SOFTWARE VALIDATION REPORT FOR VERSION 9.0

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1 SOFTWARE VALIDATION REPORT FOR VERSION 9.0 Prepared by S. Green C. Manepally Center for Nuclear Waste Regulatory Analyses San Antonio, Texas August 2006 Approved : c Manager, Hydrology ~ Date

2 CONTENTS Section Page FIGURES...iv TABLES...vi 1 SCOPE OF THE VALIDATION Natural and Forced Convection Moisture Transport With Phase Change Radiation Heat Transfer Between Surfaces Combined Convection, Radiation, and Moisture Transport With Phase Change REFERENCES ENVIRONMENT Software Standard Installation Software Modifications Hardware PREREQUISITES ASSUMPTIONS AND CONSTRAINTS NATURAL AND FORCED CONVECTION TEST CASES Laminar Natural Convection on a Vertical Surface Test Input Test Procedure Expected Test Results Test Results Turbulent Natural Convection in an Air-Filled Square Cavity Test Input Test Procedure Expected Test Results Test Results Natural Convection in an Annulus Between Horizontal Concentric Cylinders Test Input Test Procedure Expected Test Results Test Results Natural Convection Inside a Ventilated Heated Enclosure Test Input Test Procedure Expected Test Results Test Results Forced Convection Inside a Confined Structure Test Input CONTENTS (continued) ii

3 Section Page Test Procedure Expected Test Results Test Results MOISTURE TRANSPORT TEST CASES Conduction Heat Transfer and Vapor Diffusion Test Input Test Procedure Expected Test Results Test Results Moisture Transport in a Closed Container Test Input Test Procedure Expected Test Results Test Results THERMAL RADIATION TEST CASES Thermal Conduction and Radiation Between Two Surfaces Test Input Test Procedure Expected Test Results Test Results Thermal Radiation Configuration Factors Test Input Test Procedure Expected Test Results Test Results COMBINED HEAT TRANSFER TEST CASE Convection, Radiation, and Moisture Transport in an Enclosure Test Input Test Procedure Expected Test Results Test Results INDUSTRY EXPERIENCE NOTES iii

4 FIGURES Figure Page 6-1 Geometry for Natural Convection Along a Vertical Flat Plate Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection Along a Vertical Plate Nusselt Number Comparison for Vertical Flat Plate Vertical Velocity Distribution at m [0.410 ft] From the Leading Edge Temperature Distribution at m [0.410 ft] From the Leading Edge Geometry for Natural Convection in a Square Cavity Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection in a Square Cavity Local Nusselt Number of the Vertical Walls and Experiment Data Are Shown for Hot and Cold Walls Local Nusselt Number Along Upper Wall Dimensionless Temperature Distribution at the Mid-Width Velocity Distribution of the Mid-Height Dimensionless Temperature Distribution at the Mid-Height Geometry for Natural Convection Between Concentric Cylinders Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection Between Cylinders Local Value of k eff on Cylinder Walls Dimensionless Fluid Temperature Profiles Along Radial Lines Experimental Apparatus Used in Test Case Thermocouple Placement Along Midline (Depth) of System a Transient Development of Velocity Vector Flow Field, 1 8 Seconds b Transient Development of Velocity Vector Flow Field, Seconds Steady-State Velocity Vector Comparison of FLUENT 4.52 Case and FLOW-3D Case Schematic of Experimental Apparatus Used for Test Case Visualization of the Computational Model Flow Vectors in Vented Box Schematic for Heat Conduction and Species Diffusion Between Surfaces Temperature and Concentration Profile for One-Dimensional Conduction and Moisture Transport. Water Vapor Is Allowed to Be Supersaturated Temperature and Concentration Profile for One-Dimensional Conduction and Moisture Transport. Water Vapor Is Limited to 100-Percent-RH and Fog Test Setup for Natural Convection and Water Vapor Transport in a Closed Container FLOW-3D Geometry for Condensation Cell Simulations Predicted Temperature Contours and Velocity Vectors in Condensation Cell Predicted Water Mass Fraction in Condensation Cell Mid-Line Temperature Ion Condensation Cell Cold Plate Condensation Rate iv

5 FIGURES (continued) Figure Page 8-1 Schematic for Thermal Conduction and Radiation Between Opposing Surfaces Temperature Distribution for One-Dimensional Radiation and Conduction Across an Air Gap Schematic for Thermal Radiation in an Annular Gap Schematic for Thermal Radiation in a Three-Dimensional Enclosure Schematic for Convection, Radiation, and Mass Transfer in a Two-Dimensional Enclosure Wall Temperature Vectors for Scenario That Includes Convection, Radiation, and Moisture Transport v

6 TABLES Table Page 6-1 Laminar Plate FLOW-3D Verison 9.0 Input File Square Box Input File Summary Overall Nusselt Number Comparison Experiments Selected for FLOW-3D Version 9.0 Simulations FLOW-3D Version 9.0 Simulation Conditions for Concentric Cylinders Comparison of Predictions and Measurements for Concentric Cylinders Steady-State Temperature Distribution Comparison of Inlet and Outlet Flow Velocities for Test Case Condensation Cell Test Conditions and Test Results Variances of FLOW-3D Prediction From Measured Values in Condensation Cell Results Configuration Factors, F a-b, Two-Dimensional Cylinders, Exact Solution Configuration Factors, F a-b, Three-Dimensional Box, Exact Solution Configuration Factors, F a-b, Two-Dimensional Cylinders, FLOW-3D Results FLOW-3D Configuration Factor Errors, Two-Dimensional Cylinders Configuration Factors, F a-b, Three-Dimensional Box, FLOW-3D Results FLOW-3D Configuration Factor Errors, Three-Dimensional Box Analysis Results for Two-Dimensional Enclosure Heat and Mass Transfer FLOW-3D Results for Two-Dimensional Enclosure Heat and Mass Transfer vi

