Gymnasieopgave: Varmeledning

>    restart;interface(warnlevel=0):with(plots):

1. Differentialligningen

>    lign:=C*diff(T(t),t)=P-U*A*(T(t)-T0);

lign := C*diff(T(t),t) = P-U*A*(T(t)-T0)

Ligningen er en 1. ordens lineær differentialligning. Vi bruger MAPLE

>    los:=dsolve({lign,T(0)=T0},T(t));   

los := T(t) = 1/U/A*P+T0-exp(-U*A/C*t)/U/A*P

Heraf ses at for limit(T(t),t = infinity)   gælder

>    Tu:=P/U/A+T0;

Tu := 1/U/A*P+T0

Måles Tu - T0 = P/UA kan U bestemmes.

Talværdier:

>    V:=0.5^3;

>    A:=6*0.5^2;

>    lambda:=4.1*10^(-2);

>    L:=2.0*10^(-2);

>    U:=lambda/L;

>    rho:=1.20;

>    cp:=1.01*10^3;

>    m:=rho*V;

>    C:=cp*m;

>    T0:=20.0;

>    P:=60.0;

>    tau:=C/U/A;

>    cp := 1010.00;

V := .125

A := 1.50

lambda := .4100000000e-1

L := .2000000000e-1

U := 2.050000000

rho := 1.20

cp := 1010.00

m := .15000

C := 151.5000000

T0 := 20.0

P := 60.0

tau := 49.26829268

cp := 1010.00

>    los:=dsolve({lign,T(0)=T0},T(t));  

los := T(t) = 1620/41-800/41*exp(-41/2020*t)

>    los1:=subs(t=t*60,rhs(los));evalf(los1);

los1 := 1620/41-800/41*exp(-123/101*t)

39.51219512-19.51219512*exp(-1.217821782*t)

>    s0:=textplot([[0.75,41.5,"t"],[2.9,24,"  t = 0.82"]],font=[SYMBOL]):

>    s:=textplot([[1.5,32,"Temperaturkurve for flamingokasse"],[1.5,24,"Tidskonstant"],[3.4,3,"[min] , tid "],[0.1,45.0,"[C] , temperatur"],[3.4,24,"min"]],align={RIGHT}):

>    p2:=plot(20.0+19.512*60*t/tau,t=0..0.85,0..45,linestyle=2):

>    p0:=plot(40.0,t=0..4,0..40,linestyle=4):

>    p1:=plot(los1,t=0..4.5,0..40):

>    display(s0,s,p1,p2,p0);

[Maple Plot]

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