SOFTWARE VALIDATION REPORT FOR VERSION 9.0

Transcription

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

27 The Nusselt number, Nu, is described in terms of the computed or measured value of q i, which varies as a function of test conditions; a material property (k f ) that is temperature dependent, but not affected by the flow field; and geometric parameters that are constant. Equation (6-7) is not used in the postprocessing of these data. It is shown here to demonstrate that relative variances (i.e., the percent difference) between the computed and measured Nusselt numbers will be the same as the relative variances between the computed and measured values of k eff, assuming the uncertainty of the fluid thermal conductivity, k f, is neglected. This particular uncertainty is small compared to the variance between the measured and computed values for k eff shown below. The acceptance criterion for the variance between the measured and computed values of k eff was established as ± 25 percent in Section The measured values of k eff reported by Kuehn and Goldstein (1978) are compared to the values predicted by FLOW-3D in Table 6-6. For the nominal grid resolution, the variance ranges from!12.7 percent at low Ra to percent at high Ra. The variance is!15.7 percent for the refined grid resolution. All of these values are within the established acceptance criterion. The surface heat flux variations over the inner and outer cylinders are presented in terms of the local value of k eff in Figure There is excellent agreement between the predicted and measured values of local k eff for Ra = on both the inner and outer cylinders. The agreement for the outer surface is also good at Ra = At the inner surface for Ra = , there are regions where agreement is poor; however, the general trend of the measured values is matched by the computational fluid dynamics results. With the nominal mesh for Ra = , the predicted values of k eff near the top of the inner cylinder are greater than the measured values, but the predicted values are less than the measured values between about 50 and 100 from the top. From 100 to 180 the agreement is good, but this is a much less important area than the upper parts of the cylinder. With the refined mesh, the predicted values of k eff near the top of the inner cylinder are in closer agreement with the measured values than for the nominal mesh. In the area of 50 and 180, the results of the nominal and refined meshes are almost identical. Nevertheless, the refined mesh gave only slightly different local values of k eff, so, the nominal mesh is seen as adequate for this flow. The predicted fluid temperature profiles for the case of Ra = are compared to the measured values along selected radial lines around the annulus in Figure First, it is seen that the results for the nominal mesh and the refined mesh are not significantly different. This is further evidence that the nominal mesh is adequately resolving Table 6-6. Comparison of Predictions and Measurements for Concentric Cylinders Ra gap k eq experiment k eq FLOW-3D Version 9.0 Deviation Percentage ! ! ! ! (Fine Mesh) 6.63!

28 the calculated flow characteristics. The general trends of the measured temperature profiles are observed in the predicted temperature profiles except at an angle of 0. It should be noted, however, that the temperature difference between the inner and outer cylinders for this case is only 0.91 K [1.64 F]. The temperature difference between fringes on the interferogram for this case (i.e., at a pressure of 34.6 atm) is about 0.1 K [0.18 F], so there are only about nine fringe contours across the annulus gap. The uncertainty in obtaining temperature values from such an interferogram are not discussed in the paper. As a result, we cannot assess this impact on the agreement between the measured temperature profiles and the predicted temperature profiles. 6.4 Natural Convection Inside a Ventilated Heated Enclosure Test Case 4 will be a comparison of FLOW-3D results against measured data from a natural ventilation experiment (Dubovsky, et al., 2001) and against results from a different numerical model created in FLUENT 4.52 (Fluent Inc., 1994), a widely recognized and employed computational fluid dynamics code. This test, while computationally expensive, will allow the effect of walls and their interaction with a boundary of air to be examined in addition to the thermal comparatives registered by the interior thermocouples of the fixture. As suggested by the experiments, the simulation will also allow for confirmation of thermal properties. This scaled room-like natural convection experiment includes a portion of the ceiling heated by a boiling water tank and two ceiling sections open for natural ventilation through an inlet and an outlet for air flow. Figure 6-17 contains a schematic of the experiment. From the point of view of heat transfer into the enclosure, heating from the ceiling is considered a worst-case scenario. Heat transfer is primarily by conduction between the hot plate and the circulating air. The temperature differential between the walls of the room creates a natural circulation in the room, and this air motion drives the ventilation. In the experiment, the hot plate was modeled by the bottom of a metal tank filled with boiling water and maintained at one temperature by immersing two electrical heaters. The tank walls that were not part of the hot plate were insulated. Spatial uniformity of the plate temperature of 100 C [212 F] was experimentally verified and shown to be constant and uniform. The box acting as the experimental room had length, height, and width dimensions of 60, 30, and 24 cm [24, 12, and 9 in]. Along the top of the box, two 5-cm [2-in] openings running the entire width of the box acted as the air inflow and outflow regions. Also, an interior wall from the air inflow edge to 5 cm [2 in] above the bottom of the box existed. All walls of the box in the experiment were thermally insulated with a 0.2-cm [0.08-in] layer of insulation. The convective heat transfer coefficient measured outside the box was 10 to 12 W/m 2 - C [1.8 to 2.1 BTU/h-ft 2 - F] (Dubovsky, et al., 2001). The convective heat transfer coefficient assumed inside the box was 2 to 5 W/m 2 - C [0.35 to 0.88 BTU/h-ft 2 - F]. The heat transfer coefficient based on the thermal resistance of the wall and the convective resistance outside the box 0.08 W/m 2 - C [0.014 BTU/h-ft 2 - F] with an uncertainty of 15 percent. The heat transfer coefficient for the heated plate was found to be 5 W/m 2 - C [0.88 BTU/h-ft 2 - F] with a 20-percent uncertainty. Thermocouples were placed along the apparatus width midline as shown in Figure 6-18 and temperature measurements were made every 15 minutes. Steady state was determined as a point when less than 0.2- C [0.4- F] deviation from a previous measurement was made for all thermocouples in the system. Typical times to steady state were on the order of 2 hours. A more detailed accounting of the test fixture and experimental method can be obtained in Dubovsky, et al. (2001). 6-12

29 6.4.1 Test Input A comparison of the measured data with results derived from numerical model simulations using the computational fluid dynamics code FLUENT Version 4.52 is provided in Dubovsky, et al. (2001). Specifically, the simulations compare (i) a steady-state case when the whole system is sealed, (ii) a ventilated steady state when the entrance and exit windows are open, and (iii) the early transient between state (i) and state (ii). A two-dimensional grid evaluation study using FLUENT Version 4.52 that used (length height) and grid cells to determine whether temperature effects were significant was described in Dubovsky, et al. (2001). There was little difference in the comparative runs, so the coarser mesh was extended to three-dimensional calculations. The reported three-dimensional FLUENT 4.52 simulations used a grid defined as (length height width) cells, where each cell was cm 3 [ in 3 ]. A grid refinement study was conducted for one case utilizing cells. Differences between results for the grids were within experimental error; therefore, the coarser grid was maintained for the rest of the calculations. The FLOW-3D model uses the same grid scale at the coarse FLUENT grid, but additional cells will be added for the walls. Thus, the FLOW-3D model employs grid cells to include the physical nature of the walls and insulation materials of the test fixture. The original FLUENT Version 4.52 model simplified these boundaries as mesh boundaries with generalized wall properties. The boundary that incorporated the inflow and outflow condition had a pressure derivative that equaled zero the same as the continuative condition that is employed in the FLOW-3D model. The thermal properties used in the FLOW-3D model match those of the FLUENT model Test Procedure First, the simulated system was brought to a closed steady state. This means that the system is completely closed (the vents are shut) and allowed to equilibrate with the hot plate in place. Equilibration was evaluated using the temperature at history points within the system at locations shown in Figure When no change in local temperature is observed (aside from normal and regular numerical oscillation), the system is deemed steady. Then, the side vents of the system are opened, and the transient behavior is observed and compared to FLUENT results. After reaching steady state, simulated temperature results are compared to the measured data Expected Test Results The criteria for test acceptance were established in the software validation test plan (Green, et al., 2006). Results from two-dimensional plots of FLOW-3D at different times during the transient period when the vents are open are plotted for comparison with FLUENT results. Flow pattern results from FLOW-3D and FLUENT should visually match. There should be less than a 1-percent difference in aggregated velocity results for zones within the domain for the steady-state condition. Simulated temperature profiles track relative changes in measured profiles and should not differ by more than 5 percent. Some temperature variations may occur because slightly shifted flow patterns between the experiment and the numerical model can lead to markedly different 6-13

