Simulation case study: electro-thermo-mechanical cylinder

Transcription

1 Simulation case study: electro-thermo-mechanical cylinder

2 B pɺ = ( P Pa ) A mg p m A Vɺ = A Vp = p m Sɺ = f f f f f s1 s1 S 2 S 3 S 4 2 v = T R( T ) ( T T ) a fsi = Hi, i = 2,3, 4 T ki Ai Hi = x i

3 These dynamic equations can be simulated using the ODE function file shown to the right. A simulation file is shown on the following slide. Note that the parameter q_on allows heat transfer to be turned on for q_on = 1 or off for q_on = 0. function xdot = thermc(t,x) % Thermodynamic simulation example % Electro-Thermo-Mechanical System % updated 12/11/11 RGL % conduction through cylinder walls and piston global R_gas gamma global d_piston m_piston b_piston A_piston global To So Vo Po Pa Ta g global k_al dx2 dx3 dx4 q_on global m_cyl cv v_in R_c p = x(1); V = x(2); S = x(3); T_cyl = (To*(Vo/V)^(gamma-1))*exp((S-So)/(m_cyl*cv)); P_cyl = (Po*(Vo/V)^gamma)*exp((S-So)/(m_cyl*cv)); H2 = k_al*(4*v/d_piston)/dx2; H3 = k_al*a_piston/dx3; H4 = k_al*a_piston/dx4; fs1 = v_in*v_in/r_c/t_cyl; fs2 = q_on*h2*(t_cyl-ta)/t_cyl; fs3 = q_on*h3*(t_cyl-ta)/t_cyl; fs4 = q_on*h4*(t_cyl-ta)/t_cyl; pdot=a_piston*(p_cyl-pa)-m_piston*g-b_piston*p/m_piston; Vdot=A_piston*p/m_piston; Sdot=fs1-fs2-fs3-fs4; xdot=[pdot;vdot;sdot];

4 % Electro-Thermo-Mechanical System clear all global R_gas gamma global d_piston m_piston b_piston A_piston global To So Vo Po Pa Ta g global k_al dx2 dx3 dx4 q_on global m_cyl cv v_in R_c Simulation file for thermc.m g = 9.81; % Heater voltage and resistance v_in = 1*110; R_c = 100; % properties for air % molar mass = kg/mol MW_air = 1/ ; % molecular weight mol/kg R_gas = 287; % R = J/(kg*K) cv = 718; % constant volume specific heat, J/(kg*K) gamma = 1.4; k_al = 237; % Thermal conductivity of aluminum, 237 W/(m*K) rho_al = 2702; % density of aluminum, kg/m^2 % piston properties d_piston = 0.2; % diameter of piston, m l_piston = 0.05; % piston length, m rho_piston = rho_al; % density of piston material, kg/m^3 A_piston = 0.25*pi*d_piston^2; % piston area, m^2 m_piston = rho_piston*l_piston*a_piston; % piston mass, kg b_piston = 100.0; % piston damping constant, N*sec/m % cylinder properties l_cyl = 0.2; % cylinder length, m d_cyl = d_piston; % assume piston and cylinder dia are equal A_cyl = pi*d_cyl^2/4; % cylinder area, m^2 V_cyl = l_cyl*a_cyl; % volume of cylinder, m^3 dx2 = 0.01; % thickness of cylinder walls, m dx3 = 0.01; % thickness of cylinder base, m dx4 = l_piston; % thickness of piston, m Pa = ; % atmospheric pressure, Pa Ta = ; % atmospheric temperature, K To = Ta; % initial temperature of air in cylinder, K Po = Pa + m_piston*g/a_piston; % initial pressure of air in cylinder with piston weight, Pa m_cyl = Po*V_cyl/(R_gas*To); % mass of air contained in cylinder, kg (continued) % calculating initial entropy (from tables) % Interpolate - must enter values on each side of To T1 = 290; T2 = 300; s1 = ; s2 = ; % specific entropy in J/(kg*K) So = s1 + (s2-s1)*to/(t2-t1); So = So*m_cyl; % initial entropy of air in cylinder, J/K Vo = m_cyl*r_gas*to/po; % initial volume of air in cylinder, m^3 % Simulation parameters t0 = 0; tfinal = 1.0; %Define initial conditions po = 0; x0 = [po;vo;so]; % q_on = 1 for heat transfer on, q_on = 0 for insulated cylinder q_on = 0; [t,x]=ode15s(@thermc,[t0 tfinal],x0); % The outputs we want are velocity of piston % displacement (from bottom) and entropy, pressure, and temperature of air. Vp = x(:,1)/m_piston; xp = x(:,2)/a_piston; P_cyl = (Po*(Vo./x(:,2)).^gamma).*exp((x(:,3)-So)/(m_cyl*cv)); T_cyl = (To*(Vo./x(:,2)).^(gamma-1)).*exp((x(:,3)-So)/(m_cyl*cv)); See attached tables for properties of air figure(1) subplot(5,1,1), plot(t,p_cyl/po), title('relative Pressure of Air in Cylinder, P/Po') subplot(5,1,2), plot(t,x(:,3)), title('entropy of Air in Cylinder, J/K') subplot(5,1,3), plot(t,t_cyl), title('temperature of Air, K') subplot(5,1,4), plot(t,1000*xp), title('cylinder Displacemmt, mm') subplot(5,1,5), plot(t,1000*vp), title('cylinder Velocity, mm/s') Heat transfer on or off?

5 1.002 Heat transfer off (insulated) Relative Pressure of Air in Cylinder, P/Po Heat transfer on Relative Pressure of Air in Cylinder, P/Po Entropy of Air in Cylinder, J/K Entropy of Air in Cylinder, J/K Temperature of Air, K Cylinder Displacemmt, mm Cylinder Velocity, mm/s Temperature of Air, K Cylinder Displacemmt, mm Cylinder Velocity, mm/s -0.5

6 G.J. Van Wylen and R.E. Sonntag, Fundamentals of Classical Thermodynamics, Wiley, 2nd edition, 1978