FEATool Multiphysics
v1.17.0
Finite Element Analysis Toolbox
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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.
[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
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