30 temperatures. The locations to be tracked are the same as those illustrated in Figure 6-18 along the midline of the system. The cited paper incorporates an experimental comparison to a numerical model run in the computational fluid dynamics code, FLUENT Version Specifically, they compare to (i) a steady-state case when the whole system is sealed, (ii) a ventilated steady state when the entrance and exit windows are open, and (iii) the early transient between state (i) and state (ii). A two-dimensional grid evaluation was undertaken (eliminating width) using (length height) and grids to determine whether temperature effects were significant. There was little difference in the comparative runs, so the coarser mesh was extended to the three-dimensional calculations. The reported three-dimensional FLUENT Version 4.52 runs use a grid defined at (length height width), where each cell is cm 3 [ in 3 ]. A grid refinement study was conducted for one case utilizing ; differences were within experimental error, so the coarser grid was maintained for the rest of the calculations. The particular settings used in FLUENT Version 4.52 are not described in the paper Test Results The FLOW-3D model employed a grid to include the physical nature of the walls and insulation materials of the test fixture. The original FLUENT Version 4.52 model simplified these boundaries as mesh boundaries with generalized wall properties. The boundary, which incorporated the inflow and outflow condition, was specified as a fixed reference pressure, which is the same as the FLOW-3D continuative condition that we similarly employed. This test, while slightly more computationally expensive, allows us to examine the walls and their interaction with a boundary of air in addition to the thermal comparatives registered by the interior thermocouples of the fixture. It also allows us to verify thermal properties as suggested by the experiments. Hence, our results are twofold: (i) a comparison to the actual experiments by Dubovsky, et al. (2001) and (ii) a comparison to a different numerical model created in FLUENT Version 4.52, a widely recognized and employed computational fluid dynamics code. Figures 6-19a,b show a side-by-side comparison of the FLUENT Version 4.52 prediction and the FLOW-3D prediction of the velocity vectors in the flow along the mid-plane through the system over a reported 20-second transient. Figure 6-20 is a side-by-side comparison of the steady-state condition. There is less than 1 percent difference between the cases. Numerical values are not provided in the paper for FLUENT Version 4.52 velocity results; therefore, a quantitative comparison cannot be made. In addition to an evaluation of the local velocity profile as a comparison to the FLUENT Version 4.52 runs, we also measure the temperature within the fixture at discrete locations at steady state. These locations are the same as those illustrated in Figure 6-18 along the midline of the system. These values are cataloged in Table 6-7. There is good agreement between the actual temperature measurement and FLOW-3D predictions. 6-14

31 Table 6-7. Steady-State Temperature Distribution. These Temperatures Are Relative to Ambient {25 C [77 F]} Along Midline xz Plane of Test Fixture. (x,z) Notation Corresponds to Figure Thermocouple Position (x,z) = (cm,cm) Measured Temp C [ F] FLOW-3D Temp C [ F] (2.5, 26.25) 0.2 [0.36] 0.2 [0.36] (13.5, 26.25) 2.6 [4.7] 2.5 [4.5] (29.0, 26.25) 3.3 [5.9] 3.3 [5.9] (45.5, 26.25) 3.6 [6.5] 3.6 [6.5] (57.0, 26.25) 3.1 [5.6] 3.2 [5.8] (13.5, 18.75) 2.3 [4.1] 2.3 [4.1] (29.0, 18.75) 2.5 [4.5] 2.5 [4.5] (45.5, 18.75) 2.5 [4.5] 2.6 [4.7] (57.0, 18.75) 2.4 [4.3] 2.4 [4.3] (13.5, 11.25) 2.1 [3.8] 2.0 [3.6] (29.0, 11.25) 2.2 [4.0] 2.2 [4.0] (45.5, 11.25) 2.0 [3.6] 2.0 [3.6] (57.0, 11.25) 1.9 [3.4] 1.9 [3.4] (13.5, 3.75) 1.8 [3.2] 1.9 [3.4] (29.0, 3.75) 1.5 [2.7] 1.5 [2.7] (45.5, 3.75) 1.5 [2.7] 1.5 [2.7] (57.0, 3.75) 1.9 [3.4] 1.9 [3.4] 6.5 Forced Convection Inside a Confined Structure This test case involves 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 will be used to confirm that the boundary condition and overall mass balance implementation in the code are sufficient. To accomplish this check, a room having length, depth, and height dimensions of 4, 2, and 3 m [13, 6.5, and 10 ft] with a single source of forced ventilation and a single exit for natural exhaust was simulated (Figure 6-21). Forced ventilation was input through a rectangular vent of size m 2 [ ft 2 ]. Exhaust was through a similarly sized vent in the ceiling. The inlet flow was maintained at a constant temperature and pressure. Any variation in these parameters is an artifact of the compressibility of the gas employed, which in this case will be air. Physics demands that at steady state the mass of gas entering the room is equivalent to the mass of gas exiting the room. Furthermore, regardless of the physical construct of a problem, a flow can be considered one-dimensional under the following conditions: (i) the flow is normal to the boundary at locations where mass enters or exits the control volume and (ii) all intensive properties, such as velocity and density, assume a uniform position over each inlet or exit area through which matter flows (e.g., Moran and Shapiro, 2000). In particular, when flow is 6-15

32 considered one-dimensional, the mass flow rate ( ) &m at inlet and outlet is defined by &m =ρav, where D is density, A is cross sectional area, and V is velocity. Steady state in these situations, therefore, is often regarded as mass in equals mass out. Given the construct of this validation run and the definition of one-dimensional flow, both constant velocity and density are expected at the inlet and outlet. Therefore, this validation run evaluates this physical phenomenon and ascertains whether FLOW-3D accurately predicts the outcome Test Input The computational model was generated based on the physical model described above. Interior dimensions of the room will be m 3 [ ft 3 ]. Computational walls of thickness 0.2 m [8 in] were applied in each direction to simplify visualizations and properly restrict inflow and outflow. Vents were created as m 2 [ ft 2 ] openings through their respective boundaries. The full model utilizes a mesh of with a uniform grid of individual block size m 3 [ ft 3 ]. An additional run at double the resolution was also completed to support the coarse grid results. A forced air inflow condition equivalent to a constant velocity application of 0.25 m/s [0.8 ft/s] was applied to the inflow vent as shown in Figure A continuative condition was applied to the outflow boundary, which indicates that FLOW-3D will extrapolate local data upstream into appropriate conditions through the boundary. Zero normal derivatives for all quantities are implemented for continuative boundary conditions in FLOW-3D. The fluid is air having the following properties at K [68 F]: viscosity = !5 kg/m-s [ !5 lbs/ft-s] specific heat = m 2 /s 2 -K [ ft 2 /s 2 - F] thermal conductivity = kg-m/s 3 -K [ lbs-ft/s 3 - F] gas constant = m 2 /s 2 -K [1720 ft 2 /s 2 - F] density = 1.2 kg/m 3 [0.075 lbs/ft 3 ] The gas is assumed compressible so that the physical sensitivities of pressure and velocity can be included in the calculations Test Procedure History points, which are numerical markers in the flow, are located in the center of both inflow and outflow vents. These points were monitored to ascertain when the flow reaches steady state. To ascertain an average velocity across both the inflow and outflow boundary, the magnitude of total velocity was evaluated as an integral over the cross-sectional area of each vent. Simulation data were taken one grid plane from boundary; this gives a more accurate representation of velocity through the opening instead of at a discrete boundary. 6-16

