SOFTWARE VALIDATION REPORT FOR FLOW=3D ... VALIDATION REPORT FOR FLOW=3D@ VERSION 9.0 Prepared by S. Green C. Manepally Center for Nuclear Waste Regulatory Analyses San Antonio, Texas

SOFTWARE VALIDATION REPORT FOR FLOW=3D@ 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

ii

CONTENTS

Section Page

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 SCOPE OF THE VALIDATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11.1 Natural and Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.2 Moisture Transport With Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.3 Radiation Heat Transfer Between Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1-31.4 Combined Convection, Radiation, and Moisture Transport With

Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

2 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

3 ENVIRONMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13.1 Software—Standard Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13.2 Software Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13.3 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

4 PREREQUISITES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

5 ASSUMPTIONS AND CONSTRAINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

6 NATURAL AND FORCED CONVECTION TEST CASES . . . . . . . . . . . . . . . . . . . . . 6-16.1 Laminar Natural Convection on a Vertical Surface . . . . . . . . . . . . . . . . . . . . . 6-1

6.1.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.1.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.1.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.1.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

6.2 Turbulent Natural Convection in an Air-Filled Square Cavity . . . . . . . . . . . . . 6-46.2.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46.2.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46.2.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46.2.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.3 Natural Convection in an Annulus Between Horizontal Concentric Cylinders 6-66.3.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76.3.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76.3.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-86.3.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

6.4 Natural Convection Inside a Ventilated Heated Enclosure . . . . . . . . . . . . . . 6-126.4.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-136.4.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-136.4.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-136.4.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14

6.5 Forced Convection Inside a Confined Structure . . . . . . . . . . . . . . . . . . . . . . 6-156.5.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16

CONTENTS (continued)

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Section Page

6.5.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-166.5.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-176.5.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

7 MOISTURE TRANSPORT TEST CASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17.1 Conduction Heat Transfer and Vapor Diffusion . . . . . . . . . . . . . . . . . . . . . . . 7-1


7.2 Moisture Transport in a Closed Container . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-67.2.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-67.2.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-77.2.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-77.2.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

8 THERMAL RADIATION TEST CASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18.1 Thermal Conduction and Radiation Between Two Surfaces . . . . . . . . . . . . . 8-1


8.2 Thermal Radiation Configuration Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-48.2.1 Test Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-48.2.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-58.2.3 Expected Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-58.2.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

9 COMBINED HEAT TRANSFER TEST CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19.1 Convection, Radiation, and Moisture Transport in an Enclosure . . . . . . . . . . 9-1


10 INDUSTRY EXPERIENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

11 NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

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FIGURES

Figure Page

6-1 Geometry for Natural Convection Along a Vertical Flat Plate . . . . . . . . . . . . . . . . . . 6-196-2 Sample of Predicted Fluid Temperature and Velocity Vectors for Natural Convection

Along a Vertical Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-206-3 Nusselt Number Comparison for Vertical Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . 6-216-4 Vertical Velocity Distribution at 0.125 m [0.410 ft] From the Leading Edge . . . . . . . 6-216-5 Temperature Distribution at 0.125 m [0.410 ft] From the Leading Edge . . . . . . . . . . 6-226-6 Geometry for Natural Convection in a Square Cavity . . . . . . . . . . . . . . . . . . . . . . . . 6-226-7 Sample of Predicted Fluid Temperature and Velocity Vectors for Natural

Convection in a Square Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-236-8 Local Nusselt Number of the Vertical Walls and Experiment Data Are Shown for

Hot and Cold Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-246-9 Local Nusselt Number Along Upper Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-246-10 Dimensionless Temperature Distribution at the Mid-Width . . . . . . . . . . . . . . . . . . . . 6-256-11 Velocity Distribution of the Mid-Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-256-12 Dimensionless Temperature Distribution at the Mid-Height . . . . . . . . . . . . . . . . . . 6-266-13 Geometry for Natural Convection Between Concentric Cylinders . . . . . . . . . . . . . . 6-266-14 Sample of Predicted Fluid Temperature and Velocity Vectors for Natural

Convection Between Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-276-15 Local Value of keff on Cylinder Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-286-16 Dimensionless Fluid Temperature Profiles Along Radial Lines . . . . . . . . . . . . . . . . 6-286-17 Experimental Apparatus Used in Test Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-296-18 Thermocouple Placement Along Midline (Depth) of System. . . . . . . . . . . . . . . . . . . 6-296-19a Transient Development of Velocity Vector Flow Field, 1–8 Seconds . . . . . . . . . . . . 6-306-19b Transient Development of Velocity Vector Flow Field, 10–20 Seconds . . . . . . . . . . 6-316-20 Steady-State Velocity Vector Comparison of FLUENT 4.52 Case and

FLOW-3D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-316-21 Schematic of Experimental Apparatus Used for Test Case 5 . . . . . . . . . . . . . . . . . . 6-326-22 Visualization of the Computational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-326-23 Flow Vectors in Vented Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-33

7-1 Schematic for Heat Conduction and Species Diffusion Between Surfaces . . . . . . . 7-107-2 Temperature and Concentration Profile for One-Dimensional Conduction and

Moisture Transport. Water Vapor Is Allowed to Be Supersaturated . . . . . . . . . . . . 7-117-3 Temperature and Concentration Profile for One-Dimensional Conduction and

Moisture Transport. Water Vapor Is Limited to 100-Percent-RH and Fog . . . . . . . . 7-127-4 Test Setup for Natural Convection and Water Vapor Transport in a

Closed Container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-137-5 FLOW-3D Geometry for Condensation Cell Simulations . . . . . . . . . . . . . . . . . . . . . 7-137-6 Predicted Temperature Contours and Velocity Vectors in Condensation Cell . . . . . 7-147-7 Predicted Water Mass Fraction in Condensation Cell . . . . . . . . . . . . . . . . . . . . . . . 7-147-8 Mid-Line Temperature Ion Condensation Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-157-9 Cold Plate Condensation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15

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FIGURES (continued)

Figure Page

8-1 Schematic for Thermal Conduction and Radiation Between Opposing Surfaces . . . . 8-98-2 Temperature Distribution for One-Dimensional Radiation and Conduction Across

an Air Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-108-3 Schematic for Thermal Radiation in an Annular Gap . . . . . . . . . . . . . . . . . . . . . . . . 8-118-4 Schematic for Thermal Radiation in a Three-Dimensional Enclosure . . . . . . . . . . . 8-11

9-1 Schematic for Convection, Radiation, and Mass Transfer in a Two-Dimensional Enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7

9-2 Wall Temperature Vectors for Scenario That Includes Convection, Radiation, andMoisture Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8

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TABLES

Table Page

6-1 Laminar Plate FLOW-3D® Verison 9.0 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-36-2 Square Box Input File Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56-3 Overall Nusselt Number Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66-4 Experiments Selected for FLOW-3D® Version 9.0 Simulations . . . . . . . . . . . . . . . . . 6-96-5 FLOW-3D® Version 9.0 Simulation Conditions for Concentric Cylinders . . . . . . . . . 6-106-6 Comparison of Predictions and Measurements for Concentric Cylinders . . . . . . . . 6-116-7 Steady-State Temperature Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-156-8 Comparison of Inlet and Outlet Flow Velocities for Test Case 5 . . . . . . . . . . . . . . . 6-18

7-1 Condensation Cell Test Conditions and Test Results . . . . . . . . . . . . . . . . . . . . . . . . 7-67-2 Variances of FLOW-3D Prediction From Measured Values in Condensation

Cell Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

8-1 Configuration Factors, Fa-b, Two-Dimensional Cylinders, Exact Solution . . . . . . . . . . 8-68-2 Configuration Factors, Fa-b, Three-Dimensional Box, Exact Solution . . . . . . . . . . . . . 8-78-3 Configuration Factors, Fa-b, Two-Dimensional Cylinders, FLOW-3D Results . . . . . . . 8-78-4 FLOW-3D Configuration Factor Errors, Two-Dimensional Cylinders . . . . . . . . . . . . . 8-78-5 Configuration Factors, Fa-b, Three-Dimensional Box, FLOW-3D Results . . . . . . . . . . 8-88-6 FLOW-3D Configuration Factor Errors, Three-Dimensional Box . . . . . . . . . . . . . . . . 8-8

9-1 Analysis Results for Two-Dimensional Enclosure Heat and Mass Transfer . . . . . . . . 9-69-2 FLOW-3D Results for Two-Dimensional Enclosure Heat and Mass Transfer . . . . . . 9-6

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SOFTWARE VALIDATION REPORT FOR FLOW-3D® VERSION 9.0

FLOW-3D® Version 9.0 (Flow Science, Inc., 2005) is a general purpose, computational fluiddynamics simulation software package founded on the algorithms for simulating fluid flowthat were developed at Los Alamos National Laboratory in the 1960s and 1970s. The basisof the computer program is a finite volume formulation of the equations in an Eulerianframework describing the conservation of mass, momentum, and energy in a fluid. The code iscapable of simulating two-fluid problems: (i) incompressible and compressible flow and(ii) laminar and turbulent flows. The code has many auxiliary models for simulating phasechange, non-Newtonian fluids, noninertial reference frames, porous media flows, surfacetension effects, and thermoelastic behavior.

The code will be employed to simulate the flow and heat transfer processes in potentialhigh-level waste repository drifts at Yucca Mountain and support other experimental andanalytical 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-cylindricalcoordinate system. Fluid flow and heat transfer boundary conditions are applied at the sixorthogonal mesh limit surfaces. The code uses the so-called Volume of Fluid formulationpioneered by Flow Science, Inc., to incorporate solid surfaces into the mesh structure and thecomputing equations. Three-dimensional solid objects are modeled as collections of blockedvolumes and surfaces. In this way, the advantages of solving the different equations on anorthogonal, structured grid are retained.

The code includes the Boussinesq approach to modeling buoyant fluids in an otherwiseincompressible flow regime. The Boussinesq approximation neglects the effect of fluid (air)density dependence on pressure of the air phase, but includes the density dependence ontemperature. This approach will be heavily used in the simulation of in-drift air flow and heattransfer processes at Yucca Mountain. Fluid turbulence is included in the simulation equationsvia a choice of turbulence models incorporated into the software. It is up to the user to choosewhether fluid turbulence is significant and, if so, which turbulence model is appropriate for aparticular simulation.

1 SCOPE OF THE VALIDATION

FLOW-3D is capable of simulating a wide range of mass transfer, fluid flow, and heat transferprocesses. 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 exercisesas needed. FLOW-3D users are advised to perform validations pertinent to their particularneeds when applying this software to processes other than those treated here.

The validation test cases are summarized in the following subsections.

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1.1 Natural and Forced Convection

The natural and forced convection capabilities of the standard version of FLOW-3D areconsidered in this set of tests. Forced convection is another term for active ventilation. Withoutactive (or forced) ventilation, natural ventilation may occur.

