% conductionspherecooling.m Last modified 7/31/2008 % Conduction of heat in a sphere of radius R % with exterior face (r=R) at temperature of zero % and initial interior at zero temperature T0. % The diffusivity is kappa=k/(rho*c) % This is the well-known "Cooling of a can of beer..." problem posed when % investigating how long it will take to cool abeer in an ice bath. % This m-file was written at the University of Wyoming in the Electrical % and Computer Engineering Department and is to be distributed without % cost. clear all set(0,'DefaultAxesFontSize',12); set(0,'DefaultTextFontSize',12); roverR=linspace(0,1,101); nterm=20;% number of terms in the Fourier series solution. for i=1:nterm phi(i,:)=sin(i*pi*roverR); end tau=linspace(0,.2,201); T1=(ones(1,101)); F=ones(1,101)./roverR; framedata=[]; for k=1:201%time loop sum=zeros(1,101); for i=1:nterm sum=sum+(((-1)^(i+1))/i)*phi(i,:)*exp(-(i*i*pi*pi*tau(1,k))); end sum=sum*(2/pi).*F; T=sum; framedata=[framedata;T]; end framedata(1,:)=ones(1,101); %framedata(1,101)=1; figure(1);clf;% Plot T(x,t) vs x for various t axis([-.2 1.2 -.2 1.2]) hold on plot([0 0],[-.2,1.2]); hold on plot([1 1],[-.2,1.2]); hold on plot(roverR,framedata(1,:),'k') hold on xlabel('Dimensionless Radius, r/R') hold on ylabel('Dimensionless Temperature, T(r,t)/T_0') for k=21:20:201% time loop plot(roverR,framedata(k,:),'k') hold on end text(.8,1.08,['\kappat/R^2 = ' num2str(0)]) text(.72,.8,num2str(0.02)) text(.6,.75,num2str(0.04)) text(0.4,.17,num2str(0.2)) set(gca,'Box','on') text(.1,1.13,'Press Enter to Continue') hold off pause figure(2);clf;%Plot T(x,t) vs t for various x axis([0 .2 -.2 1.2]); hold on %plot(tau,ones(1,201)) hold on for j=1:10:101% space loop plot(tau,framedata(:,j)) hold on end text(.02,.03,['r/R = ' num2str(1)]) text(.03,.15,['r/R = ' num2str(0.9)]) text(.1,.72,['r/R = ' num2str(0)]) xlabel('Dimensionless Time, \kappat/R^2') ylabel('Dimensionless Temperature, T(r,t)/T_0') text(.02,1.1,'Press Enter to Continue') set(gca,'Box','on') hold off pause figure(3);clf;%now do animation axis([-.2 1.2 -.2 1.2]) hold on plot([0 0],[-.2,1.2]); plot([1 1],[-.2,1.2]); box on xlabel('Dimensionless Radius, r/R') hold on ylabel('Dimensionless Temperature, T(r,t)/T_0') hold on L=plot(roverR, framedata(1,:),'k','EraseMode','xor'); hold on texthandl=text(.4,1.1,'Press Enter to Animate'); pause set(texthandl,'String',' '); for k=2:201% time loop set(L,'Ydata',framedata(k,:)) pause(.05) end hold on plot(roverR,framedata(1,:),'k') hold off set(texthandl,'String','Press Enter to Continue'); pause figure(4);clf;%3-D Plot of T(r,t) vs x and t tau1=linspace(0,.2,201); [X,Y]=meshgrid(roverR,tau1); mesh(X,Y,framedata(1:201,:)) colormap(cool) text(.8,.2,-.2,'Dimensionless Radius, r/R','Rotation',10) text(1,.045,-.25,'Time, \kappat/R^2','Rotation',-34) zlabel('Dimensionless Temperature, T(r,t)/T_0') view(150,30) axis([0 1 0 .2 0 1]) text(.7,.2,.08,'Press Enter to Quit','Rotation',10) pause