kill(all)$

maxima: heat:heat conduction (1D),variable : k,version 1.0,part3(hollow sphere)
peter.vlasschaert@gmail.com,21/07/2021

variable thermal conductivity =k
=========================
depends on the conducting medium

model :hollow sphere (r), r1≤ r ≤ r2
=====

→ ode :

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

pb:k=α*(1+β*r);

depends(T,r);

pc:subst([pb],pa);

≥ final ode 

pc1:ev(pc,diff);

→ bc : (b)oundary (c)onditions (r1≤ r≤r2)

≥ r=r1 (begin : inner sphere)

bc1:T(r1) = T1;

≥ r=r2 (end : outer sphere)

bc2:T(r2)=T2;

→ heat flow 'through the surface hollow sphere"

pd:q=-k*A*diff(T,r);

pd1:subst([pb],pd);

solve : model ode hollow sphere .
=========================

m:ode2(pc1,T,r);

listofvars(m);

use :{bc1,bc2}

→ bc1 (r=r1,T=T1)

bc1a:subst([r=r1,T=T1],m);

→ bc2(r=r2,T=T2)

bc2:subst([r=r2,T=T2],m);

→ find

p1:solve([bc1a,bc2],[%k1,%k2]);

fnal solution : ode+bc
================

p2:subst(flatten(p1),m);

when β=0

p2a:'limit(p2,β,0)=rhs(ratsimp(limit(p2,β,0)));

heat tranfer through hollow sphere:q
===========================

first step :differentation p2

p3:'diff(T,r)=diff(rhs(p2),r);

second step : heat flow 'through the hollow sphere'

p4:ev(pd1);

p4a:subst([p3],p4);

 :A=4*π*r^2, 'surface area:hollow sphere'

A:4*%pi*r^2;

p4b:fullratsimp(ev(p4a));

p4c:facsum(p4b);

when β =0
--------------

p4d:'limit(p4c,β,0)=rhs(ratsimp(limit(p4c,β,0)));

p4e:lhs(p4d)=facsum(rhs(p4d));