FEATool Multiphysics  v1.16.1 Finite Element Analysis Toolbox
ex_heattransfer1.m File Reference

## Description

EX_HEATTRANSFER1 2D ceramic strip with radiation and convection.

[ FEA, OUT ] = EX_HEATTRANSFER1( VARARGIN ) 2D heat transfer of a ceramic strip with both radiation and convection on the top boundary.

  _                q = h*(T_inf-T) + epsilon*sigma*(T_inf^4-T^4)
^            +------------------+
|            |                  |
|            |                  |
0.01m  T=900C  |                  |  T=900C
|            |                  |
|            |                  |
v            +------------------+
dt/dn = 0
|<---- 0.02m ----->|


The ceramic has a thermal conductivity of 3 W/mC and the sides are fixed at a temperature of 900C while the bottom boundary is insulated. The surrounding temperature is 50C. The top boundary is exposed to both natural convection (with a film coefficient h=50W/m^2K) and radiation (with emissivity epsilon=0.7 and the Stefan-Boltzmann 5.669e-8 W/m^2K^4). The solution is sought at three points along the vertical symmetry line.

Reference
  [1] Holman, J. P., Heat Transfer, Fifth Edition, New York: McGraw-Hill,
1981, page 96, Example 3-8.


Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
hmax        scalar {0.001}         Grid cell size
igrid       scalar {0}/1/2         Cell type (0=quadrilaterals, 1=triangles,
solver      string fenics/{}       Use FEniCS or default solver
ischeme     scalar {0}             Time stepping scheme (0 = stationary)
sfun        string {sflag1}        Finite element shape function
iplot       scalar {1}/0           Plot solution (=1)
.
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct


# Code listing

 cOptDef = { 'hmax',     0.001;
'igrid',    0;
'solver',   '';
'ischeme',  0;
'sfun',     'sflag1';
'iplot',    1;
'tol',      0.01;
'fid',      1 };
[got,opt] = parseopt(cOptDef,varargin{:});
if( opt.ischeme==2 && ~got.tol )
opt.tol = 0.05;
end

% Geometry definition.
gobj = gobj_rectangle( 0, 0.02, 0, 0.01 );
fea.geom.objects = { gobj };

% Grid generation.
switch opt.igrid
case 0
fea.grid = rectgrid( round(0.02/opt.hmax), round(0.01/opt.hmax), [0 0.02;0 0.01] );
case 1
fea.grid = gridgen( fea, 'hmax', opt.hmax, 'fid', opt.fid );
case 2
fea.grid = rectgrid( round(0.02/opt.hmax), round(0.01/opt.hmax), [0 0.02;0 0.01] );
fea.grid = quad2tri( fea.grid, 1 );
end

% Problem definition.
fea.sdim  = { 'x', 'y' };             % Space coordinate name.
fea = addphys( fea, @heattransfer );  % Add heat transfer physics mode.
fea.phys.ht.sfun = { opt.sfun };      % Set shape function.

% Equation coefficients.
fea.phys.ht.eqn.coef{3,end} = 3;      % Thermal conductivity.

% Boundary conditions.
fea.phys.ht.bdr.sel = [3 1 4 1];
fea.phys.ht.bdr.coef{1,end}   = { [] 900+273 [] 900+273 };
fea.phys.ht.bdr.coef{4,end}{3}{2} = 50;
fea.phys.ht.bdr.coef{4,end}{3}{3} = 50+273;
fea.phys.ht.bdr.coef{4,end}{3}{4} = 0.7*5.669e-8;
fea.phys.ht.bdr.coef{4,end}{3}{5} = 50+273;

% Parse physics modes and problem struct.
fea = parsephys(fea);
fea = parseprob(fea);

% Compute solution.
if( strcmp(opt.solver,'fenics') )
fea = fenics( fea, 'fid', opt.fid, ...
'tstep', 0.1, 'tmax', 1, 'ischeme', opt.ischeme );
else
if( opt.ischeme<=0 )
fea.sol.u = solvestat( fea, 'fid', opt.fid, 'init', {'T0_ht'} );
else
[fea.sol.u,fea.sol.t] = solvetime( fea, 'fid', opt.fid, 'init', {'T0_ht'}, ...
'tstep', 0.1, 'tmax', 1, 'ischeme', opt.ischeme );
end
end

% Postprocessing.
if( opt.iplot>0 )
postplot( fea, 'surfexpr', 'T', 'isoexpr', 'T' )
title('Temperature, T')
end

% Error checking.
T2_sol = evalexpr( 'T', [0.01;0.01], fea );
T2_ref = 984;
T5_sol = evalexpr( 'T', [0.01;0.005], fea );
T5_ref = 1064;
T8_sol = evalexpr( 'T', [0.01;0], fea );
T8_ref = 1088;
out.err  = abs([T2_sol-T2_ref T5_sol-T5_ref T8_sol-T8_ref])./[T2_ref T5_ref T8_ref];
out.pass = all(out.err<opt.tol);

if( nargout==0 )
clear fea out
end