Objectives

This lab exercise explores the problem of 2-D heat conduction through the cross-section of an axisymmetric pipe. This problem is recognized as an expansion of the 1-D conduction problem that was analyzed in Lab #2. The present study investigates the effect of azimuthal and radial mesh refinement on the problem solution and convergence rate. Finally, results are compared for the simplified 1-D conduction problem solutions from Lab #2.

Problem Statement

The problem at hand considers heat conduction through a pipe wall. A liquid flows through the central core, subjecting the inner pipe wall to a convective thermal load with h = 20 Btu / hr-ft2-oF and Tr = 1500oF. The pipe is of homogeneous material and construction with a thermal conductivity that varies linearly across the pipe wall such that k(x = 0) = 10 Btu / hr-ft-oF while k(x = L) = 20 Btu / hr-ft-oF. The outside surface of the pipe is held at a constant temperature, Tb = 306.85282 oF.

With the assumption of 2-D heat conduction and with no heat sources, the mathematical description of the problem becomes Laplace's equation.

Since the problem geometry is axisymmetric, any pie-shaped piece of the pipe cross-section may be considered. This requires four boundary conditions. The convective thermal load on the inner surface requires a Robin BC.

The fixed temperature on the outer pipe surface requires a Dirichlet BC.

Finally, since the problem is axisymmetric, the two surfaces from the pie-shaped "cut" should be kept adiabatic. These two boundaries thus require a Neumann BC.

Figure 1 below shows a pictoral representation of the problem statement.


Figure 1: 2-D Heat conduction through an axisymmetric pipe.


Figure 2: The base mesh for the given problem (using Delauney triangulation).

Figure 2 above shows the base mesh over the problem domain using the instructor supplied MATLAB program file. A linear FE basis will be used to find the approximate solution. Thus, the mesh conists of triangles. As seen, the straight sided triangles poorly approximate the curved pipe surface. The convective flux applied to the inner surface should then introduce interpolation error as the boundary flux data will be incorrectly applied along the chord instead of the true curve. This error is quantified in the statement below.

It remains to be seen whether the linear FE basis (k=1) or the data interpolation issues (r-1) will dominate the asymptotic error convergence rate.

As can be seen, the base mesh has 32 elements with 4 elements spanning the theta direction and 8 elements spanning the radial direction. This lab analysis will involve a uniform mesh refinement study along the radial direction, along the theta direction, and then across the entire domain. The results will be compared to the known analytical solution for the simplified case of 1-D conduction.

Discussion of Results


Figure 3: An example of the problem domain meshing under radial mesh refinement.


Figure 4: Temperature solution across the pipe wall for all azimuthal positions under radial mesh refinement.

Figures 3-4 and Tables 1-2 give the error convergence results for a uniform mesh refinement along only the radial direction of the problem domain. Thus, there are only 4 elements across the azimuthal direction while the number of radial elements increases according to the mesh refinement. As seen in Figure 3, this results in a very coarse approximation of the pipe inner surface. As such, the convective boundary fluxes are applied along the chord of the curve instead of the true curved pipe surface. This results in an inner wall temperature solution of approximately 994.356 oF which underpredicts the known analytical solution of 1000 oF. This underprediction is expected since the chords present less surface area for the convective load than the true pipe curve. Thus less convective heat transfer occurs at that boundary.

As seen in the tables, both the Tmax and energy norm converence calculations indicate a convergence slope of 2 for the problem. This would indicate that the linear basis (k=1) is dominating the convergence as opposed to data non-smoothness (r-1).

Table 1: Taylor Series truncation error convergence results for a triangular (k=1) FE basis under radial mesh refinement.
==========================================================
 Elements  R_Mesh        Qmax  (Qmax)change   (Qmax)slope 
__________________________________________________________
     32         8     993.544             0             0 
     64        16     994.151      0.606785             0 
    128        32     994.304      0.153359       1.98427 
    256        64     994.343     0.0384471       1.99596 
    512       128     994.352    0.00961854       1.99898 
   1024       256     994.355    0.00240506       1.99975 
   2048       512     994.355   0.000601292       1.99994 
   4096      1024     994.356   0.000150325       1.99998 
==========================================================

Table 2: Energy norm error convergence results for a triangular (k=1) FE basis under radial mesh refinement.
============================================================
 Elements    R_mesh         ||Q||   change||Q||    Slope(Q) 
____________________________________________________________
     32           8    2.08342e+007           0           0 
     64          16    2.08513e+007     17111.3           0 
    128          32    2.08557e+007     4324.71     1.98427 
    256          64    2.08567e+007     1084.21     1.99596 
    512         128    2.08570e+007     271.243     1.99898 
   1024         256    2.08571e+007     67.8226     1.99975 
   2048         512    2.08571e+007     16.9564     1.99994 
   4096        1024    2.08571e+007     4.23915     1.99998 
============================================================


Figure 1a: Planform view of temperature solution (ksource=0.01 and ksink=1.0).