33 6.5.3 Expected Test Results The criteria for test acceptance were established in the software validation test plan (Green and Manepally, 2006). The simulated velocity at the inflow and outflow vents should be within 5 percent of the intended ventilation flow rate. The mass flow rate at the inlet and outlet should not differ by more than 2 percent. These acceptance criteria are acceptable for simulations of compressible flow at steady state Test Results Figure 6-22 shows a visualization of the computational model for the full model with a mesh of (An additional run at double the resolution also was completed to verify results. Only the coarse mesh solution is given here.) The wall structure (front wall removed for clarity) is shown in (a), and (b) shows the grid resolution (red is open, blue is closed). For purposes of discussion, air comes in from the left and goes out through the top. A sample of the velocity vectors predicted by FLOW-3D is shown in Figure Figure 6-23a shows the flow pattern from the inflow to the outflow in the central longitudinal (x-z) plane, while Figure 6-23b shows the flow in a transverse (y-z) plane in line with the outlet vent. In each of the two images, the colored contours display the value of the out-of-plane velocity component. The three-dimensional nature of the flow field is clearly evident in these images. If successive images are viewed, the flow is time varying and does not settle into a truly steady state. To ascertain an average velocity across both the in-flow and out-flow boundary, the magnitude of total velocity was evaluated as an integral over the cross-sectional area of each vent. The inlet average velocity and outlet average velocity are compared in Table 6-8. Mass flow then can be evaluated using the equation described above, &m =ρav. The results of this calculation for the inlet and outlet ports are likewise compared. There is a 1.60 percent difference between the inlet and outlet port mass flow rates. This variation is within acceptable limits for a compressible flow at steady state. The difference between the inflow and outflow values is attributed to the accumulation of convergence errors over all the computational cells in the FLOW-3D solution algorithm. FLOW-3D does not impose a global mass balance over the boundaries of the entire flow domain. This apparent mass flow imbalance could be reduced by decreasing the convergence criterion value at the cost of longer execution times required to meet the more stringent convergence limit. The pressure difference between the inlet and outlet parts is predicted as Pa/m. As a point of comparison, consider a long pipe elbow with a hydraulic diameter equal to that of the inlet and outlet ports and a bend radius of 2 m [6.6 ft]. The pressure drop through such a pipe is estimated to be Pa. As expected, the pressure drop across the room is greater than that of a smooth pipe. 6-17

34 Table 6-8. Comparison of Inlet and Outlet Flow Velocities for Test Case 5 Applied Condition Measured Velocity* Measured Mass Flow Inflow 0.25 m/s m/s kg/s Outflow Continuative m/s kg/s *Measured data taken 2 grid planes from boundary; this gives a more accurate representation of velocity through the opening instead of at a discrete boundary. 6-18

35 Figure 6-1. Geometry for Natural Convection Along a Vertical Flat Plate {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 6-19

36 Figure 6-2. Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection Along a Vertical Plate (Dimensions x and z in Meters) {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 6-20

37 Figure 6-3. Nusselt Number Comparison for Vertical Flat Plate {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} Figure 6-4. Vertical Velocity Distribution at m [0.410 ft] From the Leading Edge. Results Show Close Agreement Between the Nominal and Refined Mesh. [1 m = 3.28 ft] 6-21

38 Figure 6-5. Temperature Distribution at m [0.410 ft] From the Leading Edge. Results Show Close Agreement Between the Nominal and Refined Mesh. {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} Figure 6-6. Geometry for Natural Convection in a Square Cavity {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 6-22

39 Figure 6-7. Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection in a Square Cavity (Dimensions Shown in Meters) {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 6-23

40 Figure 6-8. Local Nusselt Number of the Vertical Walls and Experiment Data Are Shown for Hot and Cold Walls. Computational Fluid Dynamics Results Are Shown for Hot Wall Only. Figure 6-9. Local Nusselt Number Along Upper Wall 6-24

41 Figure Dimensionless Temperature Distribution at the Mid-Width (x/l = 0.5) Figure Velocity Distribution of the Mid-Height (y/l = 0.5) 6-25

42 Figure Dimensionless Temperature Distribution at the Mid-Height (y/l = 0.5) Figure Geometry for Natural Convection Between Concentric Cylinders [1 cm = 0.39 in] 6-26

43 Figure Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection Between Cylinders (Dimensions y and z in Meters) {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 6-27

44 Figure Local Value of k eff on Cylinder Walls [1 W/m-K = BTU/ft-h- F] Figure Dimensionless Fluid Temperature Profiles Along Radial Lines 6-28

45 Figure Experimental Apparatus Used in Test Case 4 Figure Thermocouple Placement Along Midline (Depth) of System. Left and Lower Inside Walls Shown at the Zero Axes Location. Offset Given Is in Centimeters. [1 cm = 0.39 in] 6-29

46 Figure 6-19a. Transient Development of Velocity Vector Flow Field, 1 8 Seconds 6-30

47 Figure 6-19b. Transient Development of Velocity Vector Flow Field, Seconds Figure Steady-State Velocity Vector Comparison of Fluent 4.52 Case and FLOW-3D Case 6-31

48 Figure Schematic of Experimental Apparatus Used for Test Case 5 Figure Visualization of the Computational Model 6-32

49 Figure Flow Vectors in Vented Box. Colored Contours Are the Value of the Out-of-Plane Vector Quantity in the Respective Views. [1 m/s = 3.28 ft/s] 6-33

50 7 MOISTURE TRANSPORT TEST CASES The objective of these test cases is to validate the moisture module in predicting the diffusion of water in an air/water mixture, the advection transport of water in a natural convection flow, and the phase change processes that are inherent under these conditions. 7.1 Conduction Heat Transfer and Vapor Diffusion This test case is depicted schematically in Figure 7-1. Two large, flat plates are separated by a gap filled with moist air. The top plate is held at constant temperature and the bottom plate at a lower temperature. Both surfaces provide a stationary film of water that can exchange mass with the water vapor in the air gap between the plates. It is assumed that there is no convection in the air gap. The following parameters define the necessary geometric and physical properties of the system: Gap thickness, L = 0.1 m [0.33 ft] Fluid thermal conductivity = k mix = W/(m 2 -K) [0.005 BTU/h-ft 2 - F] Top (hot) surface temperature = T h = 320 K [116 F] Bottom (cold) surface temperature = T c = 280 K [44.3 F] Pressure = 1 atm The equations describing the diffusion of thermal energy and water vapor across the gap are presented in Bird, et al. (1960) and can be solved exactly for the case in which the relative humidity is not limited to 100 percent. If the humidity is limited to 100 percent, the energy and mass transport equations must be solved numerically Test Input A FLOW-3D input file (prepin.*) was developed to model the idealized case of one-dimensional conduction heat transfer and chemical species diffusion through the air gaps. The lateral edges of the computational domain are specified as adiabatic surfaces. Convection flow was disallowed in the simulation. This portion of the problem specification is accomplished with the standard input file procedure of the basic FLOW-3D code. The standard version of FLOW-3D can simulate the diffusion of chemical species as defined in the idealized case. The unique feature of this problem is that the water at each surface must be in thermodynamic equilibrium with its vapor. The moisture transport processes were accomplished by providing user inputs to the customized portion of the code as described in Scientific Notebook 536E Test Procedure FLOW-3D was run with the input file described above until a steady-state condition was achieved. The output of the temperature profiles and water vapor concentration profiles are compared to the predictions of the mathematical analysis. 7-1

51 7.1.3 Expected Test Results The acceptance criterion for this test case is that the local temperatures and water vapor concentrations predicted by FLOW-3D shall be within 5 percent of the analytical predictions. In the problem described above, a mixture of air and water fills the space between two plates through which water can transpire. Water evaporates from the plate at a higher temperature, moves through the space, and condenses on the plate held at a lower temperature. The analysis of these processes is completely described in Scientific Notebook 536E. The following is a summary of that analysis. Bird, et al. (1960) developed the complete theory of the diffusion of chemical species in mixtures. Their treatment shows that the diffusion of water through a mixture of water and air in this one-dimensional problem can be expressed as where N w = ρ mix D wa dx w 1 xw dz (7-1) N w mole flux (mol/(m 2 sec)) x w mole fraction of water vapor (moles of vapor per total moles) D mix air/water mixture molar density (mol/m 3 ) D wa diffusion coefficient for water/air (m 2 /sec) z distance along flow path We are considering the steady diffusion of water vapor; so d ( ) dz N d ρmixd wa dx w w = dz x w dz = 0 (7-2) 1 The boundary conditions for this differential equation are x w x w,h at z = 0 (hot surface) x w x w,c at z = L (cold surface) Equation (7-2) can be readily solved for the water mole fraction profile ( ) = 1 ( 1 w, h) x z x w 1 x 1 x The steady mole flux can be obtained by substituting Eq. (7-3) for Eq. (7-1) wc, wh, zl / (7-3) N w ρ = D L 1 x, ln 1 x (7-4) mix wa wc wh, Note that the water concentration profile is nonlinear due to the presence of the air in the system. Also, as long as the thermodynamic and transport properties of the gases are independent of temperature, the concentration and mole flux of the water vapor is independent of temperature except in the specification of the boundary values. 7-2