7 SOFTWARE VALIDATION REPORT FOR FLOW-3D VERSION 9.0 FLOW-3D Version 9.0 (Flow Science, Inc., 2005) is a general purpose, computational fluid dynamics simulation software package founded on the algorithms for simulating fluid flow that were developed at Los Alamos National Laboratory in the 1960s and 1970s. The basis of the computer program is a finite volume formulation of the equations in an Eulerian framework describing the conservation of mass, momentum, and energy in a fluid. The code is capable of simulating two-fluid problems: (i) incompressible and compressible flow and (ii) laminar and turbulent flows. The code has many auxiliary models for simulating phase change, non-newtonian fluids, noninertial reference frames, porous media flows, surface tension effects, and thermoelastic behavior. The code will be employed to simulate the flow and heat transfer processes in potential high-level waste repository drifts at Yucca Mountain and support other experimental and analytical work performed by the Center for Nuclear Waste Regulatory Analyses (CNWRA). FLOW-3D uses an ordered grid scheme that is oriented along a Cartesian or a polar-cylindrical coordinate system. Fluid flow and heat transfer boundary conditions are applied at the six orthogonal mesh limit surfaces. The code uses the so-called Volume of Fluid formulation pioneered by Flow Science, Inc., to incorporate solid surfaces into the mesh structure and the computing equations. Three-dimensional solid objects are modeled as collections of blocked volumes and surfaces. In this way, the advantages of solving the different equations on an orthogonal, structured grid are retained. The code includes the Boussinesq approach to modeling buoyant fluids in an otherwise incompressible flow regime. The Boussinesq approximation neglects the effect of fluid (air) density dependence on pressure of the air phase, but includes the density dependence on temperature. This approach will be heavily used in the simulation of in-drift air flow and heat transfer processes at Yucca Mountain. Fluid turbulence is included in the simulation equations via a choice of turbulence models incorporated into the software. It is up to the user to choose whether fluid turbulence is significant and, if so, which turbulence model is appropriate for a particular simulation. 1 SCOPE OF THE VALIDATION FLOW-3D is capable of simulating a wide range of mass transfer, fluid flow, and heat transfer processes. This validation exercise considers the following four sets of test cases: Natural and forced convection Moisture transport with phase change Radiation heat transfer between surfaces Combined convection, radiation, and moisture transport with phase change Other less relevant capabilities of FLOW-3D will be addressed in additional validation exercises as needed. FLOW-3D users are advised to perform validations pertinent to their particular needs when applying this software to processes other than those treated here. The validation test cases are summarized in the following subsections. 1-1

8 1.1 Natural and Forced Convection The natural and forced convection capabilities of the standard version of FLOW-3D are considered in this set of tests. Forced convection is another term for active ventilation. Without active (or forced) ventilation, natural ventilation may occur. Five test cases for natural and forced convection are described in Section 6. The first three test cases progress from a theoretical consideration of a hypothetical laminar natural convection flow scenario to experimental treatments of heat transfer in laminar and turbulent flows. The fourth and fifth test cases address forced convection, or ventilation, in thermally perturbed enclosures. These test cases cover a range of processes and geometries relevant to preclosure and postclosure issues in facilities and drifts at Yucca Mountain and are summarized below. 1. Laminar natural convection on a vertical surface. The conservation equations for mass, momentum, and thermal energy are well known (e.g., Ostrach, 1953; Schlichting, 1968; and Incropera and Dewitt, 1996). The FLOW-3D results of a hypothetical case are compared to the semi-analytical solution of the boundary-layer type conservation equations derived specifically for this case. 2. Natural convection in a closed square cavity. This type of flow field was the subject of an experimental study reported by Ampofo and Karayiannis (2003). Fine resolution measurements of the fluid velocity, temperature, and wall heat flux are compared to the FLOW-3D simulation results. 3. Natural convection between two concentric cylinders. The experiment results reported in Kuehn and Goldstein (1978) are used here to validate FLOW-3D Version Natural convection in a room with one inlet, one outlet, and a heat source. This test case is modeled after the experiment described in Dubovsky, et al. (2001). In addition to a comparison against the measured data, FLOW-3D results are compared against the results of another widely used computational fluid dynamics code, FLUENT Version 4.52 (Fluent Inc., 1994). 5. Forced convection in a room when the fluid (air) is assumed to be compressible. A comparison of velocity and mass at the inlet and outlet of the system at steady state confirms boundary condition and overall mass balance implementation in the code. 1.2 Moisture Transport With Phase Change Two test cases are described in Section 7: a simple hypothetical case that will be solved by mathematical analysis and an experiment simulation. 1. Conduction heat transfer and vapor diffusion. In this case, the combined modes of thermal energy and mass transport by conduction and diffusion from a high temperature surface to a low temperature surface are studied. If the relative humidity is not limited to a maximum of 100 percent (i.e., a supersaturated condition is allowed), then the governing differential equations describing these processes can be solved for a one-dimensional case exactly as described by Bird, et al. (1960). Conversely, if the 1-2

