**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-ft^{2}-^{o}F and *T _{r} *= 1500

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 ^{o}F which underpredicts the known analytical solution of 1000 ^{o}F. 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 (k_{source}=0.01 and k_{sink}=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 ^{o}F which closely predicts the known analytical solution of 1000 ^{o}F. 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.