Homework 2 - ' J = "K !Ttemp !x Ttemp ( x ,t + !t...

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Unformatted text preview: ' J = "K !Ttemp !x Ttemp ( x ,t + !t ) " Ttemp ( x ,t ) !t ' dTtemp %K dx % =& J x A = J x + !x A + A!x (K dTtemp dx x Ttemp ( x ,t + !t ) ( Ttemp ( x ,t ) !t x + !x !x $ " " # !Ttemp !t =K ! 2Ttemp !x 2 t = 0, Ttemp ( x,0) = f ( x ) = TB t > 0, Ttemp (0,t) = TW TB = 37C TW = 44C For a nonhomogeneous problem, As t 0! x!L Ttemp ( L, t ) = Tw ! a steady state temperature distribution v(x) will be reached v' ' ( x) = 0 for 0<x<L v(0) = Tw v ( L ) = Tw v( x) = (Tw ! Tw ) x + Tw = Tw L Ttemp ( x, t ) = v( x) + w( x, t ) k (v + w) xx = (v + w) t v xx = 0 vt = 0 kwxx = wt w(0, t ) = Ttemp (0, t ) ! v(0) = Tw ! Tw = 0 w( L, t ) = Ttemp ( L, t ) ! v( L) = Tw ! Tw = 0 w( x,0) = Ttemp ( x,0) ! v( x) = f ( x) ! v( x) = f ( x) ! Tw w( x, t ) = X ( x)T (t ) X ( x)T !(t ) = kX !!( x)T (t ) X ##( x) T #(t ) = = "! 2 X ( x) kT (t ) X !!( x) + "2 X = 0 T ! + "# 2T = 0 X = c1 cos(!x) + c 2 sin(!x) T = c3e #!" t 2 X (0) = c1 cos(! (0)) + c 2 sin(! (0)) = c1 = 0 X ( L) = c 2 sin(!L) = 0 "= " n! L #k ( n $nx 2 ) t L ... ! w( x, t ) = ! An e n =1 sin( $nx ) L w( x,0) = " An sin( n =1 # $nx ) = f ( x) ! v( x) L 2 n#x An = ! [ f ( x) " v( x)]sin( )dx L0 L Ttemp ( x, t ) = v( x) + w( x, t ) L Ttemp ( x, t ) = Tw + ! An e n =1 " #k ( n$x 2 ) t L sin( n$x ) L 2 n#x An = ! [TB " TW ] sin( )dx L0 L L v' ' ( x) = 0 for 0<x<L v(0) = Tw L v ( ) = TB 2 v( x) = (Tw ! TB ) x + Tw L 2 0! x! L 2 t = 0, Ttemp ( x,0) = f ( x ) = TB t > 0, Ttemp (0,t) = TW L Ttemp ( , t ) = TB 2 Ttemp ( x, t ) = v( x) + w( x, t ) w(0, t ) = Ttemp (0, t ) ! v(0) = Tw ! Tw = 0 L L L w( , t ) = Ttemp ( , t ) ! v( ) = TB ! TB = 0 2 2 2 w( x,0) = Ttemp ( x,0) ! v( x) = f ( x) ! v( x) w( x, t ) = X ( x)T (t ) X ( x)T !(t ) = kX !!( x)T (t ) X ##( x) T #(t ) = = "! 2 X ( x) kT (t ) X !!( x) + "2 X = 0 T ! + "# 2T = 0 X = c1 cos(!x) + c 2 sin(!x) T = c3e #!" t 2 X (0) = c1 cos(! (0)) + c 2 sin(! (0)) = c1 = 0 L L X ( ) = c 2 sin(! ) = 0 2 2 "= " 2n! L #k ( n 2$nx 2 ) t L ... ! w( x, t ) = ! An e n =1 sin( 2$nx ) L w( x,0) = " An sin( n =1 # $nx ) = f ( x) ! v( x) L An = 4 2n#x ! [ f ( x) " v( x)]sin( L )dx L0 L 2 Ttemp ( x, t ) = v( x) + w( x, t ) " #k ( 2x Ttemp ( x, t ) = (TB # TW ) + TW + ! An e L n =1 2$nx 2 ) t