9 relative humidity is limited to a maximum of 100 percent, the governing equations are highly nonlinear and must be solved numerically. The moisture transport module is capable of solving both these scenarios and predictions will be compared to the theoretical model of each scenario. 2. Moisture transport in a closed container. This test case is the simulation of the Condensation Cell Experiment as described by Scientific Notebook 643. A closed container contains a heated pool of water at one end and a cooled wall at the other. A convection cell is established inside the container and water evaporated from the pool is advected with and diffused through the air and is condensed on the cooled plate and parts of the other walls. The FLOW-3D simulation results are compared to the measured temperatures and steady condensation rates. These cases are relevant to the postclosure issues of moisture transport in a repository drift in that the localized processes of evaporation and condensation are simulated and the thermodynamics of high-humidity air are included in the overall solution algorithm. 1.3 Radiation Heat Transfer Between Surfaces Two test cases are described in Section 8. Both of these are hypothetical cases that can be investigated using analytical solutions of thermal radiative heat transfer processes. 1. Thermal conduction and radiation between two surfaces. Simplifying assumptions lead to an exact solution for the overall heat transfer rate following the methods described by Siegel and Howell (1992). The FLOW-3D results are compared to the analytical predictions. 2. Thermal radiation configuration factors. Radiation configuration factors are an important aspect of radiation modeling, and it is important that these computations be validated along with the radiation heat transfer analysis that employs the configuration factors. The first scenario to be tested is that of radiation between two partitioned cylinders in which the geometry can be considered two-dimensional. The second scenario is a three-dimensional rectangular enclosure. The configuration factors computed by the radiation module are compared to the results of exact equations for the configuration factors for these two cases. These cases are relevant to the postclosure issues of thermal radiation exchange in a repository because it is expected that radiation heat transfer will play a significant and sometimes dominant role in the overall heat transfer processes in the drift. Radiation heat fluxes are dependent on geometry only through the configuration factors. Thus, physical size is not as important in this heat transfer mode as in convection and conduction. As long as the relative sizes of features are similar to the full scale, the geometric properties of the radiation exchange will be sufficiently tested. 1-3

10 1.4 Combined Convection, Radiation, and Moisture Transport With Phase Change A single test case with four different scenarios is described in Section 9.0. This test case is a hypothetical condition of heat transfer in a square two-dimensional enclosure. This case can be analyzed with accepted empirical correlations for the convection, radiation, and moisture transport aspects of the problem. The following four scenarios make up this test case so that the effects of moisture transport and radiation on the convection heat transfer can be investigated. Natural convection alone Natural convection with thermal radiation Natural convection with moisture transport with phase change Natural convection with radiation and moisture transport with phase change This final case is relevant to the postclosure issues of thermal radiation exchange in a repository because it embodies all the modes of heat and mass transfer that are expected in the drift. This case tests the functionality of the two software modifications for radiation and moisture transport when they are applied together in the FLOW-3D computer code. The physical scale aspects of natural convection heat transfer are adequately addressed in the convection-only tests in Section 6. The radiation heat flux exchange is relatively insensitive to physical size and the moisture transport is a local phenomenon. So, this test case is considered adequate for validating the operation of the modified FLOW-3D in a mixed-mode heat transfer process. 1-4

11 2 REFERENCES Ampofo, F. and T.G. Karayiannis. Experimental Benchmark Data for Turbulent Natural Convection in an Air-Filled Square Cavity. International Journal of Heat and Mass Transfer. Vol. 46. pp. 3,551 3, Berkovsky, B.M. and V.K. Polevikov. Numerical Study of Problems on High-Intensive Free Convection. Heat Transfer and Turbulent Bouyant Convection. Vol. II. D.B. Spalding and H. Afgan, eds. Washington, DC: Hemisphere Publishing. pp Bird, R.B., W.E. Stewart, and E.N. Lightfoot. Transport Phenomena. New York City, New York: John Wiley and Sons Churchill, S.W. and H.S. Chu. Correlating Equations for Laminar and Turbulent-Free Convection from a Vertical Plate. International Journal of Heat and Mass Transfer. Vol. 18. pp. 1,323 1, Dubovsky, V., G. Ziskand, S. Druckman, E. Moshka, Y. Weiss, and R. Letan. Natural Convection Inside Ventilated Enclosure Heated by Downward-Facing Plate: Experiments and Numerical Simulations. International Journal of Heat and Mass Transfer. Vol. 44. pp. 3,155 3, Flow Science, Inc. FLOW-3D User s Manual. Version 9.0. Sante Fe, New Mexico: Flow Science, Inc Fluent Inc. FLUENT User s Guide. Version Lebanon, New Hampshire: Fluent Inc Green, S. and C. Manepally. Software Validation Test Plan for FLOW-3D Version 9. Rev. 1. San Antonio, Texas: CNWRA Green, S. Software Requirements Description for the Modification of FLOW-3D to Include High-Humidity Moisture Transport Model and Thermal Radiation Effects Specific to Repository Drifts. San Antonio, Texas: CNWRA Howell, J.R. A Catalog of Radiation Configuration Factors. New York City, New York: McGraw-Hill Book Company Incropera, F.P. and D.P. Dewitt. Fundamentals of Heat and Mass Transfer. 4 th Edition. pp New York City, New York: John Wiley and Sons, Inc Kuehn, T.H. and R.J. Goldstein. An Experimental Study of Natural Convection Heat Transfer in Concentric and Eccentric Horizontal Cylindrical Annuli. ASME Journal of Heat Transfer. Vol pp Kuehn, T.H. and R.J. Goldstein. An Experimental Study and Theoretical Study of Natural Convection in the Annulus Between Horizontal Concentric Cylinders. Journal of Fluid Mechanics. Vol. 74, Part 4. pp

