SAFPWR home page
Thermal conduction model in pipes
The model (pic_14 ) is basically similar as that of the fuel rod, except that:
-no heat source in conductor
-no heat loss at outer pipe surface (perfect insulation)
-uniform k and c
-arbitrary number i8 of annular regions of arbitrary thickness.
In addition, the temperature distribution in each region is supposed to vary linearly with x2 instead of r2, where x ≡ r89 - r.
Thus, the temperature distribution in region 1 is normal to the outer boundary and automatically satisfy the outer boundary condition.
The model has been tested against higher order solution and also against the following analytical solution.
Benchmark test on planar form
For a plane slab (
r89 >> (r89 - r01) the thermal conservation equation
(1) accepts the particular separable solution (2, 3).
For example, for α taken as (4) with Δx = 0.5, and a large film transfer coefficient hf,
(7) gives the evolution of the driving edge temperature u5 and
(8) the corresponding heat flux represented here by -du/dx.
A test problem was setup with 4 regions (x12=.192, x23=.354, x34=.462, x45=slab thickness= .5) k=0.2, c=1, k/h=2e-11.
The initial distribution is taken as u = cos(α x) in order to avoid any departure from the particular solution.
Chart1 compares the numerically calculated
q45 with its analytical value
qth(7) for 3 values of
dsec.
It is observed that, for dsec=0.1, the results reproduce fairly well the analytical reference qth (theoretical q), which means that the spatial discretization in 4 regions with thickness diminishing towards the center is adequate.
Of course, this is not a severe test for the quadratic fitting approximation because the cos distribution can easily be fitted by a suite of arcs linear with x2
With dsec= 1 s, the error is still acceptable.
The decay time is 1/β = 2.02 s and a time step of 5 s is visibly too large, but the solution remains perfectly stable thanks to the implicit time discretization.