CHAPTER 6: Simulation of the Structural Response of a Composite SRM
6.3 Thermal Penetration Zone Element Sizing
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In the current case however, the required nozzle wall temperature and pressure distributions were calculated using Flow Model T. Temperatures and pressures were recorded at specific points in the Flow Model T results file, and spatial functions were then used to prescribe the variation of the two parameters along the wetted surfaces of the thermal and structural models.
The overall modelling approach is outlined in Figure 6.4.
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Here T is the temperature in the medium at a given position in the direction of the x dimension and at a time t, and αis the thermal diffusivity of the medium. Although strictly speaking, heat conduction within rotational structures occurs with respect to the cylindrical form of Equation 6.1, as the anticipated depth of thermal penetration was significantly smaller in dimension than the diameter at which it would occur, the application of Cartesian coordinates was deemed an acceptable approximation.
It can be shown analytically that the penetration depth, γ, which defines the distance at which the local temperature is just 0.1% above the slab’s ambient temperature at a time t may be solved as
αt
γ =4 (6.2)
Despite being independent of the magnitude of the applied thermal loading, the solution provided by Equation 6.2 predicts the thermal penetration depth in response to the instantaneous application of such loading. In relation to this condition, Figure 6.5 shows temperature histories at various points along the SRN2 wall which were derived from the flow model considered in the previous chapter. As can be seen, although at some positions there is a rapid rise in temperature, the rate of this increase is clearly not instantaneous. The specification of the instantaneous loading condition would therefore predict a slightly greater depth of penetration than would be encountered in reality, leading to a more conservative estimation.
Using the diffusivity of the 3D C-C material in the radial direction, calculated as 8.00x10-6 m2/s, and for a total simulation time period of 0.3 s, the penetration depth was calculated to be 6.2 mm from the heated surface. To account for any two-dimensional conduction occurring adjacent to contours with sharp radii or corners, the TPZ element band was specified in Flow Model T and Structural Model T to be 8 mm deep.
The next parameters that needed be resolved were the dimensions of the models’ TPZ elements in the directions normal and parallel to the heated surface. As the highest thermal gradient would exist in the normal direction, element sizing in this direction was of particular importance. A sensitivity study was therefore undertaken to determine the effect of element size on the transient
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thermoelastic response of an internally heated cylinder using the ADINA-T and ADINA modules. In doing so, the optimal dimensions for the TPZ elements in Flow Model T and Structural Model T could be estimated.
Figure 6.5 SRN2 wall temperature histories at points of interest.
As the transient thermal gradients to be encountered in the TPZ were directly proportional to the rate at which the temperature on the nozzle surface would rise, the loading condition applied in the sensitivity study was derived on the basis of the most severe temperature evolution encountered in the results shown in Figure 6.5. This was indicated to occur at a point on the exit cone positioned roughly one quarter down its length and the temperature history at this point is shown in Figure 6.6.
The transient loading condition for the sensitivity study was approximated as a ramp function based upon the parameters dT, representing the change in temperature, and dt, representing the time over which this change occurs, shown in Figure 6.6. These parameters were measured to be 2212 K and 0.014 s, respectively. It should be noted that the flow model from which this data was extracted treated the nozzle wall as an adiabatic boundary, which would inevitably lead
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to an overestimation of temperature magnitudes at the wall, and in the case of this study, a conservative estimation of thermal penetration.
Figure 6.6 Temperature history at point of maximum thermal loading rate.