Five test cases for natural and forced convection are described in Section 6. The first three testcases progress from a theoretical consideration of a hypothetical laminar natural convectionflow scenario to experimental treatments of heat transfer in laminar and turbulent flows. Thefourth and fifth test cases address forced convection, or ventilation, in thermally perturbedenclosures. These test cases cover a range of processes and geometries relevant topreclosure and postclosure issues in facilities and drifts at Yucca Mountain and are summarizedbelow.

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 arecompared to the semi-analytical solution of the boundary-layer type conservationequations derived specifically for this case.

2. Natural convection in a closed square cavity. This type of flow field was the subject ofan experimental study reported by Ampofo and Karayiannis (2003). Fine resolutionmeasurements of the fluid velocity, temperature, and wall heat flux are compared to theFLOW-3D simulation results.

3. Natural convection between two concentric cylinders. The experiment results reportedin Kuehn and Goldstein (1978) are used here to validate FLOW-3D Version 9.0.

4. Natural convection in a room with one inlet, one outlet, and a heat source. This testcase is modeled after the experiment described in Dubovsky, et al. (2001). In addition toa comparison against the measured data, FLOW-3D results are compared against theresults 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. Acomparison of velocity and mass at the inlet and outlet of the system at steady stateconfirms 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 bymathematical 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 temperaturesurface to a low temperature surface are studied. If the relative humidity is not limited toa maximum of 100 percent (i.e., a supersaturated condition is allowed), then thegoverning differential equations describing these processes can be solved for aone-dimensional case exactly as described by Bird, et al. (1960). Conversely, if the

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relative humidity is limited to a maximum of 100 percent, the governing equations arehighly nonlinear and must be solved numerically. The moisture transport module iscapable of solving both these scenarios and predictions will be compared to thetheoretical model of each scenario .

2. Moisture transport in a closed container. This test case is the simulation of theCondensation Cell Experiment as described by Scientific Notebook 643. A closedcontainer contains a heated pool of water at one end and a cooled wall at the other. Aconvection cell is established inside the container and water evaporated from the pool isadvected with and diffused through the air and is condensed on the cooled plate andparts of the other walls. The FLOW-3D simulation results are compared to themeasured temperatures and steady condensation rates.

These cases are relevant to the postclosure issues of moisture transport in a repository drift inthat the localized processes of evaporation and condensation are simulated and thethermodynamics 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 beinvestigated using analytical solutions of thermal radiative heat transfer processes.

1. Thermal conduction and radiation between two surfaces. Simplifying assumptions leadto an exact solution for the overall heat transfer rate following the methods describedby Siegel and Howell (1992). The FLOW-3D results are compared to theanalytical predictions.

2. Thermal radiation configuration factors. Radiation configuration factors are an importantaspect of radiation modeling, and it is important that these computations be validatedalong 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 inwhich the geometry can be considered two-dimensional. The second scenario is athree-dimensional rectangular enclosure. The configuration factors computed by theradiation module are compared to the results of exact equations for the configurationfactors for these two cases.

These cases are relevant to the postclosure issues of thermal radiation exchange in arepository because it is expected that radiation heat transfer will play a significant—andsometimes dominant—role in the overall heat transfer processes in the drift. Radiation heatfluxes are dependent on geometry only through the configuration factors. Thus, physical size isnot as important in this heat transfer mode as in convection and conduction. As long as therelative sizes of features are similar to the full scale, the geometric properties of the radiationexchange will be sufficiently tested.

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1.4 Combined Convection, Radiation, and Moisture Transport WithPhase Change

A single test case with four different scenarios is described in Section 9.0. This test case is ahypothetical condition of heat transfer in a square two-dimensional enclosure. This case canbe analyzed with accepted empirical correlations for the convection, radiation, and moisturetransport aspects of the problem. The following four scenarios make up this test case sothat the effects of moisture transport and radiation on the convection heat transfer canbe 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 arepository because it embodies all the modes of heat and mass transfer that are expected in thedrift. This case tests the functionality of the two software modifications for radiation andmoisture transport when they are applied together in the FLOW-3D computer code. Thephysical scale aspects of natural convection heat transfer are adequately addressed in theconvection-only tests in Section 6. The radiation heat flux exchange is relatively insensitive tophysical size and the moisture transport is a local phenomenon. So, this test case isconsidered adequate for validating the operation of the modified FLOW-3D in a mixed-modeheat transfer process.

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2 REFERENCES

Ampofo, F. and T.G. Karayiannis. “Experimental Benchmark Data for Turbulent NaturalConvection in an Air-Filled Square Cavity.” International Journal of Heat and Mass Transfer. Vol. 46. pp. 3,551–3,572. 2003.

Berkovsky, B.M. and V.K. Polevikov. “Numerical Study of Problems on High-Intensive FreeConvection.” Heat Transfer and Turbulent Bouyant Convection. Vol. II. D.B. Spalding andH. Afgan, eds. Washington, DC: Hemisphere Publishing. pp. 443–455. 1977.

Bird, R.B., W.E. Stewart, and E.N. Lightfoot. Transport Phenomena. New York City, New York: John Wiley and Sons. 1960.

Churchill, S.W. and H.S. Chu. “Correlating Equations for Laminar and Turbulent-FreeConvection from a Vertical Plate.” International Journal of Heat and Mass Transfer. Vol. 18. pp. 1,323–1,329. 1975.

Dubovsky, V., G. Ziskand, S. Druckman, E. Moshka, Y. Weiss, and R. Letan. “NaturalConvection Inside Ventilated Enclosure Heated by Downward-Facing Plate: Experiments andNumerical Simulations.” International Journal of Heat and Mass Transfer. Vol. 44. pp. 3,155–3,168. 2001.

Flow Science, Inc. “FLOW-3D® User’s Manual.” Version 9.0. Sante Fe, New Mexico: FlowScience, Inc. 2005.

Fluent Inc. “FLUENT User’s Guide.” Version 4.52. Lebanon, New Hampshire: Fluent Inc. 1994.

Green, S. and C. Manepally. “Software Validation Test Plan for FLOW-3D Version 9.” Rev. 1.San Antonio, Texas: CNWRA. 2006.

Green, S. “Software Requirements Description for the Modification of FLOW-3D to IncludeHigh-Humidity Moisture Transport Model and Thermal Radiation Effects Specific to RepositoryDrifts.” San Antonio, Texas: CNWRA. 2006.

Howell, J.R. A Catalog of Radiation Configuration Factors. New York City, New York: McGraw-Hill Book Company. 1982.

Incropera, F.P. and D.P. Dewitt. “Fundamentals of Heat and Mass Transfer. 4th Edition.” pp. 487–490. New York City, New York: John Wiley and Sons, Inc. 1996.

Kuehn, T.H. and R.J. Goldstein. “An Experimental Study of Natural Convection Heat Transfer inConcentric and Eccentric Horizontal Cylindrical Annuli.” ASME Journal of Heat Transfer. Vol. 100. pp. 635–640. 1978.

Kuehn, T.H. and R.J. Goldstein. “An Experimental Study and Theoretical Study of NaturalConvection in the Annulus Between Horizontal Concentric Cylinders.” Journal of FluidMechanics. Vol. 74, Part 4. pp. 695–719. 1976.

2-2

Moran, M.J and H.N. Shapiro. “Fundamentals of Engineering Thermodynamics. 4th Edition.” New York City, New York: John Wiley and Sons, Inc. 2000.

Ostrach, S. “An Analysis of Laminar-Free Convection Flow and Heat Transfer About a FlatPlate Parallel to the Direction of the Generating Body Force.” NASA Report 1111. 1953.

Schlichting, H. “Boundary Layer Theory.” 6th Edition. New York City, New York: McGraw-HillBook Company. pp. 300–305. 1968.

Siegel, R. and J.R. Howell. Thermal Radiation Heat Transfer. 3rd Edition. Washington, DC: Hemisphere Publishing Corporation. 1992.

3-1

3 ENVIRONMENT

3.1 Software—Standard Installation

The FLOW-3D software package has been in use since the early 1980s. It was originally basedon algorithms developed by the founders of Flow Science, Inc., when they were employed atLos Alamos National Laboratory. While the original code was a general purpose computationalfluid dynamics package that could simulate the effects of irregular solid objects, it wasespecially noted for its ability to simulate free surfaces and reduced gravity. The current versionof the code is a much-enhanced descendent of that early software package and is widely usedin industry and government agencies. A description of the software may be found at the FlowScience, Inc., website: http:/www/flow3d.com.

This software validation uses of FLOW-3D, which can operate in a Windows or Linux/UNIXenvironment. The graphical user interface is started by clicking on the executable file. The usereither creates a new simulation using the menus available in the graphical user interface, or apreviously created setup file can be opened for continued work or modification. The setup filecreated by the user completely describes the simulation and is all that is required to recreateresults for a particular scenario. Computational fluid dynamics simulations often take manyhours or even days to complete; hence, users should retain files holding simulation results forfuture analyses and postprocessing.

3.2 Software Modifications

The FLOW-3D software delivered by the vendor includes options for customizing the programfor special flow and heat transfer processes not covered in the basic code capabilities. Thebase code was modified in accordance with the requirements described by Green (2006). Anadvanced user of the FLOW-3D software can understand the code modifications; a completedescription underlying theory, and details of the modifications are described in ScientificNotebook 536E.

The basic FLOW-3D code is incapable of simulating the transport processes associated withthe 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 thesecases. The computing algorithm described by Green (2006) was programmed in accordancewith the logical framework of the FLOW-3D computer program. The moisture transport moduleadded to FLOW-3D is capable of modeling

• Water evaporation from saturated surfaces into the air when the surface temperature isabove the local dewpoint

• Water condensation to surfaces from the air when the surfaces are at a temperature lessthan 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 heattransfer cools the air to the local dewpoint

3-2

One limitation of this moisture transport module is that any water condensed as a mist will notcoalesce and rain (i.e., it is assumed that the mist diffuses and advects much like anatmospheric fog). The water that condenses on solid surfaces, however, is removed from thegas phase in the fluid cells adjacent to the surface. Water vapor is thereby transported bydiffusion 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 solidsurfaces. This heat transfer process can be a significant portion of the overall heat transfer in arepository where natural convection and conduction are the only other means of energytransport between waste packages and the drift walls. The computing algorithm described byGreen (2006) was programmed in accordance with the logical framework of the FLOW-3Dcomputer 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 onthe location and orientation with respect to the other surfaces.

• Radiation configuration factors are computed for the radiation-active surfaces or can beprovided 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 asdescribed in the FLOW-3D manual. All of the tests described here were conducted onpersonal computers with Windows 2000 or Windows XP. For the test cases using themoisture transport module and the radiation module, the software was compiled and linkedusing Compaq Visual Fortran Version 6.6c in accordance with the FLOW-3D User’s Manual(Flow Science, Inc., 2005).

4-1

4 PREREQUISITES

Users should be trained to use FLOW-3D and have experience in fluid mechanics andheat transfer.