12 Moran, M.J and H.N. Shapiro. Fundamentals of Engineering Thermodynamics. 4 th Edition. New York City, New York: John Wiley and Sons, Inc Ostrach, S. An Analysis of Laminar-Free Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force. NASA Report Schlichting, H. Boundary Layer Theory. 6 th Edition. New York City, New York: McGraw-Hill Book Company. pp Siegel, R. and J.R. Howell. Thermal Radiation Heat Transfer. 3 rd Edition. Washington, DC: Hemisphere Publishing Corporation

13 3 ENVIRONMENT 3.1 Software Standard Installation The FLOW-3D software package has been in use since the early 1980s. It was originally based on algorithms developed by the founders of Flow Science, Inc., when they were employed at Los Alamos National Laboratory. While the original code was a general purpose computational fluid dynamics package that could simulate the effects of irregular solid objects, it was especially noted for its ability to simulate free surfaces and reduced gravity. The current version of the code is a much-enhanced descendent of that early software package and is widely used in industry and government agencies. A description of the software may be found at the Flow Science, Inc., website: This software validation uses of FLOW-3D, which can operate in a Windows or Linux/UNIX environment. The graphical user interface is started by clicking on the executable file. The user either creates a new simulation using the menus available in the graphical user interface, or a previously created setup file can be opened for continued work or modification. The setup file created by the user completely describes the simulation and is all that is required to recreate results for a particular scenario. Computational fluid dynamics simulations often take many hours or even days to complete; hence, users should retain files holding simulation results for future analyses and postprocessing. 3.2 Software Modifications The FLOW-3D software delivered by the vendor includes options for customizing the program for special flow and heat transfer processes not covered in the basic code capabilities. The base code was modified in accordance with the requirements described by Green (2006). An advanced user of the FLOW-3D software can understand the code modifications; a complete description underlying theory, and details of the modifications are described in Scientific Notebook 536E. The basic FLOW-3D code is incapable of simulating the transport processes associated with the high humidity conditions that could be present in waste repository drifts. Green (2006) describes a model that addresses the primary physical processes expected to occur in these cases. The computing algorithm described by Green (2006) was programmed in accordance with the logical framework of the FLOW-3D computer program. The moisture transport module added to FLOW-3D is capable of modeling Water evaporation from saturated surfaces into the air when the surface temperature is above the local dewpoint Water condensation to surfaces from the air when the surfaces are at a temperature less than the local dewpoint Re-evaporation of water from surfaces on which water had been previously condensed Local condensation of liquid water as a mist in the bulk of the flow domain when heat transfer cools the air to the local dewpoint 3-1

14 One limitation of this moisture transport module is that any water condensed as a mist will not coalesce and rain (i.e., it is assumed that the mist diffuses and advects much like an atmospheric fog). The water that condenses on solid surfaces, however, is removed from the gas phase in the fluid cells adjacent to the surface. Water vapor is thereby transported by diffusion from the bulk of the gas phase to the walls in these regions. Likewise, the basic FLOW-3D code cannot account for radiation heat transfer between solid surfaces. This heat transfer process can be a significant portion of the overall heat transfer in a repository where natural convection and conduction are the only other means of energy transport between waste packages and the drift walls. The computing algorithm described by Green (2006) was programmed in accordance with the logical framework of the FLOW-3D computer program. The capabilities and features of this module are as follows: All surfaces are assumed to be diffuse and gray. The moist air in the drift does not affect the surface-to-surface thermal radiation. Solid obstacles may be subdivided so that radiation heat transfer can vary depending on the location and orientation with respect to the other surfaces. Radiation configuration factors are computed for the radiation-active surfaces or can be provided by the user in the problem input specifications. 3.3 Hardware The program can be run on computers with Windows or Linux/UNIX operating systems as described in the FLOW-3D manual. All of the tests described here were conducted on personal computers with Windows 2000 or Windows XP. For the test cases using the moisture transport module and the radiation module, the software was compiled and linked using Compaq Visual Fortran Version 6.6c in accordance with the FLOW-3D User s Manual (Flow Science, Inc., 2005). 3-2

15 4 PREREQUISITES Users should be trained to use FLOW-3D and have experience in fluid mechanics and heat transfer. 4-1