52 The energy flux at any position in the gap is given by where ( ) kmix ( hn a a hwnw ) ez dt = + + (7-5) dz e energy flux (W/m 2 ) k mix air/water mixture thermal conductivity [W/(m 2 -K)] h a mole specific enthalpy of dry air (J/mol) h w mole specific enthalpy of water (J/mol) N a net mole flux of air [mol/(m 2 -s)] N w net mole flux of water [mol/(m 2 -s)] from Eq. (7-4) N a was assumed to be 0 for this test case as the boundary is closed with respect to air flow. Again, we are considering steady conditions. Assuming the thermal conductivity is independent of temperature, Eq. (7-5) can be differentiated to yield d dt Nw d + ( ) dz dz k dz h w = 0 (7-6) mix The water enthalpy is a unique function of temperature, so the boundary conditions for this equation are T T h at z = 0 (hot surface) T T c at z = L (cold surface) There are two scenarios to consider in the solution of Eq. (7-6). 1. There is no limit to the water concentration. That is, Eq. (7-6) describes the conservation of energy in a system of two completely immiscible gases. In other words, the water is allowed to be supersaturated (i.e., remain in the vapor state below the dewpoint) so that liquid water does not condense. 2. The water vapor concentration is limited to one corresponding to a relative humidity of 100 percent. In this scenario, the solution to Eq. (7-6) will be obtained under the assumption that the condensed liquid remains as a fog and diffuses along with the vapor. Scenario 1 No Condensate When there is no liquid water present, the enthalpy of the water can be expressed as hw = Cpw( T Tc) (7-7) where C pw mole specific constant pressure specific heat of water vapor (J/(mol K)) 7-3

53 The temperature of the cold surface is used to define the reference temperature for the enthalpy. Eq. (7-6) can then be readily integrated to yield ( ) = h ( h c) Tz T T T NwC 1 exp k mix NwC 1 exp k mix Note that the temperature distribution is a nonlinear function of distance. The deviation from a linear profile is dictated by the mole flux and the fluid properties. Eq. (7-8) is identical to the expression defined by Bird, et al. (1960) for this scenario. Scenario 2 Water Vapor Limited by Saturation Pressure pw pw z L When there is liquid water present, the enthalpy of the water can be expressed as (7-8) where x v hw = Cpw( T Tc) 1 h vap (7-9) x w C pw mole specific constant pressure specific heat of water vapor (J/mol) x v water vapor mole fraction x w combined vapor and liquid water fog mole fraction h vap heat of vaporization of water By substituting Eq. (7-9) into (7-6), and after much algebraic manipulation, the following equation can be obtained for the temperature. d dt + g ( T z) d ( ) ( ) dz dz dz T + g T z 1, 2, = 0 (7-10) The functions g 1 and g 2 are defined as where g g 1 ( T, z) ( T z) N h w = C pw + k mix x vap d dt P ( z) P w tot ( T) tot ( w ( )) vsat, = 0 N P x x w vsat 1 1 w z 1, = h vap ln k mix P x z zh 1 x, ( ) wh, 2 2 wc, (7-11) x w(z) total water concentration from Eq. (7-3) P tot nominal total pressure of the air/water mixture P v,sat (T) saturation vapor pressure of water Note that if h vap = 0 in Eq. (7-10), it reduces to the equation for the case in which no condensate forms. 7-4

54 The functions g 1 and g 2 are highly nonlinear and a closed form solution to Eq. (7-9) is difficult, if not impossible, to obtain. Instead, an approximate approach is used to solve this equation for the temperature profile. This solution procedure, described in detail in Scientific Notebook 536E, uses central differences to approximate the derivatives and a correlation for the saturation vapor pressure of water so that the functions g 1 and g 2 can be computed. The analysis was carried out in Mathcad and the analysis results were copied into Microsoft Excel for graphing and comparison to the FLOW-3D predictions Test Results Two FLOW-3D input files (prepin.*) corresponding to the two scenarios described above were prepared. These both use the custom moisture transport model in which the required input specifications follow the instructions in Scientific Notebook 536E (see entry for March 3, 2006) for using this module. Both simulations use a uniformly spaced, one-dimensional grid of 50 computation cells to discretize the space between the hot and cold surfaces. The FLOW-3D predictions for temperature and concentration were copied into a Microsoft Excel spreadsheet for graphing and comparison to the analytical results. The FLOW-3D simulations solve for the mass fraction of the water. The conversion from mass fraction to mole fraction was done in the Microsoft Excel spreadsheet Scenario 1 No Condensate in Air/Vapor The FLOW-3D predictions are compared to the theoretical predictions for this case in Figure 7-2. The FLOW-3D predictions for both the temperature profile and the water concentration profile are close to those from the theoretical analysis. The deviation in temperature values is defined as the ratio of the local variance and the overall temperature difference between the surfaces. Similarly, the water concentration variance is defined as the ratio of the local difference in the predictions and the difference in concentration values at the two surfaces. Thus, Temperature Variance Eq. (7-12) Concentration Variance Eq. (7-13) ( ) theory ( ) TFLOW 3D z T z Temperature Variance = T T h c (7-12) Concentration Variance = ( ) theory ( ) x z x z FLOW 3D x x wh, wc, (7-13) The FLOW-3D predictions and the theoretical results agree to within 1.5 percent. Note that the concentrations values are the water vapor mole fraction with respect to the total number of moles. The Mathcad file listed above shows that the water vapor concentration is greater than that for a 100 percent relative humidity across the entire gap in this case. 7-5

55 Scenario 2 Condensate Allowed in Air/Vapor The FLOW-3D predictions are compared to the theoretical predictions for this case in Figure 7-3. The FLOW-3D predictions for both the temperature profile and the water concentration profile closely agree with those of the theoretical analysis. Recall, however, that the theoretical results in this case were obtained by a numerical approximation of the governing differential equations. The FLOW-3D predictions and the theoretical results agree to within 5 percent over the entire domain. This level of variance is within the stipulated acceptance criterion. 7.2 Moisture Transport in a Closed Container This test case is based on the condensation cell experiment specifically conducted to validate the moisture transport model. The experiments are fully described in Scientific Notebook 643. The experiment setup is depicted schematically in Figure 7-4. The walls of this container are fabricated primarily of acrylic. The aluminum pan is attached to the floor at one end of the box and extends across the width of the box. The entire opposite end of the box is an aluminum plate that is cooled with chilled water flowing through passages machined into plate. The entire container is covered with Styrofoam insulation. The water pan is maintained at a constant temperature by a heater attached to its bottom. The water is maintained at a constant level by a siphon device between the pan and a water bottle that is located outside the acrylic enclosure. Thermocouples record the temperature at the locations shown in Figure 7-4. Condensed water is collected in a graduated cylinder. The net condensation rate is estimated by knowing the time period for collecting the observed amount of water. The laboratory experiment procedure calls for the heater and chiller to be adjusted to provide for constant temperatures as measured by thermocouples immersed in the water and attached to the cold plate surfaces. The test is operated for several hours until a steady condition is achieved as shown by the air temperatures and the condensation rate. Test runs with several different combinations of heater and chiller temperatures were conducted Test Input FLOW-3D input files (prepin.*) were developed to model the idealized case of two-dimensional flow in the vertical symmetry plane of the box. The box is wide enough that two-dimensional flow very nearly exists in this cross section; therefore, the simulation for this case was two dimensional. Boundary conditions and fluid properties were based on the thermal conditions specific to each experiment test run. The convection and conduction aspects of the problem are handled by the standard portions of the FLOW-3D code. The simulation of the moisture transport processes were accomplished by providing the user inputs to the customized portion of the code as described in Scientific Notebook 536E. 7-6

