simple : heat conduction (1D), part 3, hollow sphere ,version1
peter.vlasschaert@gmail .com , 17/07/2021

kill(all)$

packages ,option
=============

load(lrats)$ratprint:false$

The hollow sphere (r1 ≤ r ≤ r2)
=======================

ode : (o)rdinary (d)ifferential (e)quation
-------

p:'diff(r^2*'diff(T,r),r)=0;

bc :(b)oundary (c)onduction (see:Figure 1)
-----
k :thermal conductivity

→ r = r1  ,(h[1],Tinf1,k)
     h[1]:convective heat transfer.
     Tinf1:hot temperature

bc1:h[1]*(T-Tinf1) - k*'diff(T,r) =0;;/*inner wall :hot */

→ r = r2  ,(h[2],Tinf1,k)
     h[1]:convective heat transfer.
     Tinf1:hot temperature

bc2:h[2]*(T-Tinf2)-k*'diff(T,r)=0;;/*outer wall cold*/

solve : ode.
=========

→ solution (general )

p1a:ode2(p,T,r);

depends(T,r);

p1a1:ev(p,nouns);

p1a2:ode2(p1a1,T,r);

rem :Maxima :log(r)  Math:ln(r)

≥ unknown

p1a3:makelist(listofvars(rhs(p1a2))[i],i,1,length(listofvars(rhs(p1a2)))-1);;

≥ variable ' one dimension '

p1a4:listofvars(rhs(p1a2))[length(listofvars(rhs(p1a2)))];

≥ differentation 'p1a2' , dT/dr ?

p1b:'diff(T,r)=diff(rhs(p1a2),r);

→ solution with bc condition(bc1,bc2) => (bc1a,bc2a)

bc1a:subst([p1b],bc1);

bc1a:subst([p1a2],bc1a);

bc1a:subst([r=r1],bc1a);

bc2a:subst([p1b],bc2);

bc2a:subst([p1a2],bc2a);

bc2a:subst([r=r2],bc2a);

≥ [%k1,%k2] ,find from {bc1a,bc2a}

p1c:flatten(solve([bc1a,bc2a],p1a3));

→ final solution:solve ode & solution with bc condition(bc1,bc2) => (bc1a,bc2a)

p1e:subst(p1c,p1a2);

p1e1:ratsimp(rhs(p1e));

→ temperature ( r=r1,linside hollow sphere)

p1e1a:ratsimp(subst([r=r1],p1e));

p1e1a1:rhs(p1e1a);

temperature ( r=r2,outside hollow sphere)

p1e1b:ratsimp(subst([r=r2],p1e));

temperature function : to analyse the 'hollow sphere, 'Temperature distribution hollow sphere'

TT[r]:=''p1e1;

we want to find temperature @( r2-r1)/2 ,the middle of the hollow sphere

temperature((r2-r1)/2) =TT[(r2-r1)/2];

Heat transfer rate:inner to the outer sphere (=L)
=====================================
A : surface inner sphere (r1)
Q :h[1]*(Tinf1-p1e1a)

rem :outer sphere (r2)

pa:A=4*%pi*r1^2;
pb:Q=h[1]*(Tinf1-p1e1a1);

p1f:q=ratsimp(subst([pa,pb] ,A*Q));