16 5 ASSUMPTIONS AND CONSTRAINTS None. 5-1

17 6 NATURAL AND FORCED CONVECTION TEST CASES 6.1 Laminar Natural Convection on a Vertical Surface Analytical results and experimental data for laminar natural convection on a flat-vertical surface provide a method to validate the accuracy of FLOW-3D for natural convection. This flow field is sketched in Figure 6-1 for the configuration and specifications used in the FLOW-3D simulations. The analytical solution documented by Incropera and Dewitt (1996) provides an expression for the local Nusselt number and average Nusselt number for laminar flow cases (Rayleigh number < 10 9 ). The Nusselt number is a dimensionless temperature gradient at a surface and provides a measure of the efficiency of convection for heat transfer relative to conduction. The empirical correlation in Churchill and Chu (1975) provides an improvement to the analytical solution for average Nusselt numbers at lower Rayleigh numbers. For this validation test case, the local and average Nusselt numbers are compared to the FLOW-3D results and these published analytical and empirical correlations Test Input The model was developed with an isothermal vertical wall with a temperature of 340 K [152 F]. The fluid is air with a free stream temperature set to 300 K [80 F]. The case is modeled as two-dimensional with an incompressible fluid and the Boussinesq approximation to capture the thermal buoyancy effects. No turbulence model was used Test Procedure FLOW-3D was run with the problem specifications described in the previous section. The output of the wall heat transfer rates was used to calculate the local and average Nusselt numbers for comparison to the benchmark correlations Expected Test Results The criteria for test acceptance were established in the software validation test plan (Green and Manepally, 2006). For the refined mesh, the benchmark and average Nusselt numbers on the vertical wall shall agree within ±10 percent. The local Nusselt number for the region from 10 to 90 percent of the length (i.e., the entry 10 percent and exit 10 percent should be neglected) should agree within 10 percent. For the coarse mesh, the benchmark and average Nusselt numbers on the vertical wall should agree within ±20 percent. The local Nusselt number for the region from 10 to 90 percent of the length (i.e., the entry 10 percent and exit 10 percent shall be neglected) should agree within 20 percent. The analytical solution documented by Incropera and Dewitt (1996) provides an expression for the local Nusselt number and average Nusselt number for laminar natural convection cases (Ra < 10 9 ) involving an isothermal vertical flat plate. The local Nusselt number expression is an empirical correlation to the theoretical solution obtained from the laminar boundary-layer theory developed by Ostrach (1953). 6-1

18 Nu ( z) z = 0.75Pr ( Pr Pr ) 0.25 Grz 4 ( z) 025. (6-1) where Nu z (z) is the local Nusselt number, Pr is the fluid Prandtl number, and Gr(z) is the local Grashof number evaluated at the distance from the leading edge. The local Grashof number (Gr z ) is defined as Gr 2 ρ gβ ( z) = ( T 2 T ) z μ z s f 3 (6-2) where D fluid density g acceleration of gravity $ fluid thermal expansion coefficient T s isothermal plate temperature T f fluid free stream temperature x distance from the leading edge μ fluid viscosity The average Nusselt number, Nu L, is an empirical correlation developed by Churchill and Chu (1975) and is also documented in Incropera and Dewitt (1996). Nu L ( Ra) = Ra Pr (6-3) where Nu L (Ra) is the effective Nusselt number of the entire plate length and Ra is the Rayleigh number (Ra = Gr*Pr) evaluated with a length scale equal to the entire plate length. The Churchill and Chu (1975) correlation agrees with the test data better than the corresponding correlation derived from the semi-analytical solution described in Eq. (6-1). For this validation test case, the local and average Nusselt numbers provided by FLOW-3D and these analytical and empirical correlations are compared Test Results A FLOW-3D input file (prepin.*) was developed to model the vertical flat plate natural convection. The simulation is of an isothermal vertical plate with a temperature of 340 K [152 F] with a fluid having a free stream temperature of 300 K [80 F]. The plate length is 0.2 m. The case is modeled as two-dimensional with an incompressible fluid. A Boussinesq 6-2

19 approximation is used to model the thermal buoyancy effects in the nominally incompressible fluid. No turbulence model is used. Three different grid resolution cases were analyzed. A refined mesh was developed based on a grid sensitivity analysis. This mesh provides more accurate results and the accuracy limits of FLOW-3D for this particular test case. A coarse mesh with grid resolution similar to what is expected to be practical for future modeling of the full-scale Yucca Mountain drifts was also specified for comparison to the more refined grids. Table 6-1 lists the prepin.* files for this test case. The grid is clustered at the leading edge in all three meshes. For the base case, the grid spacing is 0.1 mm 0.1 mm [0.004 in in], and it expands geometrically along the surface and away from the surface. The leading edge grid spacing is scaled accordingly for the coarse and refined meshes. A sample of the flow field predicted by FLOW-3D for the grid is shown in Figure 6-2. This figure shows the velocity vectors and fluid temperature contours after steady flow has been achieved. For the sake of clarity, the vectors are shown for every second mesh line in the z-direction. This figure shows that the flow domain boundary at the right side was chosen far enough away from the plate to represent a quiescent far-field condition. Also seen is the clear development of a boundary layer as the flow proceeds upward from the bottom edge of the plate. The calculations for the analytical solution for this test case are documented by David Walter in Scientific Notebook 576. The FLOW-3D predictions are compared to those obtained from the analytical expression in Figure 6-3. The local Nusselt number results for the nominal mesh were within ±10 percent of the analytical solution, and the overall effective Nusselt number was within ±8 percent of the empirical correlation. These variances are within the acceptance criteria described above; therefore, the results of this test case are acceptable for software validation. The overall Nusselt number result for the coarse mesh was 23.5 percent lower than the analytical predictions. This exceeds the acceptance criterion for this mesh which indicates that the selected mesh is too coarse for this analysis. Mesh independence of the nominal mesh solution was demonstrated by close agreement of the local Nusselt number results between the nominal and refined mesh solutions. Mesh independence is further demonstrated by the agreement of the velocity and temperature profiles shown in Figures 6-4 and 6-5. The deviation of the coarse mesh results form the others further shows that this mesh is inadequate for this analysis. This demonstrates the need to conduct grid resolution studies in computational fluid dynamics analyses to ensure grid-independent results. Table 6-1. Laminar Plate FLOW-3D Verison 9.0 Input File Summary File Name Description Comment prepin.vertsurf40k Nominal Mesh Mesh prepin.vertsurf40k-cmesh Coarse Mesh Mesh prepin.vertsurf40k-rmesh Refined Mesh Mesh(demonstrate grid independence) 6-3