56 7.2.2 Test Procedure FLOW-3D was run with the input files described above until a steady-state condition was achieved. The output of the temperatures at the center of the container are compared to the test measurements. Likewise, the output of the condensation rate at the chilled plate are compared to the test measurements Expected Test Results The acceptance criterion for temperature predictions is that the air temperatures at the selected locations should agree with the measured values to within 20 percent. Similarly, the acceptance criterion for condensation rate is that the predicted condensation rate should agree with the measured value to within 20 percent. These levels of error are consistent with generally accepted errors for turbulent convection heat transfer experiments and correlations (e.g., Incropera and DeWitt, 1996). There were 18 test runs conducted in the condensation cell experiment described above. Test runs 1 3 were used to develop the proper test procedure and to check the instrumentation. Test runs were conducted without water in the system and served as a basis for comparing other types of computational fluid dynamics simulations. The results of test runs 4 15 are used here for validating the custom moisture transport model in FLOW-3D. The pertinent test results are described in detail in Scientific Notebooks 643 and 536E. The data used for the validation test are summarized in Table 7-1 and are sorted according to the cold plate and hot water temperatures, respectively. The test results are discussed and compared to the simulation results below Test Results FLOW-3D input files (prepin.*) were prepared corresponding to each of the 12 test runs described above. All of the input files use the custom moisture transport model in which the required input specifications follow the instructions in Scientific Notebook 536E for using this module. All the simulations used a two-dimensional grid (see Figure 7-5) with 90 cells along the inner length of the box and 23 cells along the inner height of the box. The box walls and the insulation were represented by a layer of 9-cells across their 7-7

57 Test Run Condensation Rate (ml/hr) Table 7-1. Condensation Cell Test Conditions and Test Results Water Temp* (/C) Cold Plate Temp* (/C) Lower Air Temp* T3 (/C) Mid-Air Temp* T4 (/C) Upper Air Temp* T5 (/C) Lid Temp* T8 (/C) Top Ins Temp* T9 (/C) Amb Temp* (/C) *Temp = temperature F = 1.8 T C combined thickness. A two-cell thick layer was used to set the outer surface boundary temperature in accordance with the measured test conditions. The simulations were carried out until an approximately steady condition was established. All of the simulations had semiregular oscillations in the velocity and temperature fields, indicating the presence of large-scale turbulent structures. The period of oscillation was roughly 5 seconds, but this depended on the test conditions. To determine the mean value of the temperature and condensation rates, the simulation results were recorded at simulated 1-second intervals for 20 seconds. This set of results was used to compute the mean values of temperature, surface heat flux, and condensation rate for comparison to the experiment results. An example of the velocity field and the fluid and solid temperatures is shown in Figure 7-6. This image is for test run 11 and shows the temperature contours for the fluid and all solid regions on the same temperature scale. This figure clearly shows the heated water tray and the cooled end plate of the box, but not the flow pattern within the box. The fluid rises from the heated water and circulates along the upper parts of the enclosure toward the cold plate. The cold plate cools the fluid and condenses water from the flow which then returns to the heater along the bottom portion of the box. The water mass fraction is shown in Figure 7-7 along with the velocity vectors. This figure shows only the fluid portion of the model. As expected, the highest concentration of water is in the fluid as it leaves the region near the heated water tray. The lowest concentration is at the bottom of the cold end wall where the fluid ends its contact with the cold plate after water condensed along this part of the circulation path. The temperatures in the middle of the box (T3, T4, and T5 locations) are compared in Figure 7-8. The overall trend of the FLOW-3D predictions is consistent with the measured results. There is not a substantial bias in the predictions (i.e., the predictions are not 7-8

58 consistently less than or greater than the measurements). The deviation in the results is greater at the lowest sensor location (T3) than at the other two locations. Using the difference in water temperature and cold plate temperature as a basis, the deviations between the measurements and predictions are computed as shown in Table 7-2. The predicted and measured condensation rates at the cold plate surface are compared in Figure 7-9. The trends of these results are all as expected in that the condensation rate is inversely proportional to the cold plate temperature and directly proportional to the temperature difference of the heated and cooled surfaces. There is reasonable agreement between the FLOW-3D predictions and the measured condensation rates; however, the predicted condensation rate is greater than the measured value in all cases except one. The deviation between the results is listed Table 7-2 where the variance relative to the mean condensation rate is given. The FLOW-3D predictions for the gas temperature and the condensation rate are all less than 20 percent for all of the test runs. The deviations between the measured and predicted values of air temperature and condensation rate are within the acceptance criterion defined above. Table 7-2. Variances of FLOW-3D Prediction From Measured Values in Condensation Cell Results Test Run Condensation Rate Lower Air Temperature T3 Mid-Air Temperature T4 Upper Air Temperature T5 8!1.5%!12.0%!7.7% 1.2% 7 6.6%!16.0%!8.7%!6.0% %!16.2%!7.1%!5.5% %!12.9%!3.0%!1.6% 6 2.4%!9.8% 0.6% 1.7% 9 0.5%!4.1%!0.1% 4.4% %!8.4%!0.1%!0.2% %!8.4% 0.3% 1.2% %!7.9% 1.9% 2.8% % 10.9% 19.9% 13.9% % 4.3% 13.5% 10.3% %!0.4% 8.2% 7.4% 7-9

59 Figure 7-1. Schematic for Heat Conduction and Species Diffusion Between Surfaces {T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 7-10

60 Figure 7-2. Temperature and Concentration Profile for One-Dimensional Conduction and Moisture Transport. Water Vapor Is Allowed to Be Supersaturated in This Scenario. {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 7-11

61 Figure 7-3. Temperature and Concentration Profile for One-Dimensional Conduction and Moisture Transport. Water Vapor Is Limited to 100-Percent-RH and Fog Is Allowed to Form. {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 7-12

62 Figure 7-4. Test Setup for Natural Convection and Water Vapor Transport in a Closed Container Figure 7-5. FLOW-3D Geometry for Condensation Cell Simulations 7-13

63 Figure 7-6. Predicted Temperature Contours and Velocity Vectors in Condensation Cell Figure 7-7. Predicted Water Mass Fraction in Condensation Cell (Unitless) 7-14

64 Figure 7-8. Mid-Line Temperature Ion Condensation Cell {T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} Figure 7-9. Cold Plate Condensation Rate {T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 7-15

65 8 THERMAL RADIATION TEST CASES Two test cases are presented here. The first validates the capabilities of the radiation module to compute radiation heat transfer and the inclusion of this heat transfer mode into the overall FLOW-3D simulation. The second test case validates the capabilities of the radiation module in computing radiation configuration factors. 8.1 Thermal Conduction and Radiation Between Two Surfaces This test case is depicted schematically in Figure 8-1. Two large flat plates are separated by a gap. The upper plate has internal heat generation so that the heat flux into the air gap is 255 W/m 2 [80.8 BTU/h-ft 2 ]. The bottom of the lower plate is held at a lower temperature. It is assumed that there is no convection in the air gap. The following parameters define the necessary geometric and physical properties of the system: Gap thickness, t gap = 0.1 m [0.3 ft] Plate thickness, t upper = t lower = 0.02 m [0.06 ft] Emissivity, g upper = g lower = 0.9 Gap thermal conductivity = k air = 0.1 W/(m 2 -K) [ BTU/h-ft 2 - F] Plate thermal conductivity, k upper = k lower = 1 W/(m 2 -K) [ BTU/h-ft 2 - F] Upper surface heat flux = Q upper = 255 W/m 2 [80.8 BTU/h-ft 2 ] Temperature of outside surface of lower plate = T c = 300 K [80.3 F] The radiation and conduction heat transfer processes were modeled by the appropriate exact one-dimensional equations for this case Test Input A FLOW-3D input file (prepin.*) was developed to model the idealized case of one-dimensional conduction heat transfer through the three objects. The lateral edges of the computational domain were specified as adiabatic surfaces to satisfy the one-dimensional condition in the FLOW-3D simulation. Air movement was disallowed in the simulation. This portion of the test input is accomplished with the standard input file procedure of the basic FLOW-3D code. The radiation heat transfer simulation was accomplished by providing the user inputs to the customized portion of the code as described in Scientific Notebook 536E. Because this is an idealized one-dimensional case, the radiation configuration factors are Test Procedure F 1!1 = F 2!2 = 0 F 1!2 = F 2!1 = 1 FLOW-3D was run with the input file described above until a steady-state condition was achieved. The output of the temperature profiles are compared to the predictions of the mathematical analysis. The output temperature profiles are used to compute the overall heat transfer rate for comparison to the analytical solution. 8-1