5-1

5 ASSUMPTIONS AND CONSTRAINTS

None.

6-1

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 surfaceprovide a method to validate the accuracy of FLOW-3D for natural convection. This flow field issketched in Figure 6-1 for the configuration and specifications used in the FLOW-3Dsimulations. The analytical solution documented by Incropera and Dewitt (1996) provides anexpression for the local Nusselt number and average Nusselt number for laminar flow cases(Rayleigh number < 109). The Nusselt number is a dimensionless temperature gradient at asurface and provides a measure of the efficiency of convection for heat transfer relative toconduction. The empirical correlation in Churchill and Chu (1975) provides an improvement tothe analytical solution for average Nusselt numbers at lower Rayleigh numbers. For thisvalidation test case, the local and average Nusselt numbers are compared to the FLOW-3Dresults and these published analytical and empirical correlations.

6.1.1 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 astwo-dimensional with an incompressible fluid and the Boussinesq approximation to capture thethermal buoyancy effects. No turbulence model was used.

6.1.2 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 averageNusselt numbers for comparison to the benchmark correlations.

6.1.3 Expected Test Results

The criteria for test acceptance were established in the software validation test plan (Green andManepally, 2006). For the refined mesh, the benchmark and average Nusselt numbers on thevertical wall shall agree within ±10 percent. The local Nusselt number for the region from 10 to90 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 shouldagree within ±20 percent. The local Nusselt number for the region from 10 to 90 percent ofthe length (i.e., the entry 10 percent and exit 10 percent shall be neglected) should agree within20 percent.

The analytical solution documented by Incropera and Dewitt (1996) provides an expression forthe local Nusselt number and average Nusselt number for laminar natural convection cases(Ra < 109) involving an isothermal vertical flat plate. The local Nusselt number expression is anempirical correlation to the theoretical solution obtained from the laminar boundary-layer theorydeveloped by Ostrach (1953).

6-2

( )( )

( )Nu z Pr

Pr Pr

Gr zz =

+ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

0.75

0.609 1.221 4

0.5

0.5 0.25z

1238

0 25

.

.

( )Gr z g T T zz s f( ) 3= −ρ βμ

2

2

Nu Ra Ra

Pr

L ( ) = +

+ ⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0.68 0.67

1 0.492

0.25

916

49

(6-1)

where Nuz(z) is the local Nusselt number, Pr is the fluid Prandtl number, and Gr(z) is the localGrashof number evaluated at the distance from the leading edge. The local Grashof number(Grz) is defined as

(6-2)

where

D — fluid densityg — acceleration of gravity$ — fluid thermal expansion coefficientTs — isothermal plate temperatureTf — fluid free stream temperaturex — distance from the leading edge

— fluid viscosityμ

The average Nusselt number, NuL, is an empirical correlation developed by Churchill and Chu(1975) and is also documented in Incropera and Dewitt (1996).

(6-3)

where NuL(Ra) is the effective Nusselt number of the entire plate length and Ra is the Rayleighnumber (Ra = Gr*Pr) evaluated with a length scale equal to the entire plate length. TheChurchill and Chu (1975) correlation agrees with the test data better than the correspondingcorrelation 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 andthese analytical and empirical correlations are compared.

6.1.4 Test Results

A FLOW-3D input file (prepin.*) was developed to model the vertical flat plate naturalconvection. The simulation is of an isothermal vertical plate with a temperature of340 K [152 °F] with a fluid having a free stream temperature of 300 K [80 °F]. The plate lengthis 0.2 m. The case is modeled as two-dimensional with an incompressible fluid. A Boussinesq

6-3

approximation is used to model the thermal buoyancy effects in the nominally incompressiblefluid. No turbulence model is used.

Three different grid resolution cases were analyzed. A refined mesh was developed based on agrid sensitivity analysis. This mesh provides more accurate results and the accuracy limits ofFLOW-3D for this particular test case. A coarse mesh with grid resolution similar to what isexpected to be practical for future modeling of the full-scale Yucca Mountain drifts was alsospecified for comparison to the more refined grids. Table 6-1 lists the prepin.* files for this testcase. The grid is clustered at the leading edge in all three meshes. For the base case, the gridspacing is 0.1 mm × 0.1 mm [0.004 in × 0.004 in], and it expands geometrically along thesurface and away from the surface. The leading edge grid spacing is scaled accordingly for thecoarse and refined meshes.

A sample of the flow field predicted by FLOW-3D for the 50 × 100 grid is shown in Figure 6-2. This figure shows the velocity vectors and fluid temperature contours after steady flow hasbeen achieved. For the sake of clarity, the vectors are shown for every second mesh line in thez-direction. This figure shows that the flow domain boundary at the right side was chosen farenough away from the plate to represent a quiescent far-field condition. Also seen is theclear development of a boundary layer as the flow proceeds upward from the bottom edge ofthe plate.

The calculations for the analytical solution for this test case are documented by David Walter inScientific Notebook 576.

The FLOW-3D predictions are compared to those obtained from the analytical expression inFigure 6-3. The local Nusselt number results for the nominal mesh were within ±10 percent ofthe analytical solution, and the overall effective Nusselt number was within ±8 percent of theempirical correlation. These variances are within the acceptance criteria described above;therefore, the results of this test case are acceptable for software validation. The overallNusselt number result for the coarse mesh was 23.5 percent lower than the analyticalpredictions. This exceeds the acceptance criterion for this mesh which indicates that theselected mesh is too coarse for this analysis.

Mesh independence of the nominal mesh solution was demonstrated by close agreement of thelocal Nusselt number results between the nominal and refined mesh solutions. Mesh independence is further demonstrated by the agreement of the velocity and temperatureprofiles shown in Figures 6-4 and 6-5. The deviation of the coarse mesh results form theothers further shows that this mesh is inadequate for this analysis. This demonstrates theneed to conduct grid resolution studies in computational fluid dynamics analyses to ensuregrid-independent results.

Table 6-1. Laminar Plate FLOW-3D® Verison 9.0 Input File SummaryFile Name Description Comment

prepin.vertsurf40K Nominal Mesh 50 × 100 Meshprepin.vertsurf40K-CMesh Coarse Mesh 10 × 20 Meshprepin.vertsurf40K-Rmesh Refined Mesh 100 × 200 Mesh(demonstrate grid

independence)

6-4

6.2 Turbulent Natural Convection in an Air-Filled Square Cavity

An experimental study conducted by Ampofo and Karayiannis (2003) providesbenchmark data to evaluate the accuracy of FLOW-3D for natural convection in low-levelturbulence. The two-dimensional experimental work was conducted on an air-filled squarecavity of dimension 0.75 × 0.75 m2 [2.5 × 2.5 ft2] with vertical hot and cold walls maintained atisothermal temperatures of 50 and 10 °C [122 and 50 °F]. This flow field is sketched onFigure 6-6 for the configuration and specifications used in the FLOW-3D simulations. Theseconditions resulted in a Rayleigh number of 1.58 × 109. For this validation test case, the localand average heat transfer rates (described by the Nusselt number), the local velocities, and thetemperature profiles are compared between the FLOW-3D and experimental results.

6.2.1 Test Input

The experiment was modeled as two-dimensional with an incompressible fluid and theBoussinesq approximation to capture the thermal buoyancy effects. The large eddysimulation 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 experimentalapparatus described by Ampofo and Karayiannis (2003).


FLOW-3D was run with the input file, as described in the previous section. The output of thewall heat transfer rates, temperature and velocity profiles, and the mid-width and mid-heightwere compared to the experimental benchmark data.


The criteria for test acceptance were established in the software validation test plan (Green andManepally, 2006).

For the refined mesh, the experimental and numerical simulation average Nusselt numbers onthe horizontal and vertical walls should agree within ± 20 percent. The fluid temperature andvelocity profiles will be compared graphically to the measured values. The trends of the profileswill be compared for overall goodness of fit.

For the coarse mesh, the experimental and simulation average Nusselt numbers on thehorizontal and vertical walls shall agree within ± 25 percent. The fluid temperature and velocityprofiles will be compared graphically to the measured values. The trends of the profiles will becompared for overall goodness of fit.

Ampofo and Karayiannis (2003) measured the time-varying temperature and velocity at a largenumber of points in the central plane of the cavity where two-dimensional flow is closelyapproximated. Low-noise type-E thermocouples were used to measure the instantaneoustemperature, and a two-dimensional laser Doppler velocimeter was used to measure the fluidvelocity. The velocity and temperature measurements were made simultaneously, but wereseparated by a distance of about 0.4 mm [0.016 in]. This distance is far enough to prevent thetemperature and velocity probes from interacting with each other and is approximately the size

6-5

of the turbulence-length scale near the wall. This is required to ensure that the temperature andvelocity are measured as close to simultaneously as possible.

The local values of Nusselt numbers are computed using the near-wall temperature distributionto estimate the local wall heat flux. This is valid because there are 10–12 measurement pointswithin the laminar sublayer next to the wall. The overall value of the Nusselt number iscomputed from the area-weighted average of the local values.

6.2.4 Test Results

A FLOW-3D input file (prepin.*) was developed to model the square cavity experiment. Theexperiment was modeled as a two-dimensional flow of an incompressible fluid with aBoussinesq approximation to simulate the thermal buoyancy effects. The model geometry,fluid properties, and boundary conditions matched the experimental apparatus described byAmpofo and Karayiannis (2003) as closely as practical. The FLOW-3D large eddy simulationmodel was used to model fluid turbulence. The large eddy simulation model is inherentlythree-dimensional. It can be used for this two-dimensional analysis by selecting the thicknessof a two-dimensional slice consistent with the grid spacing so that the large eddy simulationlength 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 agrid sensitivity analysis. This mesh should provide more accurate results and the accuracylimits of FLOW-3D for this particular test case. A coarse mesh with grid resolution similar towhat is expected to be practical for future modeling of the full-scale Yucca Mountain drifts wasalso 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 inFigure 6-7. These results are from the simulation with the 200 × 200 mesh resolution. Thisfigure shows the overall clockwise circulation pattern and the large turbulent eddies that formalong the vertical walls. Note that the disturbances are formed at about the mid-height of eachvertical wall.

The computational fluid dynamic predictions are compared to the test results in Table 6-3 andFigures 6-8 through 6-12. All postprocessing calculations for these results are described byDavid Walter in Scientific Notebook 576.