20 6.2 Turbulent Natural Convection in an Air-Filled Square Cavity An experimental study conducted by Ampofo and Karayiannis (2003) provides benchmark data to evaluate the accuracy of FLOW-3D for natural convection in low-level turbulence. The two-dimensional experimental work was conducted on an air-filled square cavity of dimension m 2 [ ft 2 ] with vertical hot and cold walls maintained at isothermal temperatures of 50 and 10 C [122 and 50 F]. This flow field is sketched on Figure 6-6 for the configuration and specifications used in the FLOW-3D simulations. These conditions resulted in a Rayleigh number of For this validation test case, the local and average heat transfer rates (described by the Nusselt number), the local velocities, and the temperature profiles are compared between the FLOW-3D and experimental results Test Input The experiment was modeled as two-dimensional with an incompressible fluid and the Boussinesq approximation to capture the thermal buoyancy effects. The large eddy simulation model in FLOW-3D was used to model the fluid turbulence. The model geometry, fluid properties, and boundary conditions match, as closely as practical, the experimental apparatus described by Ampofo and Karayiannis (2003) Test Procedure FLOW-3D was run with the input file, as described in the previous section. The output of the wall heat transfer rates, temperature and velocity profiles, and the mid-width and mid-height were compared to the experimental benchmark data Expected Test Results The criteria for test acceptance were established in the software validation test plan (Green and Manepally, 2006). For the refined mesh, the experimental and numerical simulation average Nusselt numbers on the horizontal and vertical walls should agree within ± 20 percent. The fluid temperature and velocity profiles will be compared graphically to the measured values. The trends of the profiles will be compared for overall goodness of fit. For the coarse mesh, the experimental and simulation average Nusselt numbers on the horizontal and vertical walls shall agree within ± 25 percent. The fluid temperature and velocity profiles will be compared graphically to the measured values. The trends of the profiles will be compared for overall goodness of fit. Ampofo and Karayiannis (2003) measured the time-varying temperature and velocity at a large number of points in the central plane of the cavity where two-dimensional flow is closely approximated. Low-noise type-e thermocouples were used to measure the instantaneous temperature, and a two-dimensional laser Doppler velocimeter was used to measure the fluid velocity. The velocity and temperature measurements were made simultaneously, but were separated by a distance of about 0.4 mm [0.016 in]. This distance is far enough to prevent the temperature and velocity probes from interacting with each other and is approximately the size 6-4

21 of the turbulence-length scale near the wall. This is required to ensure that the temperature and velocity are measured as close to simultaneously as possible. The local values of Nusselt numbers are computed using the near-wall temperature distribution to estimate the local wall heat flux. This is valid because there are measurement points within the laminar sublayer next to the wall. The overall value of the Nusselt number is computed from the area-weighted average of the local values Test Results A FLOW-3D input file (prepin.*) was developed to model the square cavity experiment. The experiment was modeled as a two-dimensional flow of an incompressible fluid with a Boussinesq approximation to simulate the thermal buoyancy effects. The model geometry, fluid properties, and boundary conditions matched the experimental apparatus described by Ampofo and Karayiannis (2003) as closely as practical. The FLOW-3D large eddy simulation model was used to model fluid turbulence. The large eddy simulation model is inherently three-dimensional. It can be used for this two-dimensional analysis by selecting the thickness of a two-dimensional slice consistent with the grid spacing so that the large eddy simulation length scale is of the correct order. Other two-dimensional turbulence models could be used (e.g., K-g), but they are inadequate for modeling the turbulence in the corners of the enclosure. Three different grid resolution cases were analyzed. A refined mesh was developed based on a grid sensitivity analysis. This mesh should provide more accurate results and the accuracy limits of FLOW-3D for this particular test case. A coarse mesh with grid resolution similar to what is expected to be practical for future modeling of the full-scale Yucca Mountain drifts was also tested to determine its accuracy level. Table 6-2 provides a list of the prepin.* files for this test case. A sample of the fluid temperatures and velocity vectors predicted by FLOW-3D is shown in Figure 6-7. These results are from the simulation with the mesh resolution. This figure shows the overall clockwise circulation pattern and the large turbulent eddies that form along the vertical walls. Note that the disturbances are formed at about the mid-height of each vertical wall. The computational fluid dynamic predictions are compared to the test results in Table 6-3 and Figures 6-8 through All postprocessing calculations for these results are described by David Walter in Scientific Notebook 576. Table 6-2. Square Box Input File Summary File Name Description Comment prepinr.sqr_box-200x200 Nominal Mesh Mesh prepinr.sqr_box-75x75 Coarse Mesh Mesh prepinr.sqr_box-300x300 Refined Mesh Mesh to demonstrate grid independence 6-5