The base level element size in the radial direction was calculated as the 99.9% thermal penetration depth in the cylinder subjected to the instantaneous application of 2212 K to its inner surface after a period of 0.014 s. In this manner, the maximum element dimension in the radial direction, ∆rmax, was established to be 1.33 mm. For higher mesh densities, the particular radial dimension, ∆r , was governed by a resolution factor, N, such that
N r= ∆rmax
∆ (6.3)
The particular axial dimension of the thermal element was governed by a specified aspect ratio of 5. The test cylinder was 100m in length, and had an internal diameter of 450 mm and a wall thickness of 12 mm – the latter two dimensions being equivalent to the dimensions at the point of interest on the exit cone. Linear interpolating 4-noded axisymmetric conduction elements were used to discretize the thermal model to avoid errors typically generated by the quadratic interpolation associated with the 9-noded conduction elements, when subjected to high thermal
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gradients. To establish the sensitivity of the response of the test cylinder, thermal and structural simulations were carried out for resolution factors of 1, 2, 3 & 4. Results for the predicted temperature and hoop stress distributions within the first 8 mm of the heated boundary are shown at three instances in time in Figures 6.7-6.12.
Figure 6.7 Test cylinder temperature profile at 0.07 s.
Figure 6.8 Test cylinder hoop stress profile at 0.07 s.
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Figure 6.9 Test cylinder temperature profile at 0.083 s.
Figure 6.10 Test cylinder hoop stress profile at 0.083 s.
As expected, the mesh characterised by a resolution factor of 4 provides the smoothest and most representative thermal and hoop stress distributions at each time, especially whilst loading is still in progression at t = 0.07 s. The coarser discritizations stipulated by the lower resolution factors of 1 & 2 induce a slight overestimation of the penetration depth as demonstrated by Figure 6.9, resulting in the generation of slightly higher tensile hoop stresses in the cool region of the
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cylinder. By a time of 0.3 s however, the temperature and hoop stress profiles predicted by all four mesh densities are shown to be in good agreement. It is also interesting to note the small variation in the maximum compressive hoop stress predicted to occur at the heated surface by each mesh. Clearly however, the most accurate representation of the transient thermal penetration and stress generation is provided by a mesh resolution factor of 4.
Figure 6.11 Test cylinder temperature profile at 0.3 s.
Figure 6.12 Test cylinder hoop stress profile at 0.3 s.
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Of course, the accuracy of any numerical solution is generally related to its discretization resolution and hence computational size. In this respect, the time taken to compute solutions to the thermal and structural problems for the range of resolution factors were investigated to evaluate the cost of each solution in relation to its accuracy. The results of this assessment are displayed in Figure 6.13 and demonstrate the dramatic increase in the computational time required for an increase in resolution factor. A significant difference in the thermal and structural solution times that manifests with an increase in resolution factor can also be observed.
Figure 6.13 Variation of solution times with resolution factor.
It is useful to consider the results reflected in Figure 6.13 in the context of the relative solution error existing between hoop stress predictions using a resolution factor of 4, and those computed for factors of 1, 2 & 3. Figure 6.14 shows a comparison in hoop stresses predicted at the inner, heated surface of the cylinder, whilst Figure 6.15 makes this comparison at the outer, cool surface of the cylinder, for the three solution times.
These results demonstrate the decrease in relative error with respect to an increase in solution time and an increase in resolution factor. Interestingly, the decrease in the rate at which error decreases with resolution factor suggests that mesh independence in the solution is nearing.
Figure 6.15 also illustrates the significant hoop stress error at the cold surface, generated by the artificially deep thermal penetration zone arising from coarse discretization, particularly for
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earlier solution times. Although very large discrepancies are shown to exist for the resolution factor 4 solution, the low magnitude of the predicted stresses mitigated concern in this regard.
More significantly however, the relative error in the maximum stresses predicted at the hot surface is shown to be small, even for a resolution factor of 1 and at the earliest solution time.
Figure 6.14 Relative hoop stress error at hot surface.
Figure 6.15 Relative hoop stress error at cold surface.
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Three issues were considered in determining the most suitable TPZ element size. These comprised the cost of employing each mesh resolution factor in relation to the accuracy of the resulting solution, the restrictions that the TPZ element size placed on discretization in the flow models, and the capacity of the computational resources available for use. After careful consideration of these issues, a resolution factor of 3 was selected to scale the discretization of the TPZ bands in Flow Model T and Structural Model T.
6.4 Flow Modelling