L sin( 2$nx ) L An = 4 2x 2n#x [TB " (TB " TW ) " TW ] sin( )dx L! L L 0 L 2 ....................................................................................... Ttemp ( n ) = An e #" ( 2!nx 2 ) t L cos( 2!nx ) L " #% ( n 2$nx 2 ) t L ... ! Ttemp ( x, t ) = ! Ttemp ( n ) =! An e n =1 n =1 " cos( 2$nx ) L Ttemp ( x,0) = f ( x) = TB = ! An cos( n =1 " 2#nx ) L n=m & # $1 L' 2(nx 2 $ 1 4'nx ! L !0 % cos( L ) "dx $ 2 x + 8'n sin( L )! = 2 & # $ ! $ !0 L % " L An = 2 L 2"nx !0 f ( x) cos( L )dx L n!x 2 ) t L 2 $ , L 2!nx ) #" ( Ttemp ( x, t ) = % * & f ( x) cos( )dx 'e L n =1 + 0 L ( cos( 2!nx ) L %HW4 function HW4 clear, clc L=1.8; alpha=0.4; space_steps=50; dx=L/space_steps; time_steps=15; dt=1.5/time_steps; u=zeros(space_steps,time_steps); u(1:space_steps+1,1)=37; syms tt x for n=1:1:15 f(n)= (37/44) * exp( (-alpha)*((n*2*pi/L)^2) * tt ); end sumf=sum(f); tt=0; for t=1:time_steps+1 tt=tt+dt; x=0; for i=2:space_steps x=x+dx; u(i,t+1)=-1/0.12 * eval(sum(f)); % Constant f(x) end end % IC xx=[0:dx:L]; u(:,1) = 37; u(:, 2:size(u,2)) = u(:, 2:size(u,2)) + 44; % BC figure, plot(xx,u(:,1), xx,u(:,ceil(0.1/dt)+1), xx,u(:,ceil(0.2/dt)+1),... xx,u(:,ceil(0.3/dt)+1), xx,u(:,ceil(0.5/dt)+1),... xx,u(:,ceil(1/dt)+1),xx,u(:,ceil(1.5/dt)+1) ) axis tight; grid on; title('One-dimensional unsteady-state diffusion') xlabel('Finger thickness, X'); ylabel('Temperature, T') figure, plot(xx,u(:,1), xx,u(:,ceil(0.1/dt)+1), xx,u(:,ceil(0.2/dt)+1),... xx,u(:,ceil(0.3/dt)+1), xx,u(:,ceil(0.5/dt)+1),... xx,u(:,ceil(1/dt)+1),xx,u(:,ceil(1.5/dt)+1) ) axis tight; grid on; title('One-dimensional unsteady-state diffusion') xlim([1.780 1.8]); ylim([36.9 44.1]); xlabel('Finger thickness, X'); ylabel('Temperature, T') figure, plot(xx,u(:,1), xx,u(:,ceil(0.1/dt)+1), xx,u(:,ceil(0.2/dt)+1),... xx,u(:,ceil(0.3/dt)+1), xx,u(:,ceil(0.5/dt)+1),... xx,u(:,ceil(1/dt)+1),xx,u(:,ceil(1.5/dt)+1) ) axis tight; grid on; title('One-dimensional unsteady-state diffusion') xlim([0.605 1.01]); ylim([36.9 44.1]); xlabel('Finger thickness, X'); ylabel('Temperature, T') ...
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This note was uploaded on 04/17/2008 for the course BMED 2200 taught by Professor Robertspilker during the Spring '08 term at Rensselaer Polytechnic Institute.

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