Fenics Heat conduction-convection problem
5 months ago by
I am new to Fenics and I am trying to solve the simple heat conduction - convection problem of a cube with a prescribed temperature at left end and convective heat transfer in the right end. Using the fenics tutorial chap 3 I have tried to frame the heat equation pde in variational form.
To verify the pde I have used a simple square plate with a prescribed temperature of 873K at left end and specified a convective zone with htc = 0.01 at the right end of the plate.
The free stream temperature is set at 293K.
All the other constants e.g. thermal conductivity, density, are set at 1.00
I am running the model under steady state conditions by removing the ‘dt’ from the pde.
On solving the problem for T ( temperature at the right edge) I should get the following : K*A * dT/dx = htc * A *(T-To) : T = 873+293 / 2 = 583 K
However fenics provides an answer of 129.9 K
If anyone has any idea where I am going wrong, please let me know. I have pasted the code for reference.
Thank you very much.
### Heat Conduction and Convection problem - simple 2d plate from __future__ import print_function from fenics import * rho = 1.0 # density k = 1.0 # thermal conductivity coefficient C_p = 1.0 # specific heat capacity source = 0 T_left = 873.0 T_amb = 293.0 h_c = 0.01 # convection heat transfer coefficient flux = 0.0 # Create mesh and define function space nx = ny = 4 mesh = RectangleMesh(Point(0, 0),Point(100, 100), nx, ny,"right/left") V = FunctionSpace(mesh, 'P', 2) # Define boundary conditions class LeftEdge(SubDomain): def inside(self, x, on_boundary): tol = 1e-6 return on_boundary and abs(x) < tol class RightEdge(SubDomain): def inside(self, x, on_boundary): tol = 1e-6 return on_boundary and near(x, 100, tol) left_edge = LeftEdge() right_edge = RightEdge() sub_domains = FacetFunction("size_t", mesh, mesh.topology().dim() - 1) sub_domains.set_all(0) right_edge.mark(sub_domains, 1) left_edge.mark(sub_domains, 2) ds = Measure('ds', domain = mesh, subdomain_data = sub_domains) dx = Measure('dx', domain = mesh, subdomain_data = sub_domains) # Applying a prescribed temperature at left edge bc = DirichletBC(V, T_left, left_edge) # Define initial value u_n = project(T_amb,V) # Define variational problem u = TrialFunction(V) v = TestFunction(V) K_o = k / (rho*C_p) # Define the pde #F_tran = (u*v*dx) + (K_o*inner(grad(u), grad(v))*dx) + ((u_n + dt*source)*v*dx) + (dt*h_c*u*v*ds(2)) +(dt*h_c*T_amb*v*ds(2)) F = (u*v*dx) + (K_o*inner(grad(u), grad(v))*dx) - ((u_n + source)*v*dx) + (h_c*u*v*ds(2)) + (h_c*T_amb*v*ds(2)) # removing the dt term for steady state soln. a, L = lhs(F), rhs(F) # Compute solution u = Function(V) t = 0 solve(a == L, u, bc) print(u.vector().array()) print(u.vector().array().min(),u.vector().array().max()) interactive()
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