Table 6-2. Square Box Input File SummaryFile Name Description Comment

prepinr.sqr_box-200X200 Nominal Mesh 200 × 200 Meshprepinr.sqr_box-75X75 Coarse Mesh 75 × 75 Meshprepinr.sqr_box-300X300 Refined Mesh 300 × 300 Mesh

to demonstrate grid independence

6-6

Table 6-3. Overall Nusselt Number ComparisonLower Wall Upper Wall Hot Wall Cold Wall

Experimental Data 13.9 14.4 62.9 62.6Computation Fluid Dynamics Results

Mesh Size Nu Error Nu Error Nu Error Nu Error75 × 75 10.7 !23% 10.6 !36% 58.6 !7.3% 59.4 !5.1%

200 × 200 12.9 !7.8% 12.8 !13% 55.9 !13% 56.0 !12%300 × 300 12.0 !16% 11.9 !21% 54.1 !16% 53.8 !16%

The predicted Nusselt number values for all of the enclosure surfaces are less than themeasured values. The variances for all of the predictions from the various mesh resolutions are25 percent of the values measured in the experiments, except for the upper wall using thecoarse mesh. The nominal and refined mesh results meet the acceptance criteria listed inSection 6.2.3.

The local Nusselt number values for the sidewalls are compared to the measured values inFigure 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 towardsthe corner (ie., y/L < 0.1), whereas the measured results show a monotonic increase in Nusseltnumbers near the corner. This difference in Nusselt number values in the corners is stronglyaffected by the specification of the thermal conditions for the walls. A perfectly conducting wallyields a high value of Nusselt number for the vertical wall in the corner, while a perfectlyabiabatic wall yields a nearly zero value for the Nusselt number for the corner. The Nusseltnumber value for real conditions is between these limits. The thermal conditions are dictated bythe precision of the isothermal surface condition, the interface conductance between thevertical and horizontal walls, and the thermal conductivity of the horizontal wall. The sourcearticle did not provide sufficient detail to allow for a high-fidelity simulation of the cornerconditions. The assumptions described for this simulation were not altered to home in on amore agreeable match.

The fine-grid results presented in Figures 6-8 through 6-12 are not substantially different fromthe nominal-grid results. It is counterintuitive that the refined mesh results show a largerdiscrepancy in the Nusselt numbers than the nominal mesh results. This is thought to berelated to the problem of modeling the heat transfer within the walls near the corner. The smalldifferences in the temperature profiles of the two sets of predictions is amplified in the Nusseltnumber calculations. Based on the quality of agreement in the temperature profile, the nominalmesh resolution is considered to be adequate for this analysis.

6.3 Natural Convection in an Annulus Between HorizontalConcentric Cylinders

Kuehn and Goldstein (1978) conducted experiments on the temperature and heat fluxmeasurements of the thermal behavior of a gas in an annulus between concentric and eccentriccircular cylinders. Only the concentric configuration (Figure 6-13) was considered for thisvalidation activity. This is a widely referenced article for empirical correlations and validations ofcomputational fluid dynamics calculations of natural convection flows. The experimenters usedthermal conductivity for the condition in which heat transfer only occurs by conduction acrossthe radial gap between the cylinders.The equivalent thermal conductivity of the annulus gas is defined as

6-7

( )k

Q D DY T

o ieff

ln=

2 π Δ

Ra g T L PrL = ρ βμ

2

23Δ

(6-4)

where

Q — heat transfer rate at inner cylinderDo — inner diameter of the outer cylinder Di — outer diameter of the inner cylinderY — length of annulus

— temperature difference between cylindersΔT

For pure conduction, keq = 1 and keff increases to nearly 20 for the most turbulent flow reportedby Kuehn and Goldstein (1978).

The results are correlated by the Rayleigh number for gap width

(6-5)

where

p — gas densityg — acceleration due to gravity

— thermal expansion coefficient of gasβ— dynamic viscosityμ

L — 0.5(Do ! Di) = gap width delineated by the diameters of the cylindersPr — gas Prandtl number

6.3.1 Test Input

FLOW-3D input files were developed for the cases described by Kuehn and Goldstein (1978) forRaL = 1.31 × 103, RaL = 6.19 × 104, RaL = 6.81 × 105, RaL = 2.51 × 106, RaL = 1.90 × 107, andRaL = 6.60 × 107. These represent laminar to fully turbulent flow regimes as RaL increases.


FLOW-3D was run using an identical grid for all cases. In addition, the flow with theRaL = 2.51 × 106 was simulated with a finer grid resolution to demonstrate that the simulationresults are approximately grid-independent. The FLOW-3D results were used to compute theeffective overall equivalent thermal conductivity for comparison to the experiment results ofKuehn and Goldstein (1978). The calculated fluid temperature profiles across the gap werecompared to the available experiment results.


The criteria for test acceptance were established in the software validation test plan (Green andManepally, 2006).

6-8

The acceptance criterion for the simulated overall equivalent thermal conductivity will be adeviation of no more than 25 percent of the measured value. The fluid temperature profilesacross the gap will be compared graphically to the measured values. The trends of the profileswill be compared for overall goodness of fit.

The experiment facility consisted of two concentric cylinders sealed in a pressure vessel. Theouter diameter of the inner cylinder was 3.56 cm [1.40 in], and the inner diameter of the outercylinder was 9.25 cm [3.64 in]. The annular gap between the cylinders was 2.845 cm [1.12 in]. The cylinders were 20.8 cm [8.19 in] in length. The inner cylinder was electrically heated, whilethe outer cylinder was cooled with a chilled water loop. The test chamber was filled withnitrogen as the test fluid. The nitrogen pressure was varied between 0.071 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 2.2 × 102 to 7.74 × 107. Temperatures in the annulus were measured via Mach-Zender interferometry, and surfacetemperatures were measured with thermocouples.

This range of Rayleigh number values does not represent the expected Rayleigh number rangeof the Yucca Mountain drifts when there is no drip shield. When a drip shield is used, the rangeof Rayleigh number values covered by the experiments includes an expected range of Rayleighnumbers 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 theannulus in terms of an effective thermal conductivity, as described in Section 6.3. The value ofkeff can be either a local value or a net value for a large surface, depending on whether thevalue 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 orthe derived values of keff. In an earlier paper, Kuehn and Goldstein (1976) state that thevariance between values of keff derived from interferogram contours and those obtained fromoverall heat transfer rates was less than 3 percent. The difference between the derived value ofthe Rayleigh number was about 7 percent. It is noted that the results reported in Kuehn andGoldstein (1976) do not extend to the highest Rayleigh numbers reported in Kuehn andGoldstein (1978).

For nitrogen, a single fringe shift in the interferogram corresponds to about 35 K [63 °F] at apressure of 0.15 atm and about 0.1 K [0.18 °F] at a pressure of 35 atm.

6.3.4 Test Results

The details of the validation test for this case are described in Scientific Notebook 536E. Thetest 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

6-9

Table 6-4. Experiments Selected for FLOW-3D Simulations

Ragap

Patm

ΔT°C [°F]

½(Ti + To)°C [°F]

1.31 × 103 0.110 53.5 [128.5] 51.1 [124.2]6.19 × 104 0.977 38.0 [100.6] 44.4 [112.1]6.81 × 105 8.46 4.29 [39.9] 27.3 [81.3]2.51 × 106 34.6 0.91 [33.8] 27.7 [82.0]1.90 × 107 34.7 7.01 [44.8] 29.1 [84.6]6.60 × 107 35.0 28.7 [83.9] 40.8 [105.6]

a fine resolution mesh at Ra = 2.51 × 106 since Kuehn and Goldstein (1978) provide somedetails of the temperature profiles along the cylinder surfaces and between the cylinders for thiscase. The conditions used for the seven computational fluid dynamics simulations are listed inTable 6-5.

The properties of nitrogen at the selected test conditions were provided from the computerprogram NIST12. This program is equivalent to the web-based property information softwareavailable at the National Institute of Standards and Technology.

FLOW-3D uses a structured Cartesian orthogonal mesh and the volume of fluid method todefine cells blocked by solid obstacles. Because the cylinder surfaces are curved, the fractionalblockage of cells containing the solid surface varies greatly. Grid refinement in the boundarylayers is of limited use; consequently, a uniform mesh was selected. The mesh resolution waschosen to adequately capture the expected temperature and velocity distributions forRa < 1 × 105 (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 . 2 × 105. At increasing Rayleigh numbers, more of the flow is turbulent; at Ra . 108, the upper half of theannulus is clearly in turbulent flow, but the lower half is in laminar flow. The large eddysimulation turbulence model within FLOW-3D was used for simulations of the higher Rayleighnumber flows, and a consistent mesh was used under the assumption that the selected meshadequately captures the flow structures of interest. The large eddy simulation is inherentlythree-dimensional. It can be used for this two-dimensional analysis by selecting the thicknessof the two-dimensional slice consistent with the grid spacing so that the large eddy simulationlength scale is of the correct order.

A sample of the fluid temperatures and velocity vectors predicted by FLOW-3D is shown inFigure 6-14. These results are from the simulation with the 72 × 72 mesh resolution for thecase of Ra = 6.6 × 107. This figure shows the circulation pattern between the cylinders. Theflow field is nearly symmetric, but by viewing several files in the time sequence, it was found thatthe plume oscillates slightly from side-to-side with attendant oscillations in the circulationpatterns on either side of the cylinder.

The grid resolution was refined by a factor of 2 for the case of Ra = 2.5 × 106 to determine theadequacy of the chosen grid discretization.

The FLOW-3D postprocessor will provide the user with the total heat transfer rate from solidobjects to the fluid. This is the value of Q in Eq. (6-4) to compute the effective overall thermal

6-10

( ) ( )′′ = − = −q h T T NukD

T Ti i i oi

i oeff

( )Nu q Dk T T

k DDD

ii

f i o

i

o

i

= ′′−

=⎛

⎝⎜

⎞

⎠⎟

2 eff

ln

Table 6-5. FLOW-3D Simulation Conditions for Concentric Cylinders

Ragap

Mesh(Uniform)

Dkg/m3

$1/K

:Pa × sec

CvJ/(kg × K)

kW/(m × K)

FLOW-3DInput File

Name1.31 × 103 72 × 72 0.1158 3.08 × 10!3 1.903 × 10!5 742.6 0.0274 prepin.K-G_

Validation_Ra1-3e03

6.19 × 104 72 × 72 1.051 3.15 × 10!3 1.875 × 10!5 742.6 0.0270 prepin.K-G_Validation_Ra6-2e04



2.51 × 106

(FineMesh)144 × 144 prepin.K-G_

Validation_Ra2-5e06_fine



conductivity between the two cylinders. These calculations were performed in Microsoft® Excelas described in Scientific Notebook 536E.

The convection heat transfer can also be expressed in terms of a Nusselt number. First, theheat flux at the inner surface can be expressed as

(6-6)

where

— heat flux at inner surface′′qi

hi — heat transfer coefficient [e.g., W/(m2 × K)]Nu — Nusselt numberkf — fluid material thermal conductivity

Comparing Eq. (6-3) can be rearranged to yield the expression

(6-7)

6-11

The Nusselt number, Nu, is described in terms of the computed or measured value of which′′qi ,varies as a function of test conditions; a material property (kf) that is temperature dependent, butnot affected by the flow field; and geometric parameters that are constant. Equation (6-7) is notused in the postprocessing of these data. It is shown here to demonstrate that relativevariances (i.e., the percent difference) between the computed and measured Nusselt numberswill be the same as the relative variances between the computed and measured values of keff,assuming the uncertainty of the fluid thermal conductivity, kf, is neglected. This particularuncertainty is small compared to the variance between the measured and computed values forkeff shown below. The acceptance criterion for the variance between the measured andcomputed values of keff was established as ± 25 percent in Section 6.3.1.