22 Table 6-3. Overall Nusselt Number Comparison Lower Wall Upper Wall Hot Wall Cold Wall Experimental Data Computation Fluid Dynamics Results Mesh Size Nu Error Nu Error Nu Error Nu Error !23% 10.6!36% 58.6!7.3% 59.4!5.1% !7.8% 12.8!13% 55.9!13% 56.0!12% !16% 11.9!21% 54.1!16% 53.8!16% The predicted Nusselt number values for all of the enclosure surfaces are less than the measured values. The variances for all of the predictions from the various mesh resolutions are 25 percent of the values measured in the experiments, except for the upper wall using the coarse mesh. The nominal and refined mesh results meet the acceptance criteria listed in Section The local Nusselt number values for the sidewalls are compared to the measured values in Figure 6-8. There is good agreement, except in the region of the wall closest to the bottom wall. The computational fluid dynamics predictions shows a decrease in Nusselt numbers towards the corner (ie., y/l < 0.1), whereas the measured results show a monotonic increase in Nusselt numbers near the corner. This difference in Nusselt number values in the corners is strongly affected by the specification of the thermal conditions for the walls. A perfectly conducting wall yields a high value of Nusselt number for the vertical wall in the corner, while a perfectly abiabatic wall yields a nearly zero value for the Nusselt number for the corner. The Nusselt number value for real conditions is between these limits. The thermal conditions are dictated by the precision of the isothermal surface condition, the interface conductance between the vertical and horizontal walls, and the thermal conductivity of the horizontal wall. The source article did not provide sufficient detail to allow for a high-fidelity simulation of the corner conditions. The assumptions described for this simulation were not altered to home in on a more agreeable match. The fine-grid results presented in Figures 6-8 through 6-12 are not substantially different from the nominal-grid results. It is counterintuitive that the refined mesh results show a larger discrepancy in the Nusselt numbers than the nominal mesh results. This is thought to be related to the problem of modeling the heat transfer within the walls near the corner. The small differences in the temperature profiles of the two sets of predictions is amplified in the Nusselt number calculations. Based on the quality of agreement in the temperature profile, the nominal mesh resolution is considered to be adequate for this analysis. 6.3 Natural Convection in an Annulus Between Horizontal Concentric Cylinders Kuehn and Goldstein (1978) conducted experiments on the temperature and heat flux measurements of the thermal behavior of a gas in an annulus between concentric and eccentric circular cylinders. Only the concentric configuration (Figure 6-13) was considered for this validation activity. This is a widely referenced article for empirical correlations and validations of computational fluid dynamics calculations of natural convection flows. The experimenters used thermal conductivity for the condition in which heat transfer only occurs by conduction across the radial gap between the cylinders. The equivalent thermal conductivity of the annulus gas is defined as 6-6

23 where k eff ( o i) Q ln D D = 2 π Y ΔT (6-4) Q heat transfer rate at inner cylinder D o inner diameter of the outer cylinder D i outer diameter of the inner cylinder Y length of annulus ΔT temperature difference between cylinders For pure conduction, k eq = 1 and k eff increases to nearly 20 for the most turbulent flow reported by Kuehn and Goldstein (1978). The results are correlated by the Rayleigh number for gap width where 2 g RaL = ρ β ΔT L 2 μ 3 Pr (6-5) p gas density g acceleration due to gravity β thermal expansion coefficient of gas μ dynamic viscosity L 0.5(D o! D i ) = gap width delineated by the diameters of the cylinders Pr gas Prandtl number Test Input FLOW-3D input files were developed for the cases described by Kuehn and Goldstein (1978) for Ra L = , Ra L = , Ra L = , Ra L = , Ra L = , and Ra L = These represent laminar to fully turbulent flow regimes as Ra L increases Test Procedure FLOW-3D was run using an identical grid for all cases. In addition, the flow with the Ra L = was simulated with a finer grid resolution to demonstrate that the simulation results are approximately grid-independent. The FLOW-3D results were used to compute the effective overall equivalent thermal conductivity for comparison to the experiment results of Kuehn and Goldstein (1978). The calculated fluid temperature profiles across the gap were compared to the available experiment results Expected Test Results The criteria for test acceptance were established in the software validation test plan (Green and Manepally, 2006). 6-7