66 8.1.3 Expected Test Results The acceptance criterion for this test case is that the local temperatures predicted by FLOW-3D shall be within 5 percent (relative to the overall temperature difference between the two plates surfaces) of the analytical predictions. Similarly, the overall heat transfer rates should be within 5-percent analytical prediction. In the problem described above, dry air fills the space between two plates. The upper plate has a heat flux of 255 W/m 2 [80.8 BTU/h-ft 2 ] at its surface while the back side of the lower plate is held at a constant 300 K [80.3 F]. Heat is transferred by conduction and radiation across the air gap; convection is not allowed. The analysis of these processes is completely described in Scientific Notebook 536E. The following is a summary of that analysis. The conservation of energy for the air gap dictates that the upper plate heat flux is balanced by the heat transfer due to conduction and radiation across the air gap where Qupper = Qlower = Qair, cond + Qrad (8-1) Q upper heat flux at upper plate 255 W/m 2 [80.8 BTU/h-ft 2 ] Q air,cond heat flux through air via conduction Q rad heat flux across gap via radiation Q lower heat flux through the lower plate via conduction The conduction heat transfer through the air can be expressed as Q air, cond = k air T t T s, upper s, lower gap (8-2) where k air air thermal conductivity t gap gap width T s,upper temperature of upper plate surface in contact with air T s,lower temperature of lower plate surface in contact with air Siegel and Howell (1992) describe the governing equations for radiation in enclosures with diffuse, gray surfaces. For the one-dimensional case here, the radiation heat flux from the upper to lower surface is σ 4 4 Qrad = ( Ts, upper Ts, lower ) ε ε (8-3) upper lower 8-2

67 where F Stefan-Boltzmann constant = !8 W/(m 2 -K 4 ) [ BTU/(h-ft 2 - R 4 )], upper emissivity of upper surface = 0.9, lower emissivity of lower surface = 0.9 Finally, the heat transfer by conduction through the lower plate is Q lower = k lower T s, lower t lower T C (8-4) where k lower lower plate thermal conductivity t lower lower plate thickness T C fixed temperature at the back side of the lower plate = 300 K [80.3 F] Equations (8-1) and (8-4) form a nonlinear system of two equations with two unknowns, T s,upper and T s,lower. The solution procedure for this system of equations is described fully in Scientific Notebook 536E. It is found that T s,upper = K Q rad = 10.0 W/m 2 [3.2 BTU/ft 2 h] T s,lower = K Q cond,air = 245 W/m 2 [77.7 BTU/ft 2 h] Now that the plates surface temperatures have been determined, the temperature profiles in the plates and in the air gap can be specified. Define the location z = 0 at the center of the air gap, positive upward. The interface locations are z H m [+2.76 in] z s,upper m [+1.97 in] z s,lower!0.05 m [!1.97 in] z C!0.07 m [!2.76 in] The temperature distribution in the lower plate and the air gap follow the typical linear form for materials with constant thermal conductivity Tz ( ) T ( T T) z = z s, lower s, lower C t s, lower lower z z z, (8-5) c s lower ( ) = s, upper ( s, upper s, lower ) Tz T T T z z, z z s upper s, lower s, upper z z z (8-6) s, lower s, upper The upper plate has a uniform internal heat generation. It can be shown that the temperature distribution in this material is given by ( ) Tz ( ) Q z z = Ts, upper + 2k z z 1 2H z upper s, upper H s, upper upper s, upper 2 zs, upper z zh (8-7) 8-3

68 The temperature profiles in the three regions will be compared to the FLOW-3D predictions below Test Results FLOW-3D input files (prepin.*) were prepared corresponding to the one-dimensional problem description. The input files use the custom radiation module in which the required input specifications follow the instructions in Scientific Notebook 536E (see entry for March 3, 2006) for using this module. Two grid resolutions were specified for this problem. First, 28 cells were specified for the entire one-dimensional domain; 20 cells across the air gap and 4 cells through the thickness of each of the two plates. The second grid specified was at half the spacing of the first (i.e., a total of 56 uniformly spaced cells was used). The simulations were conducted until a steady condition was achieved. The FLOW-3D results for the temperature distribution are compared to the theoretical distributions in Figure 8-2. The FLOW-3D predictions for the two different grid resolutions are in close agreement, and both the predicted curves show slightly lower temperatures than the theory. The 28-cell predictions show a worst-case deviation of less than 5 percent from the theoretical values. The 56-cell predictions are slightly better with a worst-case deviation of just over 3 percent. The heat transfer rates predicted by FLOW-3D are 240 W/m 2 [76.1 BTU/ft 2 -h] by radiation and 15 W/m 2 [4.8 BUT/ft 2 -h] by conduction through the air. These differ from the corresponding theoretical values by about 2 percent of the total heat flux. The deviations between the theoretical and predicted values of air temperature and heat flux are within the acceptance criterion defined above. 8.2 Thermal Radiation Configuration Factors This test case demonstrates the computation of configuration factors, which is a part of the overall radiation module created for FLOW-3D. There are two scenarios in this test case. The first scenario is the two-dimensional geometry of concentric cylinders (Figure 8-3). The outer cylinder has an inner diameter of 1.0 m [3.3 ft], and the inner cylinder has an outer diameter of 0.3 m [0.98 ft]. In this scenario, each of the cylinder surfaces is divided into four subsurfaces of equal size for which the configuration factors are to be computed. The Hottel method [as described by Siegel and Howell (1992)] will be used to compute the configuration factors between each pair of surfaces for this case. In the radiation module, the radiative exchange from one part of a sector to another part of the same sector is neglected; therefore, these self-referenced configuration factors will be neglected here. The essential point of this scenario is to test the capability of the radiation module to account for blockages between surfaces so that the configuration factor is less than that for the condition in which the surfaces have an otherwise unimpeded view of each other. The second scenario is the radiation within a three-dimensional enclosure (Figure 8-4). This enclosure is 2 m 1 m 0.5 m. (Configuration factors are dimensionless so, the units of these dimensions are, in fact, not relevant.) The exact configuration factors can be computed from published literature for this geometry (e.g., Howell, 1982). The essential point of this scenario is to validate the radiation module configuration factor computations for three-dimensional problems. 8-4

69 8.2.1 Test Input FLOW-3D input files (prepin.*) were developed to model the idealized cases described above. The radiation heat transfer simulation was accomplished by providing the user inputs to the customized portion of the code as described in Scientific Notebook 536E. Only the configuration factors are to be validated here; the description of the fluid and other heat transfer related parameters is not necessary Test Procedure FLOW-3D was executed as required for only two time steps to allow the radiation module initialization to be executed. The computed configuration factors are recorded to a file as part of the initialization sequence. These values are compared to the exact values computed for the respective scenarios Expected Test Results The acceptance criterion for this test case are that the configuration factors predicted by FLOW-3D shall be within 5 percent of the analytical predictions. The radiation configuration factors for the two-dimensional geometry of the concentric cylinders are easily computed by the Hottel Crossed-String Method (Siegel and Howell, 1992). This method compares the lengths of line segments between combinations of the respective edges of two surfaces. The method is entirely geometric and does not involve the complicated integration procedure required of general three-dimensional problems. The computations for this case are described in detail in Scientific Notebook 536E in the entry for March 20, The results of the calculations are summarized in Table 8-1. It is noted that the configuration factors for a surface viewing itself (e.g., F1 1) are not zero for surfaces 1, 2, 3, and 4. Because the radiation module does not compute the self-viewing configuration factors, these values are not listed here. In the case of the three-dimensional rectangular box, there are only two general types of configurations for the surfaces for all non-zero factors. First, there are six cases in which the surfaces directly oppose each other. The radiation configuration factor for this configuration is given by Howell (1982) F a b = 2 2 ( 1+ X )( 1+ Y ) X + X( 1+ Y ) ln tan X + Y 1 2 ( 1+ Y ) πxy Y 2 + Y( 1+ X ) tan X tan ( Y) Y tan ( X) 2 ( + X ) (8-8) 8-5