The measured values of keff reported by Kuehn and Goldstein (1978) are compared to thevalues predicted by FLOW-3D in Table 6-6.

For the nominal grid resolution, the variance ranges from !12.7 percent at low Ra to +11.7percent at high Ra. The variance is !15.7 percent for the refined grid resolution. All of thesevalues are within the established acceptance criterion.

The surface heat flux variations over the inner and outer cylinders are presented in terms of thelocal value of keff in Figure 6-15. There is excellent agreement between the predicted andmeasured values of local keff for Ra = 6.19 × 104 on both the inner and outer cylinders. Theagreement for the outer surface is also good at Ra = 2.51 × 106. At the inner surface forRa = 2.51 × 106, there are regions where agreement is poor; however, the general trend of themeasured values is matched by the computational fluid dynamics results.

With the nominal mesh for Ra = 2.51 × 106, the predicted values of keff near the top of the innercylinder are greater than the measured values, but the predicted values are less than themeasured values between about 50° and 100° from the top. From 100° to 180° the agreementis good, but this is a much less important area than the upper parts of the cylinder. With therefined mesh, the predicted values of keff near the top of the inner cylinder are in closeragreement 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 refinedmesh gave only slightly different local values of keff, so, the nominal mesh is seen as adequatefor this flow. The predicted fluid temperature profiles for the case of Ra = 2.51 × 106 arecompared to the measured values along selected radial lines around the annulus in Figure 6-16. First, it is seen that the results for the nominal mesh and the refined mesh are not significantlydifferent. This is further evidence that the nominal mesh is adequately resolving

Table 6-6. Comparison of Predictions and Measurements for Concentric CylindersRagap keq experiment keq FLOW-3D Version 9.0 Deviation Percentage

1.31 × 103 1.14 1.04 !9.06.19 × 104 3.32 2.90 !12.76.81 × 105 5.6 5.27 !5.82.51 × 106 7.88 7.87 !0.1

2.51 × 106 (Fine Mesh) 6.63 !15.91.90 × 107 13.27 14.52 9.56.60 × 107 18.65 20.83 11.7

6-12

the calculated flow characteristics. The general trends of the measured temperature profilesare 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 outercylinders for this case is only 0.91 K [1.64 °F]. The temperature difference between fringes onthe interferogram for this case (i.e., at a pressure of 34.6 atm) is about 0.1 K [0.18 °F], so thereare only about nine fringe contours across the annulus gap. The uncertainty in obtainingtemperature values from such an interferogram are not discussed in the paper. As a result, wecannot assess this impact on the agreement between the measured temperature profiles andthe 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 naturalventilation experiment (Dubovsky, et al., 2001) and against results from a different numericalmodel created in FLUENT 4.52 (Fluent Inc., 1994), a widely recognized and employedcomputational fluid dynamics code. This test, while computationally expensive, will allow theeffect of walls and their interaction with a boundary of air to be examined in addition to thethermal comparatives registered by the interior thermocouples of the fixture. As suggested bythe 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 bya boiling water tank and two ceiling sections open for natural ventilation through an inlet and anoutlet for air flow. Figure 6-17 contains a schematic of the experiment. From the point of viewof 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. Thetemperature 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 boilingwater and maintained at one temperature by immersing two electrical heaters. The tank wallsthat were not part of the hot plate were insulated. Spatial uniformity of the plate temperature of100 °C [212 °F] was experimentally verified and shown to be constant and uniform. The boxacting 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 ofthe box acted as the air inflow and outflow regions. Also, an interior wall from the air inflowedge to 5 cm [2 in] above the bottom of the box existed. All walls of the box in the experimentwere thermally insulated with a 0.2-cm [0.08-in] layer of insulation. The convective heattransfer coefficient measured outside the box was 10 to 12 W/m2-°C [1.8 to 2.1 BTU/h-ft2-°F](Dubovsky, et al., 2001). The convective heat transfer coefficient assumed inside the box was2 to 5 W/m2-°C [0.35 to 0.88 BTU/h-ft2-°F]. The heat transfer coefficient based onthe thermal resistance of the wall and the convective resistance outside the box 0.08 W/m2-°C[0.014 BTU/h-ft2-°F] with an uncertainty of 15 percent. The heat transfer coefficient for theheated plate was found to be 5 W/m2-°C [0.88 BTU/h-ft2-°F] with a 20-percent uncertainty.

Thermocouples were placed along the apparatus width midline as shown in Figure 6-18 andtemperature measurements were made every 15 minutes. Steady state was determined as apoint when less than 0.2-°C [0.4-°F] deviation from a previous measurement was made for allthermocouples 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 inDubovsky, et al. (2001).

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6.4.1 Test Input

A comparison of the measured data with results derived from numerical model simulationsusing 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 wholesystem 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 evaluationstudy using FLUENT Version 4.52 that used 60 × 30 (length × height) and 120 × 60 grid cells todetermine 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 extendedto three-dimensional calculations. The reported three-dimensional FLUENT 4.52 simulationsused a grid defined as 60 × 30 × 8 (length × height × width) cells, where each cell was1 × 1 × 3 cm3 [0.4 × 0.4 × 1.3 in3]. A grid refinement study was conducted for one case utilizing60 × 30 × 24 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 cellswill be added for the walls. Thus, the FLOW-3D model employs 64 × 34 × 12 grid cells toinclude the physical nature of the walls and insulation materials of the test fixture. The originalFLUENT Version 4.52 model simplified these boundaries as mesh boundaries with generalizedwall properties. The boundary that incorporated the inflow and outflow condition had a pressurederivative that equaled zero—the same as the continuative condition that is employed in theFLOW-3D model. The thermal properties used in the FLOW-3D model match those of theFLUENT model.


First, the simulated system was brought to a closed steady state. This means that the system iscompletely 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 atlocations shown in Figure 6-18. When no change in local temperature is observed (aside fromnormal and regular numerical oscillation), the system is deemed steady. Then, the side ventsof the system are opened, and the transient behavior is observed and compared toFLUENT results. After reaching steady state, simulated temperature results are compared tothe measured data.


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 transientperiod when the vents are open are plotted for comparison with FLUENT results. Flowpattern results from FLOW-3D and FLUENT should visually match. There should be less thana 1-percent difference in aggregated velocity results for zones within the domain for thesteady-state condition.

Simulated temperature profiles track relative changes in measured profiles and should not differby more than 5 percent. Some temperature variations may occur because slightly shifted flowpatterns between the experiment and the numerical model can lead to markedly different

6-14

temperatures. The locations to be tracked are the same as those illustrated in Figure 6-18along the midline of the system.

The cited paper incorporates an experimental comparison to a numerical model run in thecomputational fluid dynamics code, FLUENT Version 4.52. Specifically, they compare to (i) asteady-state case when the whole system is sealed, (ii) a ventilated steady state when theentrance and exit windows are open, and (iii) the early transient between state (i) andstate (ii). A two-dimensional grid evaluation was undertaken (eliminating width) using60 × 30 (length × height) and 120 × 60 grids to determine whether temperature effects weresignificant. There was little difference in the comparative runs, so the coarser mesh wasextended to the three-dimensional calculations. The reported three-dimensionalFLUENT Version 4.52 runs use a grid defined at 60 × 30 × 8 (length × height × width), whereeach cell is 1 × 1 × 3 cm3 [0.39 × 0.39 × 1.18 in3]. A grid refinement study was conducted forone case utilizing 60 × 30 × 24; differences were within experimental error, so the coarser gridwas maintained for the rest of the calculations.

The particular settings used in FLUENT Version 4.52 are not described in the paper.

6.4.4 Test Results

The FLOW-3D model employed a 64 × 34 × 12 grid to include the physical nature of the wallsand insulation materials of the test fixture. The original FLUENT Version 4.52 model simplifiedthese boundaries as mesh boundaries with generalized wall properties. The boundary, whichincorporated 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 theirinteraction with a boundary of air in addition to the thermal comparatives registered by theinterior thermocouples of the fixture. It also allows us to verify thermal properties as suggestedby the experiments. Hence, our results are twofold: (i) a comparison to the actual experimentsby Dubovsky, et al. (2001) and (ii) a comparison to a different numerical model created inFLUENT 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 andthe FLOW-3D prediction of the velocity vectors in the flow along the mid-plane through thesystem over a reported 20-second transient. Figure 6-20 is a side-by-side comparison of thesteady-state condition. There is less than 1 percent difference between the cases. Numericalvalues are not provided in the paper for FLUENT Version 4.52 velocity results; therefore, aquantitative comparison cannot be made.

In addition to an evaluation of the local velocity profile as a comparison to the FLUENTVersion 4.52 runs, we also measure the temperature within the fixture at discrete locations atsteady state. These locations are the same as those illustrated in Figure 6-18 along the midlineof the system. These values are cataloged in Table 6-7.

There is good agreement between the actual temperature measurement andFLOW-3D predictions.

6-15

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 6-18.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 becompressible. A comparison of velocity and mass at the inlet and outlet of the system at steadystate will be used to confirm that the boundary condition and overall mass balanceimplementation 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 exhaustwas simulated (Figure 6-21). Forced ventilation was input through a rectangular vent of size0.4 × 0.4 m2 [1.3 × 1.3 ft2]. Exhaust was through a similarly sized vent in the ceiling. The inletflow was maintained at a constant temperature and pressure. Any variation in these parametersis 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 themass of gas exiting the room. Furthermore, regardless of the physical construct of a problem, aflow can be considered one-dimensional under the following conditions: (i) the flow is normal tothe boundary at locations where mass enters or exits the control volume and (ii) all intensiveproperties, such as velocity and density, assume a uniform position over each inlet or exit areathrough which matter flows (e.g., Moran and Shapiro, 2000). In particular, when flow is

6-16

considered one-dimensional, the mass flow rate at inlet and outlet is defined by( )&m &m AV ,= ρ

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, bothconstant velocity and density are expected at the inlet and outlet. Therefore, this validation runevaluates this physical phenomenon and ascertains whether FLOW-3D accurately predictsthe outcome.

