kill(all)$

maxima: heat:heat conduction (1D), heat generation:g(T),version 1.0,part1b(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))
θ :temperature:T-Tamb
x : location in the wall (-L,+L) = 2L
AA' = mid line ( = center line)
Tamb = fixed

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

→ ode :

pa:'diff(k*'diff(θ,x),x)+g=0;

nonuniform heat generation:Temperature dependent = g(T)

assume(a>0,b>0);

pb:g=a*(1+b*θ);

depends(θ,x);

pc:subst([pb],pa);

≥ final ode 

pc1:expand(ev(pc,diff)/k);

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

≥ x=0 (center line) 

bc1:diff(θ,x)= 0;

≥ x=L → θ=0

bc2:θ(L)=0;

→ 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,θ,x);

listofvars(m);

use :{bc1,bc2}

→ bc1

bc1a:diff(rhs(m),x)=0;

→ bc2

bc2:subst([x=L],rhs(m))=0;

→ find

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

p1a:flatten(p1)[1];

p1b:flatten(p1)[2];

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

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

p2:trigreduce(p2);

p2:trigsimp(p2);

p21:θ = T-Tamb;

p22:subst([p21],p2);

p23:p22+Tamb;

rem : T dependent on x

depends(T,x);

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

p2a:θ[-L]=trigsimp(ratsimp(subst([x=-L],rhs(p2))));
p2aa: T[-L]=subst([x=-L],rhs(p23));
p2b:θ[L]=trigsimp(ratsimp(subst([x=L],rhs(p2))));
p2bb:T[L]=subst([x=L],rhs(p23));

temperature x=0

θ[0]=expand(subst([x=0],rhs(p2)));
T[0]=subst([x=0],rhs(p23));

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

first step :differentation p2

p3:'diff(θ,x)=trigsimp(diff(rhs(p2),x));

p3:subst([p21],p3);

p3:ev(p3,diff); /*Tamb =fixed*/

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;

rem : use ' simplifications wth trig function', 'trigsimp & trigreduce'