70 Table 8-1. Configuration Factors, F a-b, Two-Dimensional Cylinders, Exact Solution Surface b n/a n/a n/a* n/a n/a n/a n/a n/a *n/a = not applicable Surface a where F a!b radiation configuration factor from surface a to surface b L distance between surfaces W width of the surfaces H height of the surfaces AN A L BN B L The other general type of configuration for the enclosure is for orthogonal faces with a common edge. All of the remaining 24 non-zero factors fall into this category. The configuration factors are defined by Howell (1982) as F a b = A tan + B tan ( B + A ) tan A B 1 πa W ( + A )( + B ) A ( + A + B ) ln A + B ( 1+ A )( A + B ) ( B + A ) B ( 1+ B + A ) ( 1+ B )( B + A ) H2 (8-9) where L common edge of the two surfaces A dimension of surface a normal to common edge B dimension of surface b normal to common edge The configuration factors for this rectangular enclosure are summarized in Table Test Results FLOW-3D input files (prepin.*) were prepared for each of these scenarios. The input files use the custom radiation module in which the required input specifications follow the instructions in 8-6

71 Table 8-2. Configuration Factors, F a-b, Three-Dimensional Box, Exact Solution Surface b n/a n/a* n/a n/a n/a n/a *n/a = not applicable Surface a Table 8-3. Configuration Factors, F a-b, Two-Dimensional Cylinders, FLOW-3D Results Surface b n/a n/a n/a n/a n/a n/a n/a n/a *n/a = not applicable Surface a Surface a Table 8-4. FLOW-3D Configuration Factor Errors, Two-Dimensional Cylinders Surface b %!3.81% 0.08% 0.13%!0.82%!0.82% % 0.08%!3.81%!0.82% 0.14%!0.82% 3!3.81% 0.08% 0.08%!0.82% 0.15%!0.82% %!3.81% 0.08%!0.82%!0.82% 0.14% % 0.11% 0.11% % 0.77% 0.10% % 0.77% 0.10% % 0.10% 0.77% Scientific Notebook 536E (see entry for March 3, 2006) for using this module. The fluid flow and heat transfer are not of interest here; the solutions were executed for only one time step so that the radiation module initialization sequence would be executed. The radiation configuration factors are computed during this initialization step. A grid of cells was used to simulate the concentric cylinder geometry. The results of the FLOW-3D computations for this case are summarized in Table 8-3. The variances between the exact values listed in Table 8-1 and the values computed by FLOW-3D are listed in Table 8-4. It is shown that the errors are all less than 5 percent. Also, note that the configuration factor F 1 2 should be the same as F 2 3. The FLOW-3D computations do not show these as being identical. To date, there has not been a reason determined for this discrepancy. It is perhaps related to a 8-7

72 slight error in the geometry specifications relative to grid line locations or round-off in the computations. Nonetheless, this is considered to be good agreement in light of the fact that most of these factors are affected by partial or full blockage by the inner cylinder. The FLOW-3D input file for the three-dimensional rectangular enclosure used a grid of cells to discretize the air space in the enclosure. A single layer of cells was used to define the six sides of the box. The results of the FLOW-3D computations for the three-dimensional rectangular enclosure configuration factors are summarized in Table 8-5. The variances between the exact values listed in Table 8-2 and the values computed by FLOW-3D are listed in Table 8-6. It is shown that the maximum error is just over 2 percent. Also, note that there is reasonable symmetry to the factors. If the grid resolution is halved in all three directions, the errors in the configuration factors are approximately double those listed here. This case is described in Scientific Notebook 536E. Surface a Table 8-5. Configuration Factors, F a-b, Three-Dimensional Box, FLOW-3D Results Surface b Surface a Table 8-6. FLOW-3D Configuration Factor Errors, Three-Dimensional Box Surface b % 2.02% 2.02% 2.14% 2.14% % 2.02% 2.02% 2.14% 2.14% % 2.02% 0.02% 2.01% 2.01% % 2.02% 0.02% 2.01% 2.01% % 2.14% 2.02% 2.02% 0.02% % 2.14% 2.02% 2.02% 0.02% 8-8

73 Figure 8-1. Schematic for Thermal Conduction and Radiation Between Opposing Surfaces {1 W/m-K = BTU/ft-h- F; 1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 8-9

74 Figure 8-2. Temperature Distribution for One-Dimensional Radiation and Conduction Across an Air Gap {1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 8-10

75 Figure 8-3. Schematic for Thermal Radiation in an Annular Gap [1 m = 3.28 ft] Figure 8-4. Schematic for Thermal Radiation in a Three-Dimensional Enclosure 8-11

76 9 COMBINED HEAT TRANSFER TEST CASE The test case described in this section validates the FLOW-3D code with the custom modules for moisture transport and radiation when all the pertinent modes of energy transfer conduction, convection, phase change, and radiation are present. 9.1 Convection, Radiation, and Moisture Transport in an Enclosure A two-dimensional enclosure measuring 0.1 m 0.1 m [3.9 in 3.9 in] is depicted in Figure 9-1. The left vertical wall is m [1 in] thick and has an internal heat generation rate such that the heat flux at the inner surface is 200 W/m 2 [63.4 BTU/ft 2 -h]. The outer surface of this wall is adiabatic. The right vertical wall is m [1 in] thick and its outer surface is held constant at 300 K [80.3 F]. The emissivity of both the left and right walls is 0.9. The vertical walls provide for the evaporation and condensation of water as needed under the existing temperature and concentration conditions in the flow. The upper and lower walls are adiabatic and do not exchange heat with the vertical walls. These walls are assumed to be transparent to radiation and therefore do not interact with the other walls via this mode. These walls are furthermore assumed not to be a source or a sink for water. The only interaction of these walls in the test case is to bound the flow and provide for viscous drag. The acceleration due to gravity is assumed to be only g so that the flow field for these geometric and thermal conditions will be laminar. The objective here is to compare the effects of radiation and moisture transport, not to accurately model a turbulent flow scenario. The FLOW-3D predictions will be compared to the predictions of an analysis of this scenario using a empirical heat transfer correlation approach. This approach is based on the equations described below. The Nusselt number correlation for natural convection in a two-dimensional square enclosure described by Berkovsky and Polevikov (1977) is a widely used relationship for this case. The net mass transfer rate of water vapor through the enclosure will be estimated using the analogy of heat and mass transfer (Incropera and Dewitt, 1996). This is a common practice based on the fundamentals of heat and mass transfer theory and similarity principles. Finally, the radiation heat transfer will be analyzed using the methods of Siegel and Howell (1992) for gray diffuse surfaces in an enclosure. The following properties are to be used for the fluid and wall. Viscosity, : = 2 10!5 Pa*sec Fluid thermal conductivity, k air = W/(m 2 -K) [ BTU/h-ft 2 - F] Air/Vapor diffusivity, D = !5 m 2 /sec [1 ft 2 /hr] Density, D = kg/m 3 (nominal value) [ lbs/ft 3 ] Wall thermal conductivity, k w = 1 W/(m 2 -K) [0.176 BTU/h-ft 2 - F] The density value listed above is used as the nominal density in the conservation of energy equation. In keeping with the moisture transport model, the incompressible ideal gas model is 9-1

77 used for this test case for the temperature and concentration dependent density that is used for the momentum equation. The moisture model parameters pertinent to this case are Water heat of vaporization, u fg = 2,304,900 J/kg [993 BTU/lb] Water vapor specific heat, C vv = 1370 J/(kg*K) [0.328 BTU/(lb R)] Water liquid specific heat, C vl = 4186 J/(kg*K) [1.00 BTU/(lb R)] Water vapor gas constant, R v = 416 J/(kg*K) [0.100 BTU/(lb R)] Air gas constant, R a = 289 J/(kg*K) [0.069 BTU/(lb R)] Test Input FLOW-3D input files (prepin.*) were developed to model the idealized cases as follows: 1. Convection only 2. Convection with radiation 3. Convection with moisture transport 4. Convection, radiation, and moisture transport The convection and conduction aspects of the problem are handled by the standard portions of the FLOW-3D code. The moisture transport and thermal radiation processes are accomplished by providing the user inputs to the customized portion of the code as described in Scientific Notebook 536E Test Procedure FLOW-3D was run with the input files described above until a steady-state condition was achieved. An analysis using the empirical heat and mass transfer correlations was developed for each of the four scenarios described above. The output of the heat transfer rates predicted by FLOW-3D is compared to the predictions of the engineering heat transfer analysis. The average wall surface temperatures and heat transfer rates predicted by FLOW-3D are compared to those resulting from the engineering analysis Expected Test Results The acceptance criterion for this test case is that the local temperatures predicted by FLOW-3D shall be within 20 percent (relative to the overall temperature difference between the two isothermal surfaces) of the analytical predictions. Similarly, the overall heat transfer rates should be within 20 percent of the analytical prediction. This is a generally accepted criterion in light of the approximate nature of the available empirical correlations for natural convection heat and mass transfer, especially in combination with radiation. The analytical solutions for the four different scenarios investigated here are all described in detail in Scientific Notebook 536E. The analysis approach is outlined here. The analysis seeks to determine the average temperatures of the vertical side walls and the heat transfer rates for the active heat transfer modes in each scenario. Consider a control volume enclosing the air in the box. The conservation of energy for this control volume dictates that 9-2

