The particle trajectory function 'plotmass'
The function plotmass allows one to do a particle trajectory plot using Natural Coordinates, as defined by Allan Organ: 'Natural' coordinates for analysis of the practical Stirling cycle (ImechE, 1992). Refer also to Chapter 9 of his book: The Regenerator and the Stirling Engine (John Wiley, 1997).
Essentially, for an ideal gas the mass in any volume is inversely proportional to the temperature. Thus for the particle mass trajectory plot we normalize the volumes with respect to the local temperature. In the typical particle trajectory plot shown below the reduced volumes of the expansion and heater spaces are much smaller than those of the cooler and compression spaces. The regenerator is normalized with respect to the regenerator mean effective temperature.
This
plot shows particles of equal mass flowing through the engine over a
cycle. In this design it appears as though the cooler volume can be
reduced, since some mass particles never leave the cooler. This
information is not intuitively obvious, and enables a better
understanding of the process, however at this stage it should not be
used to influence the design of the machine.
The following MATLAB function plotmass was developed by students of the Stirling Cycle Machine Analysis course, and is invoked by the function Schmidt.
function plotmass % Kyle Wilson 10-2-02 % Stirling Cycle Machine Analysis ME 589 % Particle Trajectory Map % Equations from Organ's "'Natural' coordinates for analysis of the practical % Stirling cycle" and Oegik Soegihardjo's 1993 project on the same topic % Modified by Israel Urieli (11/27/2010) to obtain correct phase advance % angle alpha subsequent to error determined by Zack Alexy (March 2010) % Global values from define program global vclc vcle % compression,expansion clearance vols [m^3] global vswc vswe % compression, expansion swept volumes [m^3] global alpha % phase angle advance of expansion space [radians] global vk % cooler void volume [m^3] global vh % heater void volume [m^3] global vr % regen void volume [m^3] global pmean % mean (charge) pressure [Pa] global tk tr th % cooler, regenerator, heater temperatures [K] NT = th/tk; % Temperature ratio Vref = vswe; % Reference volume (m^3) %% Fixed reduced volumes (dimensionless) vswe_r = (vswe/Vref)/NT; % Reduced expansion swept volume vcle_r = (vcle/Vref)/NT; % Reduced expansion clearance volume vh_r = (vh/Vref)/NT; % Reduced heater void volume vr_r = (vr/Vref)*log(NT)/(NT-1); % Reduced regenerator void volume vk_r = (vk/Vref); % Reduced cooler void volume vswc_r = (vswc/Vref); % Reduced compression swept volume vclc_r = (vclc/Vref); % Reduced compression clearance volume %% Phase domain angi = 0; angf = 2*pi; dang = 0.1; ang = [angi:dang:angf]; n = size(ang); %% Reduced volume variations for i = 1:n(2) deg(i) = ang(i)*180/pi; Ve(i) = (vswe/2)*(1- cos(ang(i))); % Expansion volume vs phase Vc(i) = (vswc/2)*(1+ cos(ang(i) - alpha)); % Compression volume vs phase ve(i) = (Ve(i)/Vref)/NT; % Reduced expansion vs phase vc(i) = Vc(i)/Vref; % Reduced compression vs phase vt(i) = vswe_r + vcle_r + vh_r + vr_r + vk_r + vclc_r + vc(i); % Total volume vs phase end figure step = 30; for m = 1:step-1 for i = 1:n(2) v(i) = ve(i) + (m/step)*(vt(i)-ve(i)); % Reduced volume segments end hold on plot(v,deg, 'k:') end hold on plot(ve,deg,'k') plot(vt,deg, 'k') %% Vertical lines L1 = vswe_r; % Boundary of reduced expansion swept volume L2 = L1 + vcle_r; % Boundary of reduced expansion clearance volume L3 = L2 + vh_r; % Boundary of reduced heater void volume L4 = L3 + vr_r; % Boundary of reduced regenerator void volume L5 = L4 + vk_r; % Boundary of reduced cooler void volume L6 = min(vt); % Boundary of reduced expansion swept volume point1 = [L1;L1]; % Preparing for plot point2 = [L2;L2]; point3 = [L3;L3]; point4 = [L4;L4]; point5 = [L5;L5]; point6 = [L6;L6]; point = [0;deg(n(2))]; plot(point1, point, 'r--', point2, point, 'r--', point3, point, 'g--') plot(point4, point, 'g--', point5, point, 'b--', point6, point, 'b--') axis([0 max(vt) 0 deg(n(2))]) xlabel('Reduced volume') ylabel('Crank Angle (deg)') title('Particle mass plot') hold off |
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Stirling Cycle Machine Analysis by
Israel
Urieli is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 3.0 United States
License