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| {{UFR|front=UFR 3-07|description=UFR 3-07 Description|references=UFR 3-07 References|testcase=UFR 3-07 Test Case|evaluation=UFR 3-07 Evaluation|qualityreview=UFR 3-07 Quality Review|bestpractice=UFR 3-07 Best Practice Advice|relatedACs=UFR 3-07 Related ACs}}
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| {{Status|checked=no|by= |date= }}
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| = Natural and mixed convection boundary layers on<br />vertical heated walls (B) =
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| Underlying Flow Regime 3-07 <font size="-2" color="#888888"> © copyright ERCOFTAC 2004</font>
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| = Test Case =
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| == Brief description of the study test case ==
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| The test case consists of either upward or downward flow of air in a heated tube. The inside diameter of the tube is 76.12mm. The unheated length before the heated test section is either 3.805 or 3.926m long, depending on whether the air flow is downwards or upwards respectively. The heated test section has a length of 8.08m.
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| The parameters that define the flow regime are the inlet Reynolds number, Grashof number (based on wall heat flux), buoyancy parameter, wall temperature and ambient temperature. The principal derived quantity, by which the success, or failure, of CFD calculations is judged, is Nusselt number. This is derived from the heat flux, wall temperature and bulk air temperature, defined as
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| The numerical results are mainly presented in terms of Nusselt number development along with the experimental data for each test condition.
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| == Test Case Experiments ==
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| The basic test section is the same for all cases. The heated length is 8.08m long and consists of two stainless steel tube sections carefully welded together. The main upper part is 6.571m long with inside diameter 76.12mm and wall thickness 1.90mm. The lower part is 1.51m long and has the same inside diameter as the upper part but a slightly different wall thickness of about 1.83mm.
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| The heated length is resistance heated using AC electrical power from the mains supplied by a variable voltage system. 58 chromel-alumel thermocouples were resistance welded on the outside tube wall of the heated section. In total there are 43 measuring positions along the test section, 38 positions with one thermocouple at each position and another five positions with four thermocouples at each position to check the circumferential uniformity of temperature around the tube.
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| In the experiments, the thermal boundary condition was, strictly speaking, that of a slightly non-uniform prescribed heat flux pattern as a result of the temperature dependence of the resistivity of the stainless steel tube. However, a uniform wall heat flux was attempted by carefully selecting a tube of very uniform wall thickness. Careful ultra-sonic measurements of tube wall thickness were made to check its uniformity.
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| Accurate hot film calibration was a necessary pre-cursor to flow rate measurement. A resulting explicit equation provided the required correction for each case.
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| Appendix 5 of Li (1994) provides estimates of the accuracy of the wall temperature, the ambient air temperature (the air temperature at each section is derived from the known heat flux distribution), the local heat flux and the local Nusselt number. The random errors in temperatures, local heat flux and local Nusselt number are 0.45<sup>o</sup>C, 1%, and generally less than 5% respectively.
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| == CFD Methods ==
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| '''''5.1 Li'''''
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| Li (1994) describes calculations of the experiments using the Launder-Sharma two equation k-ε turbulence model (Launder and Sharma, 1974). Temperature dependent property functions for viscosity, thermal conductivity and density are incorporated.
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| In his calculations, the differential equations were discretised following the finite volume/finite difference scheme of Leschziner (1982) under which the differential equations are integrated over a control volume prior to the application of the finite difference approximation. The discretisation procedure converts the set of differential equations into a set of algebraic equations which are then solved using the Gaussian elimination method. Solution of the parabolic equation set is obtained by starting with an assumed distribution of the dependent variables at the first station and marching in the streamwise direction.
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| Li (1994) does not describe the numerical mesh he uses. However, as is well known, it is necessary to solve the equations down to y<sup>+</sup> values below about two everywhere on the walls, otherwise the Launder-Sharma model gives poor predictions.
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| '''''5.2 Gerasimov'''''
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| Standard wall functions usually give unreliable prediction of heat transfer, in conjunction with any turbulence model. UMIST (Craft et al, 2002) have recently developed an analytic wall function that includes a more physical representation of the near-wall region compared with the standard approach. The reader is referred to Craft et al (2002) or to the Application Challenge documentation provided by British Energy and Magnox Electric for more details. Broadly, the analytic wall function incorporates the effects of,
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| (a) convection parallel and normal to the wall,
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| (b) fluid property variation across the wall layer, including a parabolic variation of molecular viscosity across the viscous sub-layer,
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| (c) the inclusion of any pressure gradient and buoyancy force,
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| (d) viscous sub-layer thickening and thinning,
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| (e) a Prandtl number which may not be close to unity (e.g. water at room conditions) and its significant variation with temperature.
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| Gerasimov (2002) has undertaken calculations of the test case using the analytic wall function approach. He used the TEAM code (Huang and Leschziner, 1983), which uses TDMA to resolve a set of discretised equations. Initially, he repeated some of Li (1994)s cases, using the Launder-Sharma low Reynolds number turbulence model, for both buoyancy-opposed and buoyancy-aided flows and results were identical. The predictions using the analytic wall function are compared with the experimental results and results from solving the Launder-Sharma k-ε turbulence model up to the wall. Note that the analytic wall function is used in conjunction with the solution of the low-Reynolds number equations outside the wall layer.
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| Entry conditions correspond to isothermal fully developed flow. The outlet boundary was positioned well downstream of the heated section to ensure that the results are not affected by the outlet boundary condition. Thermal boundary conditions of uniform wall heat flux are imposed.
