified waveguide representation, including arbitrary shunt and series impedances Input and output pressure and volume velocity of each element can then be represented:

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Propagation Matrix Feedback system representation of thermoacoustic loop Whole loop is represented by a cascade of the S-parameter matrices for each element The poles of the closed-loop transfer function are determined by solving the characteristic equation: System can be solved in full to determine impulse response, frequency response etc. using Laplace solution methods e.g. poles and residues

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Stability Analysis In solving for the transfer function poles (two equations, two unknowns) we gain far more information about the system response than can be obtained from a simple frequency domain representation Can explore the system stability characteristics in response to changes in geometrical and other system parameters For a loop containing a number of undetermined parameters e.g. feedback pipe length, feedback pipe diameter, it is possible to find the region of instability and then determine an optimum operating point Boundary of the region of instability corresponds to the condition for steady oscillations (a conjugate pair of poles placed on the imaginary axis) and thus represents the limiting case for the onset of oscillation.

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Root Locus Method Conditions for instability: Temperature > 550K Alternator load > 0.6W Combination of pipe T=650K, RL=1W Solve characteristic equation to find dominant pole-pair and plot as function of selected parameters Positive real part indicates oscillatory behaviour Amplitude determined by e.g. onset of non-linearity, heat transfer limitations etc.

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Effect of Pipe Lengths Real part of dominant pole for combination of pipe T=650K, RL=1W a=0 “Strength” of instability related to positive real part of pole, a Area inside the a=0 contour is unstable and gives an indication of the margin of error permissible in construction or the range needed for tuning purposes

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**Engine Operating Conditions**

Magnitude of the impedance for the unstable region corresponds to values of engine pressure ratio Ae below 4 Phase plot shows pressure and velocity to be close to “in phase” with optimum loop parameters

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**Coupling with Heat Transfer**

Heat transfer added through auxiliary equations to represent heat flow through regenerator and transfer from HHX Lumped parameter representation of alternator and load Frequency domain solution Additional analyses enabled: Engine performance (efficiency, output) under varying acoustic and heat transfer conditions Power output (effect of changing parameters, optimisation) Tuning (using stubs to optimise output in presence of uncertainty)

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**Example Engine 1 Stub 1 Pipe E Engine 2 Pipe 1 Stub 2 Pipe 2**

Alternator Dual series engine loop with radiant HHX (loosely based on SCORE demo 0_3) Two tuning stubs, nominal adjustment range 0.1 to 0.6 m Feedback pipes 100 mm, stubs 75 mm diameter Alternator based on idealised SCORE design

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Engine Performance Available acoustic power strong function of regenerator temperature, radiant HHX temperature, acoustic conditions (enforced pressure ratio Ae) Drop in power/efficiency at high temperatures due to solid conduction through regenerator

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Stability Real part of dominant pole for combination of pipe T=750K, RL=1W Real part of dominant pole-pair as function of feedback pipe lengths Significant unstable region reflecting additional gain from series engines

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**Power Response Electrical power output Hot end temperature**

Simultaneously solve acoustic loop and heat transfer equations: hot end temperature varies Peak power output at regenerator hot end temperature of ~550K corresponds to peak engine efficiency Optimum pipe lengths do not correspond to those for greatest instability as temperature is now a variable

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**Tuning Power response Tuned response**

Power response (left) determined with fixed side branch length Optimised response (right) determined with varying side branch lengths Possible to maintain virtually constant output power by tuning

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Conclusions Frequency domain techniques can be difficult to apply to design of thermoacoustic loops Laplace domain techniques offer an attractive alternative by allowing use of standard methods such as root locus for determining feasible operating regimes Optimisation and parametric sensitivity analysis can be performed without explicitly solving for acoustic variables Coupling with heat transfer equations allows complete steady-state solution to be determined Application to e.g. output power optimisation, sensitivity analysis and tuning

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