Figure 6: Temperature solution across the pipe wall for all azimuthal positions under azimuthal mesh refinement.

Figures 5-6 and Tables 3-4 give the error convergence results for a uniform mesh refinement along only the azimuthal direction of the problem domain. Thus, there are only 8 elements across the radial direction while the number of azimuthal elements increases according to the mesh refinement. As seen in Figure 5, this results in a very good approximation of the pipe inner surface. As such, the convective boundary fluxes are applied over the correct area resulting in an inner wall temperature solution of approximately 999.186 oF which closely predicts the known analytical solution of 1000 oF. It is interesting to note that, even thought there are only 8 elements across the radial direction, the radial temperature profile shows very little deviation from the previous study as seen in Figure 2.

Again, the tables indicate that both the Tmax and energy norm converence calculations yield a convergence slope of 2 for the problem, indicating that the linear basis (k=1) is dominating the convergence as opposed to data non-smoothness (r-1).

Table 3: Taylor Series truncation error convergence results for a triangular (k=1) FE basis under azimuthal mesh refinement.
==============================================================
 Elements  Theta_Mesh        Qmax  (Qmax)change   (Qmax)slope 
______________________________________________________________
     32             8     993.544             0             0 
     64            16     997.788       4.24352             0 
    128            32     998.841       1.05296       2.01082 
    256            64     999.103      0.262747        2.0027 
    512           128     999.169     0.0656561       2.00067 
   1024           256     999.186     0.0164121       2.00017 
   2048           512     999.190    0.00410291       2.00004 
   4096          1024     999.191    0.00102572       2.00001 
==============================================================

Table 4: Energy norm error convergence results for a triangular (k=1) FE basis under azimuthal mesh refinement.
==============================================================
 Elements  Theta_Mesh         ||Q||   change||Q||    Slope(Q) 
______________________________________________________________
     32             8  2.08342e+007             0           0 
     64            16  2.10553e+007        221054           0 
    128            32  2.11105e+007       55265.6     1.99995 
    256            64  2.11244e+007       13816.6     1.99998 
    512           128  2.11278e+007       3454.15           2 
   1024           256  2.11287e+007       863.538           2 
   2048           512  2.11289e+007       215.885           2 
   4096          1024   2.1129e+007       53.9711           2 
==============================================================

Tables 5 and 6 list the results of a full,uniform mesh refinement study. Since the mesh refinement occurs in both the azimuthal and radial directions, the number of elements increases by a factor of four at every iteration. The results indicate an inner wall temperature of 999.994 F with a convergence slope of 2 with 1.9% convergence accuracy. This solution required a computing time of several minutes for the last case of 32,768 elements. The azimuthal mesh refinement, in comparison, generated an equivalent max temperature with 0.1% convergence accuracy using only a 4,096 element mesh.

Table 5: Taylor Series truncation error convergence results for a triangular (k=1) FE basis under full, uniform mesh refinement.
===============================================
 Elements     Qmax  (Qmax)change   (Qmax)slope 
_______________________________________________
     32    993.544             0             0 
    128    998.393       4.84894             0 
    512    999.599       1.20553         2.008 
   2048    999.9        0.300969       2.00198 
   8192    999.975     0.0752165       2.00049 
  32768    999.994     0.0188025       2.00012 
===============================================

Table 6: Energy norm error convergence results for a triangular (k=1) FE basis under full, uniform mesh refinement.
==================================================
 Elements         ||Q||   change||Q||    Slope(Q) 
__________________________________________________
     32    2.08342e+007             0           0 
    128    2.10724e+007        238209           0 
    512    2.11321e+007       59614.8     1.99849 
   2048    2.11470e+007       14907.7     1.99961 
   8192    2.11507e+007       3727.19      1.9999 
  32768    2.11516e+007       931.813     1.99998 
==================================================

Finally, Table 7 relists the results from the 1-D conduction problem previously analyzed in Lab 2. In terms of computational savings, it is clearly of benefit to model the 2-D axisymmetric case as a 1-D model.

Table 7: Lab #2 1-D conduction Taylor Series truncation error convergence results for a linear FE basis.
===============================================
 Elements      Qmax  (Qmax)change  (Qmax)slope 
_______________________________________________
      8    999.7960             0            0 
     16    999.9489      0.152892            0 
     32    999.9872     0.0383301      1.99597 
     64    999.9968    0.00958926      1.99899 
    128    999.9992    0.00239774      1.99975 
    256    999.9998   0.000599461      1.99994 
    512   1000.0000   0.000149867      1.99999 
===============================================

Conclusion

In conclusion, a verification study has been completed for radial heat conduction through a pipe flowing hot fluid. The convective heat transfer at the inner boundary has introduced additional interpolation error in the data. This interpolation error does not show itself through a degredation of the error convergence since the problem exists in the application of a boundary condiction. Instead, the engineer must recognize the difficiency and compensate to avoid incorrect temperature solutions. The most efficient FE solution then requires a fine mesh in the azimuthal direction with a more course mesh in the radial direction.