6.5.1 Test Input

The computational model was generated based on the physical model described above. Interior dimensions of the room will be 4 × 2 × 3 m3 [13 × 6.5 × 10 ft3]. Computational walls ofthickness 0.2 m [8 in] were applied in each direction to simplify visualizations and properlyrestrict inflow and outflow. Vents were created as 0.4 × 0.4 m2 [1.3 × 1.3 ft2] openings throughtheir respective boundaries. The full model utilizes a mesh of 44 × 24 × 34 with a uniform gridof individual block size 0.1 × 0.1 × 0.1 m3 [0.3 × 0.3 × 0.3 ft3]. An additional run at double theresolution 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 6-21. A continuative condition was applied tothe outflow boundary, which indicates that FLOW-3D will extrapolate local data upstream intoappropriate conditions through the boundary. Zero normal derivatives for all quantities areimplemented for continuative boundary conditions in FLOW-3D.

The fluid is air having the following properties at 293.15 K [68 °F]:

viscosity = 1.86 × 10!5 kg/m-s [1.25 × 10!5 lbs/ft-s]specific heat = 1883.7 m2/s2-K [1.126 × 104 ft2/s2-°F]thermal conductivity = 0.0264 kg-m/s3-K [9.86 × 104 lbs-ft/s3-°F]gas constant = 287.0 m2/s2-K [1720 ft2/s2-°F]density = 1.2 kg/m3 [0.075 lbs/ft3]

The gas is assumed compressible so that the physical sensitivities of pressure and velocity canbe included in the calculations.


History points, which are numerical markers in the flow, are located in the center of both inflowand outflow vents. These points were monitored to ascertain when the flow reachessteady state.

To ascertain an average velocity across both the inflow and outflow boundary, the magnitude oftotal 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 accuraterepresentation of velocity through the opening instead of at a discrete boundary.

6-17


The criteria for test acceptance were established in the software validation test plan (Green andManepally, 2006). The simulated velocity at the inflow and outflow vents should be within5 percent of the intended ventilation flow rate. The mass flow rate at the inlet and outlet shouldnot differ by more than 2 percent. These acceptance criteria are acceptable for simulations ofcompressible flow at steady state.

6.5.4 Test Results

Figure 6-22 shows a visualization of the computational model for the full model with a mesh of 44 × 24 × 34. (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 gridresolution (red is open, blue is closed). For purposes of discussion, air comes in from the leftand goes out through the top.

A sample of the velocity vectors predicted by FLOW-3D is shown in Figure 6-23. Figure 6-23ashows the flow pattern from the inflow to the outflow in the central longitudinal (x-z) plane, whileFigure 6-23b shows the flow in a transverse (y-z) plane in line with the outlet vent. In each ofthe 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 successiveimages 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 magnitudeof total velocity was evaluated as an integral over the cross-sectional area of each vent. Theinlet average velocity and outlet average velocity are compared in Table 6-8. Mass flow thencan be evaluated using the equation described above, The results of this calculation&m AV.= ρfor the inlet and outlet ports are likewise compared.

There is a 1.60 percent difference between the inlet and outlet port mass flow rates. Thisvariation is within acceptable limits for a compressible flow at steady state. Thedifference between the inflow and outflow values is attributed to the accumulation ofconvergence 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 flowdomain. This apparent mass flow imbalance could be reduced by decreasing the convergencecriterion value at the cost of longer execution times required to meet the more stringentconvergence limit.

The pressure difference between the inlet and outlet parts is predicted as 0.041 Pa/m. As apoint of comparison, consider a long pipe elbow with a hydraulic diameter equal to that of theinlet and outlet ports and a bend radius of 2 m [6.6 ft]. The pressure drop through such a pipe isestimated to be 0.036 Pa. As expected, the pressure drop across the room is greater than thatof a smooth pipe.

6-18

Table 6-8. Comparison of Inlet and Outlet Flow Velocities for Test Case 5Applied Condition Measured Velocity* Measured Mass Flow

Inflow 0.25 m/s 0.2499 m/s 0.047990 kg/sOutflow Continuative 0.2539 m/s 0.048759 kg/s*Measured data taken 2 grid planes from boundary; this gives a more accurate representation of velocity throughthe opening instead of at a discrete boundary.

6-19

Figure 6-1. Geometry for Natural Convection Along a Vertical Flat Plate {1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

6-20

Figure 6-2. Sample of Predicted Fluid Temperature and Velocity Vectors for NaturalConvection Along a Vertical Plate (Dimensions x and z in Meters)

{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

6-21

Figure 6-3. Nusselt Number Comparison for Vertical Flat Plate{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

Figure 6-4. Vertical Velocity Distribution at 0.125 m [0.410 ft] From the LeadingEdge. Results Show Close Agreement Between the Nominal and Refined Mesh.

[1 m = 3.28 ft]

6-22

Figure 6-6. Geometry for Natural Convection in a Square Cavity{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

Figure 6-5. Temperature Distribution at 0.125 m [0.410 ft] From the LeadingEdge. Results Show Close Agreement Between the Nominal and Refined Mesh.

{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

6-23

Figure 6-7. Sample of Predicted Fluid Temperature and Velocity Vectors for NaturalConvection in a Square Cavity (Dimensions Shown in Meters)

{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

6-24

Figure 6-8. Local Nusselt Number of the Vertical Walls and Experiment Data Are Shownfor Hot and Cold Walls. Computational Fluid Dynamics Results Are Shown

for Hot Wall Only.

Figure 6-9. Local Nusselt Number Along Upper Wall

6-25

Figure 6-10. Dimensionless Temperature Distribution at the Mid-Width (x/L = 0.5)

Figure 6-11. Velocity Distribution of the Mid-Height (y/L = 0.5)

6-26

Figure 6-12. Dimensionless Temperature Distribution at the Mid-Height (y/L = 0.5)

Figure 6-13. Geometry for Natural Convection Between Concentric Cylinders [1 cm = 0.39 in]

6-27

Figure 6-14. Sample of Predicted Fluid Temperature and Velocity Vectors for NaturalConvection Between Cylinders (Dimensions y and z in Meters)

{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

6-28

Figure 6-15. Local Value of keff on Cylinder Walls [1 W/m-K = 0.578 BTU/ft-h-°F]

Figure 6-16. Dimensionless Fluid Temperature Profiles Along Radial Lines

6-29

Figure 6-17. Experimental Apparatus Used in Test Case 4

Figure 6-18. Thermocouple Placement Along Midline (Depth) of System. Left and LowerInside Walls Shown at the Zero Axes Location. Offset Given Is in Centimeters.

[1 cm = 0.39 in]

6-30

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

6-31

Figure 6-19b. Transient Development of Velocity Vector Flow Field, 10–20 Seconds

Figure 6-20. Steady-State Velocity Vector Comparison of Fluent 4.52 Case and FLOW-3D Case

6-32

Figure 6-21. Schematic of Experimental Apparatus Used for Test Case 5

Figure 6-22. Visualization of the Computational Model

6-33

Figure 6-23. 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]

7-1

7 MOISTURE TRANSPORT TEST CASES

The objective of these test cases is to validate the moisture module in predicting the diffusion ofwater in an air/water mixture, the advection transport of water in a natural convection flow, andthe 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 agap filled with moist air. The top plate is held at constant temperature and the bottom plate at alower temperature. Both surfaces provide a stationary film of water that can exchange masswith the water vapor in the air gap between the plates. It is assumed that there is no convectionin the air gap. The following parameters define the necessary geometric and physicalproperties of the system:

• Gap thickness, L = 0.1 m [0.33 ft]• Fluid thermal conductivity = kmix = 0.026 W/(m2-K) [0.005 BTU/h-ft2-°F]• Top (hot) surface temperature = Th = 320 K [116 °F]• Bottom (cold) surface temperature = Tc = 280 K [44.3 °F]• Pressure = 1 atm

The equations describing the diffusion of thermal energy and water vapor across the gap arepresented in Bird, et al. (1960) and can be solved exactly for the case in which the relativehumidity is not limited to 100 percent. If the humidity is limited to 100 percent, the energy andmass transport equations must be solved numerically.

7.1.1 Test Input

A FLOW-3D input file (prepin.*) was developed to model the idealized case of one-dimensionalconduction heat transfer and chemical species diffusion through the air gaps. The lateral edgesof the computational domain are specified as adiabatic surfaces. Convection flow wasdisallowed in the simulation. This portion of the problem specification is accomplished with thestandard input file procedure of the basic FLOW-3D code.

The standard version of FLOW-3D can simulate the diffusion of chemical species as defined inthe idealized case. The unique feature of this problem is that the water at each surface mustbe in thermodynamic equilibrium with its vapor. The moisture transport processes wereaccomplished by providing user inputs to the customized portion of the code as described inScientific Notebook 536E.


FLOW-3D was run with the input file described above until a steady-state condition wasachieved. The output of the temperature profiles and water vapor concentration profiles arecompared to the predictions of the mathematical analysis.

7-2


The acceptance criterion for this test case is that the local temperatures and water vaporconcentrations 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 platesthrough 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. Theanalysis of these processes is completely described in Scientific Notebook 536E. The followingis a summary of that analysis.

Bird, et al. (1960) developed the complete theory of the diffusion of chemical species inmixtures. Their treatment shows that the diffusion of water through a mixture of water and air inthis one-dimensional problem can be expressed as

Nw = −−

ρmix wa wDx

dxdz1 w

(7-1)

where

Nw — mole flux (mol/(m2 × sec))xw — mole fraction of water vapor (moles of vapor per total moles)Dmix — air/water mixture molar density (mol/m3)Dwa — diffusion coefficient for water/air (m2/sec)z — distance along flow path

We are considering the steady diffusion of water vapor; so

( )ddz

N ddz

Dx

dxdzw

mix wa

w

w= −−

⎡

⎣⎢

⎤

⎦⎥ =

ρ1

0 (7-2)

The boundary conditions for this differential equation are

xw — xw,h at z = 0 (hot surface)xw — xw,c at z = L (cold surface)

Equation (7-2) can be readily solved for the water mole fraction profile

( ) ( )x z xxxw w h

w c

w h

z L

= − −−−

⎛

⎝⎜

⎞

⎠⎟1 1

11,

,

,

/

(7-3)

The steady mole flux can be obtained by substituting Eq. (7-3) for Eq. (7-1)

N DL

xxw

mix wa w c

w h=

−−

⎛

⎝⎜

⎞

⎠⎟

ρ ln ,

,

11 (7-4)

Note that the water concentration profile is nonlinear due to the presence of the air in thesystem. Also, as long as the thermodynamic and transport properties of the gases areindependent of temperature, the concentration and mole flux of the water vapor is independentof temperature except in the specification of the boundary values.

7-3

The energy flux at any position in the gap is given by

( ) ( )e z k dTdz

h N h Nmix a a w w= − + + (7-5)

where

e — energy flux (W/m2)kmix — air/water mixture thermal conductivity [W/(m2-K)]ha — mole specific enthalpy of dry air (J/mol)hw — mole specific enthalpy of water (J/mol)Na — net mole flux of air [mol/(m2-s)]Nw — net mole flux of water [mol/(m2-s)] from Eq. (7-4)

Na 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 independentof temperature, Eq. (7-5) can be differentiated to yield

( )ddz

dTdz

Nk

ddz

hw

mixw

⎛⎝⎜

⎞⎠⎟

+ = 0 (7-6)

The water enthalpy is a unique function of temperature, so the boundary conditions for thisequation are

T — Th at z = 0 (hot surface)T — Tc 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 theconservation 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 thedewpoint) so that liquid water does not condense.

