kill(all)$

maxima: heat:heat conduction (1D),non uniform heat generation:g(r),version 1.0,part3(solid sphere)
peter.vlasschaert@gmail.com,21/07/2021

properties
---------------

k: thermalbconductivity ('fixed')
g : heat generation in the medium (=g(r))
h: convective heat transfer coefficient.
T :temperature
r : location in the solid sphere(r)

model :solid sphere (r), 0≤ r ≤ R
=====

→ ode :

pa:1/r^2*'diff(k*r^2*'diff(T,r),r)+g=0;

pb:g=g0*(r);

depends(T,r);

pc:subst([pb],pa);

≥ final ode 

pc1:ev(pc,diff);

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

maxima: heat:heat conduction (1D),non uniform heat generation 1.0,part2(solid cylinder)
dT/dr =0, infinity of T, see : file above

≥ r=0, dT/dr=0,                    (begin : inside solid sphere)

bc1:diff(T,r)= 0;

≥ r=R,h*(T-Tinf)+k*dT/dr =0 (end : surface solid sphere)

bc2:h*(T-Tinf)+k*diff(T,r) =0 ;

→ heat flow 'through the surface solid sphere"

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

pd1:subst([pb],pd);

solve : model ode solid sphere .
=========================

m:ode2(pc1,T,r);

rem : thermal symmetric : % k2=0

m1:subst([%k2=0],m);

listofvars(m);

use :{bc1,bc2}

→ bc1 (r=0,dT/dr=0)

bc1a:diff(T,r)=0; /* thermal symmetric */

→ bc2(r=R,h*(T-Tinf)+k*dT/dr =0)

bc2a:bc2;
bc21:diff(T,r)=diff(rhs(m1),r);
bc2a:subst([m1,bc21],bc2a);
bc2a:ev(bc2a,diff);
bc2a:subst([r=R],bc2a);

→ find

p1:solve([bc2a],[%k1]);

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

p2:subst(flatten(p1),m1);

heat tranfer conduction through solid 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:subst([r=R],p4b);