kill(all)$

maxima: heat:heat conduction (1D), heat generation:g(T),version 1.0,part1a(plane wall)
peter.vlasschaert@gmail.com,26/07/2021

Plane Wall (2*L), x: -L< x= 0 (center line)< L

properties :
---------------

k: thermalconductivity ('fixed')
g : heat generation in the medium (=g(T))
T :temperature:T(x)
x : location in the wall (-L,+L) = 2L
AA' = mid line ( = center line)

model :plane wall (2*L)
=====

→ ode :

pa:'diff(k*'diff(T,x),x)+g=0;

nonuniform heat generation:Temperature dependent = g(T)

assume(a>0,b>0);

pb:g=a*(1+b*T);

depends(T,x);

pc:subst([pb],pa);

≥ final ode 

pc1:ev(pc,diff);

→ bc : (b)oundary (c)onditions (2*L =length of the wall)

≥ x=0 (begin : wall) ,T=Tamb

bc1:T(-L) = Tamb;

≥ x=L (end : wall),T=Tamb

bc2:T(L)=Tamb;

→ heat flow 'through the wall'

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

pd1:subst([pb],pd);

solve : model ode plane wall (2*L)
=======================
use :pc1

assume(k>0);

m:ode2(pc1,T,x);

listofvars(m);

use :{bc1,bc2}

→ bc1

bc1a:subst([x=-L,T=Tamb],m);

→ bc2

bc2:subst([x=L,T=Tamb],m);

→ find

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

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

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

→temperature  x= -L & L (check for temperature )

p2a:T[-L]=expand(subst([x=-L],rhs(p2)));
p2b:T[L]=expand(subst([x=L],rhs(p2)));

→temperature x=0

p2c:T[0]=expand(subst([x=0],rhs(p2)));

p2c1:ratsimp(p2c);

what when : (x= 0   & b=0) → g=fixed

p2c2:'limit(lhs(p2c1),b,0)=limit(rhs(p2c1),b,0);

heat tranfer to wall:q
================

first step :differentation p2

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

second step: heat flow 'through the wall'

p4:ev(pd1);

p4a:subst([p3],p4);

suppose : A=1, 'surface area plane wall'

A:1;

p4b:ratsimp(ev(p4a));

when x=-L,begin of the wall (domain), heat transfer rate:q

p4c1:subst([x=-L],p4b),expand;

when x=L,end of the wall (domain), heat transfer rate:q

p4c2:subst([x=L],p4b),expand;