2. The water vapor concentration is limited to one corresponding to a relative humidity of100 percent. In this scenario, the solution to Eq. (7-6) will be obtained under theassumption that the condensed liquid remains as a “fog” and diffuses along withthe vapor.

Scenario 1—No Condensate

When there is no liquid water present, the enthalpy of the water can be expressed as

( )h C T Tw pw c= − (7-7)

where

Cpw — mole specific constant pressure specific heat of water vapor (J/(mol × K))

7-4

The temperature of the cold surface is used to define the reference temperature for theenthalpy. Eq. (7-6) can then be readily integrated to yield

( ) ( )T z T T T

N Ck

z

N Ck

Lh h c

w pw

mix

w pw

mix

= − −−

⎛

⎝⎜

⎞

⎠⎟

−⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

1

1

exp

exp(7-8)

Note that the temperature distribution is a nonlinear function of distance. The deviation from alinear profile is dictated by the mole flux and the fluid properties. Eq. (7-8) is identical to theexpression defined by Bird, et al. (1960) for this scenario.

Scenario 2—Water Vapor Limited by Saturation Pressure

When there is liquid water present, the enthalpy of the water can be expressed as

( )h C T T xx

hw pw cv

wvap= − − −

⎛

⎝⎜

⎞

⎠⎟1 (7-9)

where

Cpw — mole specific constant pressure specific heat of water vapor (J/mol)xv — water vapor mole fraction xw — combined vapor and liquid water “fog” mole fraction hvap — heat of vaporization of water

By substituting Eq. (7-9) into (7-6), and after much algebraic manipulation, the followingequation can be obtained for the temperature.

( ) ( ) ( )ddz

dTdz

g T z ddz

T g T z⎛⎝⎜

⎞⎠⎟

+ + =1 2 0, , (7-10)

The functions g1 and g2 are defined as

( ) ( )

( ) ( )( )( )

g T z Nk

Ch

x z Pd

dTP

g T z Nk

hP T

P x z

xz

xx

w

mixpw

vap

w totv sat

w

mixvap

v sat

tot w

w z

h

w h

w c

1

2 2

0

1 1 11

,

, ln

,

, ( ) ,

,

= − + ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=

= −− −

−⎛

⎝⎜

⎞

⎠⎟

(7-11)

where

xw(z) — total water concentration from Eq. (7-3)Ptot — nominal total pressure of the air/water mixturePv,sat(T) — saturation vapor pressure of water

Note that if hvap = 0 in Eq. (7-10), it reduces to the equation for the case in which nocondensate forms.

7-5

The functions g1 and g2 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 equationfor the temperature profile. This solution procedure, described in detail in ScientificNotebook 536E, uses central differences to approximate the derivatives and a correlation forthe saturation vapor pressure of water so that the functions g1 and g2 can be computed.

The analysis was carried out in Mathcad© and the analysis results were copied into MicrosoftExcel for graphing and comparison to the FLOW-3D predictions.

7.1.4 Test Results

Two FLOW-3D input files (prepin.*) corresponding to the two scenarios described above wereprepared. These both use the custom moisture transport model in which the required inputspecifications 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 of50 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 Excelspreadsheet for graphing and comparison to the analytical results. The FLOW-3D simulationssolve for the mass fraction of the water. The conversion from mass fraction to mole fraction wasdone 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 inFigure 7-2. The FLOW-3D predictions for both the temperature profile and the waterconcentration profile are close to those from the theoretical analysis. The deviation intemperature values is defined as the ratio of the local variance and the overall temperaturedifference between the surfaces. Similarly, the water concentration variance is defined as theratio of the local difference in the predictions and the difference in concentration values at thetwo surfaces. Thus,

Temperature Variance — Eq. (7-12)Concentration Variance — Eq. (7-13)

( ) ( )Temperature Variance

T z T zT T

FLOW D theory

h c=

−−

3(7-12)

( ) ( )Concentration Variance

x z x zx x

FLOW D theory

w h w c

=−

−3

, ,(7-13)

The FLOW-3D predictions and the theoretical results agree to within 1.5 percent. Note that theconcentrations values are the water vapor mole fraction with respect to the total number ofmoles. The Mathcad© file listed above shows that the water vapor concentration is greater thanthat for a 100 percent relative humidity across the entire gap in this case.

7-6

Scenario 2—Condensate Allowed in Air/Vapor

The FLOW-3D predictions are compared to the theoretical predictions for this case inFigure 7-3. The FLOW-3D predictions for both the temperature profile and the waterconcentration profile closely agree with those of the theoretical analysis. Recall, however, thatthe theoretical results in this case were obtained by a numerical approximation of the governingdifferential equations. The FLOW-3D predictions and the theoretical results agree to within 5percent 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 validatethe 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 arefabricated primarily of acrylic. The aluminum pan is attached to the floor at one end of the boxand extends across the width of the box. The entire opposite end of the box is an aluminumplate that is cooled with chilled water flowing through passages machined into plate. The entirecontainer is covered with Styrofoam™ insulation.

The water pan is maintained at a constant temperature by a heater attached to its bottom. Thewater is maintained at a constant level by a siphon device between the pan and a water bottlethat is located outside the acrylic enclosure. Thermocouples record the temperature at thelocations shown in Figure 7-4. Condensed water is collected in a graduated cylinder. The netcondensation rate is estimated by knowing the time period for collecting the observed amountof water.

The laboratory experiment procedure calls for the heater and chiller to be adjusted to provide forconstant temperatures as measured by thermocouples immersed in the water and attached tothe cold plate surfaces. The test is operated for several hours until a steady condition isachieved as shown by the air temperatures and the condensation rate.

Test runs with several different combinations of heater and chiller temperatureswere conducted.

7.2.1 Test Input

FLOW-3D input files (prepin.*) were developed to model the idealized case of two-dimensionalflow in the vertical symmetry plane of the box. The box is wide enough that two-dimensionalflow very nearly exists in this cross section; therefore, the simulation for this case was twodimensional. Boundary conditions and fluid properties were based on the thermal conditionsspecific to each experiment test run.

The convection and conduction aspects of the problem are handled by the standard portions ofthe FLOW-3D code. The simulation of the moisture transport processes were accomplished byproviding the user inputs to the customized portion of the code as described in ScientificNotebook 536E.

7-7


FLOW-3D was run with the input files described above until a steady-state condition wasachieved. The output of the temperatures at the center of the container are compared to thetest measurements. Likewise, the output of the condensation rate at the chilled plate arecompared to the test measurements.


The acceptance criterion for temperature predictions is that the air temperatures at the selectedlocations should agree with the measured values to within 20 percent. Similarly, theacceptance criterion for condensation rate is that the predicted condensation rate should agreewith the measured value to within 20 percent. These levels of error are consistent withgenerally 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. Testruns 1–3 were used to develop the proper test procedure and to check the instrumentation. Test runs 16–18 were conducted without water in the system and served as a basis forcomparing other types of computational fluid dynamics simulations. The results of testruns 4–15 are used here for validating the custom moisture transport model in FLOW-3D. Thepertinent 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 tothe cold plate and hot water temperatures, respectively.

The test results are discussed and compared to the simulation results below.

7.2.4 Test Results

FLOW-3D input files (prepin.*) were prepared corresponding to each of the 12 test runsdescribed above. All of the input files use the custom moisture transport model in which therequired input specifications follow the instructions in Scientific Notebook 536E for using thismodule. All the simulations used a two-dimensional grid (see Figure 7-5) with 90 cells along theinner length of the box and 23 cells along the inner height of the box. The box walls and theinsulation were represented by a layer of 9-cells across their

7-8

Table 7-1. Condensation Cell Test Conditions and Test Results

TestRun

Condensation Rate

(ml/hr)

WaterTemp*(/C)†

ColdPlate

Temp*(/C)

LowerAir

Temp*T3 (/C)

Mid-AirTemp*T4 (/C)

UpperAir

Temp*T5 (/C)

LidTemp*T8 (/C)

TopIns‡

Temp*T9 (/C)

Amb§Temp*

(/C)8 2.5 19.5 5.2 14.3 16.3 17.4 18.2 24.2 24.97 4.7 25.3 5.2 15.6 17.9 19.2 18.7 23.3 23.74 7.2 32.3 5.3 18.8 21.6 23.2 22.4 24.5 24.75 12.1 39.4 5.4 21.5 24.9 26.6 24.7 23.1 23.16 20.9 47.7 5.5 25.1 29.1 30.9 29.0 23.7 23.19 1.7 26.2 19.1 22.6 23.5 24.1 24.0 25.4 25.610 0.0 38.2 19.1 26.3 28.3 29.5 27.9 25.1 24.812 12.8 45.0 19.2 29.2 32.1 33.4 31.5 26.2 25.611 20.8 50.9 19.3 32.1 35.3 36.8 34.4 25.1 23.915 0.5 34.0 29.4 29.7 30.0 30.5 29.6 25.5 24.814 4.2 39.7 29.4 31.3 32.2 33.1 31.5 23.9 23.013 8.1 46.1 29.5 33.7 35.5 36.6 34.8 25.1 23.9*Temp = temperature†°F = 1.8 × T °C + 32.

combined thickness. A two-cell thick layer was used to set the outer surface boundarytemperature in accordance with the measured test conditions.

The simulations were carried out until an approximately steady condition was established. All ofthe simulations had semiregular oscillations in the velocity and temperature fields, indicating thepresence 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 andcondensation rates, the simulation results were recorded at simulated 1-second intervals for20 seconds. This set of results was used to compute the mean values of temperature, surfaceheat flux, and condensation rate for comparison to the experiment results. An example of thevelocity field and the fluid and solid temperatures is shown in Figure 7-6. This image is for testrun 11 and shows the temperature contours for the fluid and all solid regions on the sametemperature scale. This figure clearly shows the heated water tray and the cooled end plate ofthe box, but not the flow pattern within the box. The fluid rises from the heated water andcirculates along the upper parts of the enclosure toward the cold plate. The cold plate cools thefluid and condenses water from the flow which then returns to the heater along the bottomportion of the box.

The water mass fraction is shown in Figure 7-7 along with the velocity vectors. This figureshows only the fluid portion of the model. As expected, the highest concentration of water is inthe fluid as it leaves the region near the heated water tray. The lowest concentration is at thebottom of the cold end wall where the fluid ends its contact with the cold plate after watercondensed along this part of the circulation path.