78 where Qh = Qc = Qconv + Qvap + Qrad (9-1) Q h specified heat flux from the left hot surface Q c heat flux through the right cold wall Q conv heat flux via convection across the enclosure Q vap heat flux via phase change at the side walls (if present) Q rad heat flux via radiation across enclosure (if present) The heat flux via convection is computed from where h heat transfer coefficient T s,h hot wall surface temperature T s,c cold wall surface temperature ( ) Qconv = h Ts, h Ts, c (9-2) The heat transfer coefficient is estimated from the definition of the Nusselt number, hl Nu = (9-3) where L enclosure dimension = 0.1 m [3.9 in] k air air thermal conductivity = W/(m 2 -K) [0.004 BTU/h-ft 2 -F] k air The Nusselt number for a square enclosure is given by Berkovsky and Polevikov (1977) as 029. Pr Nu = 018. Ra 02 (. + Pr) (9-4) where Pr air Prandtl number Ra Rayleigh number = Ra = ( s,h s,c ) aβ ν 2 T T L 3 Pr a g = acceleration due to gravity for this test case $ 2/(T s,h + T s,c ) approximate thermal expansion coefficient for air v air kinematic viscosity Pr fluid Prandtl number 9-3

79 The heat flux via conduction through the right wall is Q k T T sc, c = w t w C (9-5) where k w wall thermal conductivity = 1 W/(m 2 -K) [0.176 BTU/h-ft 2 - F] t w wall thickness = m T C cold wall outer surface temperature = 300 K [80.3 F] For the case when Q vap = Q rad = 0, Eqs. (9-1) and (9-5) form a nonlinear set of two equations with two unknowns, T s,h and T s,c, after the appropriate substitutions of Eqs. (9-2) through (9-4). The nonlinearity of the solution is increased if radiation is considered. The heat flux via radiation for this case is where Q rad σfh c = 1 1 ε + Fh c ε ε h c c 4 4 ( Ts,h Ts,c ) (9-6) F Stefan-Boltzmann constant = !8 W/(m 2 -K 4 ) [ BTU/(h-ft 2 - R 4 )], h, c = surface emissivity F h!c configuration factor = When Q rad is included in Eq. (9-1), the equations still form a system of two equations in two unknowns, but the nonlinearity of the system is increased because of the T 4 terms. Finally, when moisture transport is considered, the heat flux due to evaporation and condensation is approximated as where Q = m& h (9-7) rad v vap h vap heat of vaporization of water = J/kg [993 BTU/lb] &m v mass flux of water [kg/(m 2 sec)] The mass transfer rate is obtained by applying the analogy between heat and mass transfer (Incropera and DeWitt, 1992). In this method, the Nusselt number correlation can be used to define the Sherwood number with the substitution of the Schmidt number for the Prandlt number where 029. h L m Sc Sh = = D Sc Ra 018. wa 02 (. + ) (9-8) 9-4

80 Sh Sherwood number h m mass transfer coefficient (m/sec) D wa diffusion coefficient for air and water = !5 m 2 /sec [1.0 ft 2 /hr] Sc Schmidt number The water mass flux rate is then given by where ( ) & v m w, h w, c m = h c c (9-9) c w,h water vapor concentration at the hot wall surface (kg/m 3 ) c w,c water vapor concentration at the cold wall surface (kg/m 3 ) It is assumed here that the water vapor concentrations are those corresponding to air at 100-percent humidity at the wall temperature. So, adding Eq. (9-7) through Eq. (9-1) brings another term with the unknown temperatures into the analysis. The analysis results are summarized in Table 9-1 for each of the four different scenarios. The heat transfer rates and the mass transfer rate are expressed in terms of power per unit depth and mass flow per unit depth. Note the significant effect of the radiation and phase change on the hot wall surface temperature. When these modes are not considered, the hot wall temperature is about 300 K [80.3 F] hotter than when radiation and moisture transport are considered. In this case, energy transfer by radiation or phase change modes accounts for several times more energy flow than the convection heat transfer Test Results FLOW-3D input files (prepin.*) were prepared for each of these scenarios. The input files use the custom radiation module and moisture transport module as required. The input specifications follow the instructions in Scientific Notebook 536E (see entry for March 3, 2006) for using these modules. The specification of these files is described in detail in Scientific Notebook 536E. All of the simulations were conducted using a grid mesh in the flow field. Five cells were used through the thicknesses of the left and right walls, so the entire grid was cells. The FLOW-3D predictions for the four scenarios are summarized in Table 9-2. The values in parentheses are the relative variances from the corresponding value from the analytical results. The basis for the heat transfer variance is the total heat transfer rate of 20 W/m [19.22 BTU/h-ft] of depth. The basis for the mass transfer variance is the value from the empirical analysis. The basis for the temperature variance is the overall temperature difference between the hot and cold surfaces. The FLOW-3D results agree with the analytical results for this case. All of the deviations are less than 5 percent of their basis values and meet the acceptance criterion for this test case. 9-5

81 Scenario Convection Only Convection + Radiation Convection + Moisture Transport Convection + Radiation + Moisture Transport Table 9-1. Analysis Results for Two-Dimensional Enclosure Heat and Mass Transfer T h, average K* T c, average K Convection W/m Heat Transfer Rate Radiation W/m Phase Change W/m Total W/m Mass Transfer kg/sec n/a n/a 20 n/a n/a 20 n/a n/a ! !6 *T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32] 1 W/m = BTU/h-ft 1 kg/sec = 2.2 lb/sec n/a = not applicable Scenario Convection Only Convection + Radiation Convection + Moisture Transport Convection + Radiation + Moisture Transport Table 9-2. FLOW-3D Results for Two-Dimensional Enclosure Heat and Mass Transfer T h, average K* (!1.1%) (1.1%) (!4.4%) (!1.5%) T c, average K (0.1%) (!1.0%) (!1.4%) (!1.7%) Convection W/m Heat Transfer Rate Radiation W/m Phase Change W/m Total W/m Mass Transfer kg/sec 20 n/a2 n/a 20 n/a 2.3 (!2.0%) 1.8 (0.2%) 1.3 (0.3%) 17.7 (2.0%) n/a 18.2 (!0.4%) 7.9 (0.6%) n/a 20 n/a 11.0 (0%) !6 (0%) 20.2 (1.0%) !6 (0%) *These values are the average of all surface cells where the temperature values correspond to the FLOW-3D cell center located half the cell width from the obstacle surface. T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32] 1 W/m = BTU/h-ft 1 kg/sec = 2.2 lb/sec 2n/a = not applicable. 9-6

82 Figure 9-1. Schematic for Convection, Radiation, and Mass Transfer in a Two-Dimensional Enclosure {1 W/m-K = BTU/ft-h- F; 1 W/m 3 = BTU/(hr-ft 3 ); 1 m = 3.28 ft; T ( C) = (T (K)! ); T ( F) = [1.80 (T (K)! ) + 32]} 9-7

83 Figure 9-2. Wall Temperature Vectors for Scenario That Includes Convection, Radiation, and Moisture Transport (Dimensions x and z in Meters) [1 m = 3.28 ft] 9-8

84 10 INDUSTRY EXPERIENCE FLOW-3D is used widely in the casting industry because of its solid-liquid phase change capabilities and in the aerospace industry for its free surface, surface tension (i.e., zero gravity considerations), and noninertial reference frame capabilities. 10-1

85 11 NOTES None. 11-1