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| '''Regarding grids for buoyancy-affected air flows in pipes:'''
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| WF calculations grid: 12<span lang="EN-US" style="layout-grid-mode: line"><font face="Symbol">´</font></span>190 nodes
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| LRN calculations grid: 12 sections of 45<span lang="EN-US" style="layout-grid-mode: line"><font face="Symbol">´</font></span>61 nodes, superimposed by 6 nodes, which is equivalent to 45<span lang="EN-US" style="layout-grid-mode: line"><font face="Symbol">´</font></span>660 nodes. The whole domain was split into 12 sections to make it converge faster.
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| Regarding grid independence:
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| WF calculations: number of nodes in axial direction; 300, 250, 200, 175 and 150 were tried. There was no difference between the results of the 300 to 175 node cases. 190 was chosen for reassurance. In the radial direction, it is normal to have 10 nodes between the axis and the near-wall cell to capture all of the outer gradients within the mean flow. Most of the variation for the number of nodes in the radial direction was related to the change of the near-wall cell size. This variation is shown in the Figures. For higher Reynolds numbers (50000-100000) the number of nodes in the radial direction, within the main flow, was increased to 15. Results were pretty much the same as with 10-11 nodes.
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| LRN Calculations: At the highest Reynolds number, the first node needed to be placed within y+=1. This is why it was necessary to have 45 nodes in the radial direction. In the axial direction, 61 nodes were needed for each section to obtain grid-independent results and a reasonable convergence rate (to avoid extremely elongated cells).
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| The calculations were repeated with the domain fully resolved (without sectioning) as well, but convergence was very slow. The sectioning was suitable only forbuoyancy-aided flows, as velocity vectors are always aligned with the axis of the domain.
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| The buoyancy-opposed flows, however, required calculations throughout the whole domain due to the deviation of velocity vectors from the axial direction or even due to possible reversal of the flow.
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| <font size="-2" color="#888888">© copyright ERCOFTAC 2004</font><br />
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| ----
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| Contributors: Mike Rabbitt - British Energy
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| {{UFR|front=UFR 3-07|description=UFR 3-07 Description|references=UFR 3-07 References|testcase=UFR 3-07 Test Case|evaluation=UFR 3-07 Evaluation|qualityreview=UFR 3-07 Quality Review|bestpractice=UFR 3-07 Best Practice Advice|relatedACs=UFR 3-07 Related ACs}}
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| [[Category:Underlying Flow Regime]]
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| {{UFR|front=UFR 3-07|description=UFR 3-07 Description|references=UFR 3-07 References|testcase=UFR 3-07 Test Case|evaluation=UFR 3-07 Evaluation|qualityreview=UFR 3-07 Quality Review|bestpractice=UFR 3-07 Best Practice Advice|relatedACs=UFR 3-07 Related ACs}} | | {{UFR|front=UFR 3-07|description=UFR 3-07 Description|references=UFR 3-07 References|testcase=UFR 3-07 Test Case|evaluation=UFR 3-07 Evaluation|qualityreview=UFR 3-07 Quality Review|bestpractice=UFR 3-07 Best Practice Advice|relatedACs=UFR 3-07 Related ACs}} |
Natural and mixed convection boundary layers on
vertical heated walls (B)
Underlying Flow Regime 3-07 © copyright ERCOFTAC 2004
Evaluation
Comparison of CFD calculations with Experiments
The predictions of heat transfer behaviour for cases in which (a) the heat flux distribution applied is irregular and (b) uniform are virtually the same (Li, 1994). Therefore most calculations have been performed with a uniform heat flux, using the mean experimental value rather than the actual slightly non-uniform distribution.
6.1 Downward Flow
This case is buoyancy-opposed. Simulations with the Launder-Sharma two-equation k-ε turbulence model have been undertaken to cover the range of flows from forced convection to strongly buoyancy-influenced conditions for the heated downward flow of air (Li, 1994). The predicted Nusselt number development is in particularly good agreement with the experimental data for the cases of Re<9000, when the buoyancy effects are strong. There are discrepancies between the predictions and the experimental data for the cases with relatively weak buoyancy influences but significant property variation. These are probably due to the over-response of the model to the laminarisation caused by the property variation. In these cases the model under-predicts the Nusselt number by up to 20% compared with the experimental data.
Generally, the Nusselt number for air flow in the tube decreases along the heated section, which can be explained by the decreasing turbulent viscosity with the increase of x/D. In downward flow the buoyancy forces generally increase the turbulence level, and hence cause enhancement of heat transfer.
Gerasimov (2002) describes several of the downward flow cases obtained by Li (1994), using the analytic wall function. Figure 1 shows results for a case in which the Launder-Sharma model under-predicts heat transfer by ~20% (Re=14815). The analytic wall function performs slightly better than the low-Reynolds number model solution down to the wall for this case.
Figure 2 shows another case (Re=7044), which is one for which Li (1994) shows a good prediction using the Launder-Sharma model. Results from the analytic wall function are just as good. Figure 3 shows results from cases in which the Reynolds number is low, and enhancement of heat transfer, compared with forced flow, is significant. The results are good.
Results using the analytic wall function are good, and almost independent of the size of the wall layer.
6.2 Upward Flow
This case is buoyancy-aided. The simulated results, using the Launder-Sharma low Reynolds number model, agree well with the experimental data under the conditions where the flow is laminarised and the heat transfer is severely impaired by buoyancy forces. Some discrepancies between the model predictions and the experimental data are evident for the cases of weak buoyancy influence, where the model under-predicts the experimental values by as much as 20%, as was also found in the downward flow case.
Gerasimov (2002) also describes calculations of upward flow in a heated tube, using the analytic wall function. Figure 4 shows results for a case in which there is weak buoyancy influence, i.e. one in which the low-Reynolds number model under-predicts the experimental values by ~20%. The analytic wall function model performs slightly better than the low Reynolds number model, and results are almost independent of the size of the wall layer.
© copyright ERCOFTAC 2004
Contributors: Mike Rabbitt - British Energy