24 The acceptance criterion for the simulated overall equivalent thermal conductivity will be a deviation of no more than 25 percent of the measured value. The fluid temperature profiles across the gap will be compared graphically to the measured values. The trends of the profiles will be compared for overall goodness of fit. The experiment facility consisted of two concentric cylinders sealed in a pressure vessel. The outer diameter of the inner cylinder was 3.56 cm [1.40 in], and the inner diameter of the outer cylinder was 9.25 cm [3.64 in]. The annular gap between the cylinders was cm [1.12 in]. The cylinders were 20.8 cm [8.19 in] in length. The inner cylinder was electrically heated, while the outer cylinder was cooled with a chilled water loop. The test chamber was filled with nitrogen as the test fluid. The nitrogen pressure was varied between atm and 35.2 atm, and the temperature difference between the two cylinders was varied between 0.83 K [1.5 F] and 60.1 K [108.2 F]. This provided for a Rayleigh number range of to Temperatures in the annulus were measured via Mach-Zender interferometry, and surface temperatures were measured with thermocouples. This range of Rayleigh number values does not represent the expected Rayleigh number range of the Yucca Mountain drifts when there is no drip shield. When a drip shield is used, the range of Rayleigh number values covered by the experiments includes an expected range of Rayleigh numbers for convection heat transfer between the waste package and the drip shield. Kuehn and Goldstein (1978) reported the experiment results for the heat transfer across the annulus in terms of an effective thermal conductivity, as described in Section 6.3. The value of k eff can be either a local value or a net value for a large surface, depending on whether the value of the heat flux is a local value of the net value for the surface. Kuehn and Goldstein (1978) do not describe the measurement uncertainty for temperature or the derived values of k eff. In an earlier paper, Kuehn and Goldstein (1976) state that the variance between values of k eff derived from interferogram contours and those obtained from overall heat transfer rates was less than 3 percent. The difference between the derived value of the Rayleigh number was about 7 percent. It is noted that the results reported in Kuehn and Goldstein (1976) do not extend to the highest Rayleigh numbers reported in Kuehn and Goldstein (1978). For nitrogen, a single fringe shift in the interferogram corresponds to about 35 K [63 F] at a pressure of 0.15 atm and about 0.1 K [0.18 F] at a pressure of 35 atm Test Results The details of the validation test for this case are described in Scientific Notebook 536E. The test results are summarized here. Six of the reported 40 sets of test measurements were selected for simulation with FLOW-3D. These six cases are described in Table 6-4. A seventh case was selected for a simulation with 6-8

25 Table 6-4. Experiments Selected for FLOW-3D Simulations Ra gap P atm ΔT C [ F] ½(T i + T o ) C [ F] [128.5] 51.1 [124.2] [100.6] 44.4 [112.1] [39.9] 27.3 [81.3] [33.8] 27.7 [82.0] [44.8] 29.1 [84.6] [83.9] 40.8 [105.6] a fine resolution mesh at Ra = since Kuehn and Goldstein (1978) provide some details of the temperature profiles along the cylinder surfaces and between the cylinders for this case. The conditions used for the seven computational fluid dynamics simulations are listed in Table 6-5. The properties of nitrogen at the selected test conditions were provided from the computer program NIST12. This program is equivalent to the web-based property information software available at the National Institute of Standards and Technology. FLOW-3D uses a structured Cartesian orthogonal mesh and the volume of fluid method to define cells blocked by solid obstacles. Because the cylinder surfaces are curved, the fractional blockage of cells containing the solid surface varies greatly. Grid refinement in the boundary layers is of limited use; consequently, a uniform mesh was selected. The mesh resolution was chosen to adequately capture the expected temperature and velocity distributions for Ra < (i.e., truly laminar flow throughout the entire flow domain). Kuehn and Goldstein (1978) report that turbulent eddies are observed at the top of the inner cylinder for Ra At increasing Rayleigh numbers, more of the flow is turbulent; at Ra. 10 8, the upper half of the annulus is clearly in turbulent flow, but the lower half is in laminar flow. The large eddy simulation turbulence model within FLOW-3D was used for simulations of the higher Rayleigh number flows, and a consistent mesh was used under the assumption that the selected mesh adequately captures the flow structures of interest. The large eddy simulation is inherently three-dimensional. It can be used for this two-dimensional analysis by selecting the thickness of the two-dimensional slice consistent with the grid spacing so that the large eddy simulation length scale is of the correct order. A sample of the fluid temperatures and velocity vectors predicted by FLOW-3D is shown in Figure These results are from the simulation with the mesh resolution for the case of Ra = This figure shows the circulation pattern between the cylinders. The flow field is nearly symmetric, but by viewing several files in the time sequence, it was found that the plume oscillates slightly from side-to-side with attendant oscillations in the circulation patterns on either side of the cylinder. The grid resolution was refined by a factor of 2 for the case of Ra = to determine the adequacy of the chosen grid discretization. The FLOW-3D postprocessor will provide the user with the total heat transfer rate from solid objects to the fluid. This is the value of Q in Eq. (6-4) to compute the effective overall thermal 6-9

26 Ra gap Table 6-5. FLOW-3D Simulation Conditions for Concentric Cylinders Mesh (Uniform) D $ kg/m 3 1/K : Pa sec C v J/(kg K) k W/(m K) FLOW-3D Input File Name ! ! prepin.k-g_ Validation_ Ra1-3e ! ! prepin.k-g_ Validation_ Ra6-2e ! ! prepin.k-g_ Validation_ Ra6-8e ! ! prepin.k-g_ Validation_ Ra2-5e (FineMesh) prepin.k-g_ Validation_ Ra2-5e06_fine ! ! prepin.k- G_Validation_ Ra1-9e ! ! prepin.k- G_Validation_ Ra6-6e07 conductivity between the two cylinders. These calculations were performed in Microsoft Excel as described in Scientific Notebook 536E. The convection heat transfer can also be expressed in terms of a Nusselt number. First, the heat flux at the inner surface can be expressed as Nuk qi = hi Ti To = D eff ( ) ( Ti To) i (6-6) where q i heat flux at inner surface h i heat transfer coefficient [e.g., W/(m 2 K)] Nu Nusselt number k f fluid material thermal conductivity Comparing Eq. (6-3) can be rearranged to yield the expression Di Nu = q i k T T ( ) f i o Di = 2k eff Do ln D i (6-7) 6-10

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