The temperatures in the middle of the box (T3, T4, and T5 locations) are compared inFigure 7-8. The overall trend of the FLOW-3D predictions is consistent with the measuredresults. There is not a substantial bias in the predictions (i.e., the predictions are not

7-9

consistently less than or greater than the measurements). The deviation in the results is greaterat the lowest sensor location (T3) than at the other two locations. Using the difference in watertemperature and cold plate temperature as a basis, the deviations between the measurementsand predictions are computed as shown in Table 7-2.

The predicted and measured condensation rates at the cold plate surface are compared inFigure 7-9. The trends of these results are all as expected in that the condensation rate isinversely proportional to the cold plate temperature and directly proportional to the temperaturedifference of the heated and cooled surfaces. There is reasonable agreement between theFLOW-3D predictions and the measured condensation rates; however, the predictedcondensation rate is greater than the measured value in all cases except one. The deviationbetween the results is listed Table 7-2 where the variance relative to the mean condensationrate is given .

The FLOW-3D predictions for the gas temperature and the condensation rate are all less than20 percent for all of the test runs. The deviations between the measured and predicted valuesof air temperature and condensation rate are within the acceptance criterion defined above.

Table 7-2. Variances of FLOW-3D Prediction From Measured Values in CondensationCell Results

TestRun

Condensation Rate

Lower AirTemperature T3

Mid-AirTemperature T4

Upper AirTemperature T5

8 !1.5% !12.0% !7.7% 1.2%7 6.6% !16.0% !8.7% !6.0%4 15.2% !16.2% !7.1% !5.5%5 12.2% !12.9% !3.0% !1.6%6 2.4% !9.8% 0.6% 1.7%9 0.5% !4.1% !0.1% 4.4%10 18.5% !8.4% !0.1% !0.2%12 18.0% !8.4% 0.3% 1.2%11 10.0% !7.9% 1.9% 2.8%15 5.1% 10.9% 19.9% 13.9%14 0.1% 4.3% 13.5% 10.3%13 19.8% !0.4% 8.2% 7.4%

7-10

Figure 7-1. Schematic for Heat Conduction and Species Diffusion Between Surfaces{T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

7-11

Figure 7-2. Temperature and Concentration Profile for One-Dimensional Conduction andMoisture Transport. Water Vapor Is Allowed to Be Supersaturated in This Scenario.

{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

7-12

Figure 7-3. Temperature and Concentration Profile for One-Dimensional Conduction andMoisture Transport. Water Vapor Is Limited to 100-Percent-RH and

Fog Is Allowed to Form. {1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

7-13

Figure 7-5. FLOW-3D Geometry for Condensation Cell Simulations

Figure 7-4. Test Setup for Natural Convection and Water Vapor Transport in a Closed Container

7-14

Figure 7-6. Predicted Temperature Contours and Velocity Vectors in Condensation Cell

Figure 7-7. Predicted Water Mass Fraction in Condensation Cell (Unitless)

7-15

Figure 7-8. Mid-Line Temperature Ion Condensation Cell {T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

Figure 7-9. Cold Plate Condensation Rate {T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

8-1

8 THERMAL RADIATION TEST CASES

Two test cases are presented here. The first validates the capabilities of the radiation moduleto compute radiation heat transfer and the inclusion of this heat transfer mode into the overallFLOW-3D simulation. The second test case validates the capabilities of the radiation module incomputing 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 agap. The upper plate has internal heat generation so that the heat flux into the air gap is255 W/m2 [80.8 BTU/h-ft2]. The bottom of the lower plate is held at a lower temperature. It isassumed that there is no convection in the air gap.

The following parameters define the necessary geometric and physical properties of the system:

• Gap thickness, tgap = 0.1 m [0.3 ft]• Plate thickness, tupper = tlower = 0.02 m [0.06 ft]• Emissivity, gupper = glower = 0.9• Gap thermal conductivity = kair = 0.1 W/(m2-K) [0.0176 BTU/h-ft2-°F]• Plate thermal conductivity, kupper = klower = 1 W/(m2-K) [0.0176 BTU/h-ft2-°F]• Upper surface heat flux = Qupper = 255 W/m2 [80.8 BTU/h-ft2]• Temperature of outside surface of lower plate = Tc = 300 K [80.3 °F]

The radiation and conduction heat transfer processes were modeled by the appropriate exactone-dimensional equations for this case.

8.1.1 Test Input

A FLOW-3D input file (prepin.*) was developed to model the idealized case of one-dimensionalconduction heat transfer through the three objects. The lateral edges of the computationaldomain were specified as adiabatic surfaces to satisfy the one-dimensional condition in theFLOW-3D simulation. Air movement was disallowed in the simulation. This portion of the testinput 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 thecustomized portion of the code as described in Scientific Notebook 536E. Because this is anidealized one-dimensional case, the radiation configuration factors are

F1!1 = F2!2 = 0F1!2 = F2!1 = 1


FLOW-3D was run with the input file described above until a steady-state condition wasachieved. The output of the temperature profiles are compared to the predictions of themathematical analysis. The output temperature profiles are used to compute the overall heattransfer rate for comparison to the analytical solution.

8-2


The acceptance criterion for this test case is that the local temperatures predicted by FLOW-3Dshall 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 within5-percent analytical prediction.

In the problem described above, dry air fills the space between two plates. The upper plate hasa heat flux of 255 W/m2 [80.8 BTU/h-ft2] at its surface while the back side of the lower plate isheld at a constant 300 K [80.3 °F]. Heat is transferred by conduction and radiation across theair gap; convection is not allowed. The analysis of these processes is completely described inScientific 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 bythe heat transfer due to conduction and radiation across the air gap

Q Q Q Qupper lower air cond rad= = +, (8-1)

where

Qupper — heat flux at upper plate 255 W/m2 [80.8 BTU/h-ft2]Qair,cond — heat flux through air via conductionQrad — heat flux across gap via radiation Qlower — heat flux through the lower plate via conduction

The conduction heat transfer through the air can be expressed as

Q kT T

tair cond airs upper s lower

gap,

, ,=−

(8-2)

where

kair — air thermal conductivitytgap — gap widthTs,upper — temperature of upper plate surface in contact with airTs,lower — temperature of lower plate surface in contact with air

Siegel and Howell (1992) describe the governing equations for radiation in enclosures withdiffuse, gray surfaces. For the one-dimensional case here, the radiation heat flux from theupper to lower surface is

( )Q T Trad

upper lower

s upper s lower=+ −

−σ

ε ε1 1 1

4 4, ,

(8-3)

8-3

where

F — Stefan-Boltzmann constant = 5.67 × 10!8 W/(m2-K4) [1.17 × 10-9 BTU/(h-ft2-°R4)] ,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 kT T

tlower lowers lower C

lower

=−,

(8-4)

where

klower — lower plate thermal conductivitytlower — lower plate thicknessTC — 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, Ts,upperand Ts,lower. The solution procedure for this system of equations is described fully in ScientificNotebook 536E. It is found that

Ts,upper = 343.6 K Qrad = 10.0 W/m2 [3.2 BTU/ft2h]Ts,lower = 305.1 K Qcond,air = 245 W/m2 [77.7 BTU/ft2h]

Now that the plates’ surface temperatures have been determined, the temperature profiles inthe plates and in the air gap can be specified. Define the location z = 0 at the center of the airgap, positive upward. The interface locations are

zH — +0.07 m [+2.76 in]zs,upper — +0.05 m [+1.97 in]zs,lower — !0.05 m [!1.97 in]zC — !0.07 m [!2.76 in]

The temperature distribution in the lower plate and the air gap follow the typical linear form formaterials with constant thermal conductivity

( ) ( )T z T T Tz z

ts lower s lower Cs lower

lower

= − −−

, ,, z z zc s lower≤ ≤ , (8-5)

( ) ( )T z T T Tz z

z zs upper s upper s lowers upper

s lower s upper

= − −−

−, , ,,

, ,

z z zs lower s upper, ,≤ ≤ (8-6)

The upper plate has a uniform internal heat generation. It can be shown that the temperaturedistribution in this material is given by

( ) ( )T z T

Q z zk

z zzs upper

upper s upper H

upper

s upper

H s upper

= +−

−−−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥,

, ,

,21

2

2

z z zs upper H, ≤ ≤ (8-7)

8-4

The temperature profiles in the three regions will be compared to the FLOW-3Dpredictions below.

8.1.4 Test Results

FLOW-3D input files (prepin.*) were prepared corresponding to the one-dimensional problemdescription. The input files use the custom radiation module in which the required inputspecifications 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 werespecified for the entire one-dimensional domain; 20 cells across the air gap and 4 cells throughthe thickness of each of the two plates. The second grid specified was at half the spacing of thefirst (i.e., a total of 56 uniformly spaced cells was used).

The simulations were conducted until a steady condition was achieved. The FLOW-3D resultsfor the temperature distribution are compared to the theoretical distributions in Figure 8-2. TheFLOW-3D predictions for the two different grid resolutions are in close agreement, and both thepredicted curves show slightly lower temperatures than the theory. The 28-cell predictionsshow a worst-case deviation of less than 5 percent from the theoretical values. The 56-cellpredictions are slightly better with a worst-case deviation of just over 3 percent. The heattransfer rates predicted by FLOW-3D are 240 W/m2 [76.1 BTU/ft2-h] by radiation and 15 W/m2

[4.8 BUT/ft2-h] by conduction through the air. These differ from the corresponding theoreticalvalues by about 2 percent of the total heat flux.

The deviations between the theoretical and predicted values of air temperature and heat flux arewithin 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 theoverall 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). Theouter cylinder has an inner diameter of 1.0 m [3.3 ft], and the inner cylinder has an outerdiameter of 0.3 m [0.98 ft]. In this scenario, each of the cylinder surfaces is divided into foursubsurfaces of equal size for which the configuration factors are to be computed. The Hottelmethod [as described by Siegel and Howell (1992)] will be used to compute the configurationfactors between each pair of surfaces for this case. In the radiation module, the radiativeexchange 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 thisscenario is to test the capability of the radiation module to account for blockages betweensurfaces so that the configuration factor is less than that for the condition in which the surfaceshave an otherwise unimpeded view of each other.

The second scenario is the radiation within a three-dimensional enclosure (Figure 8-4). Thisenclosure is 2 m × 1 m × 0.5 m. (Configuration factors are dimensionless so, the units of thesedimensions are, in fact, not relevant.) The exact configuration factors can be computedfrom published literature for this geometry (e.g., Howell, 1982). The essential point of thisscenario is to validate the radiation module configuration factor computations forthree-dimensional problems.

8-5

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 thecustomized portion of the code as described in Scientific Notebook 536E. Only theconfiguration factors are to be validated here; the description of the fluid and other heat transferrelated parameters is not necessary.


FLOW-3D was executed as required for only two time steps to allow the radiation moduleinitialization to be executed. The computed configuration factors are recorded to a file as part ofthe initialization sequence. These values are compared to the exact values computed for therespective scenarios.


The acceptance criterion for this test case are that the configuration factors predicted byFLOW-3D shall be within 5 percent of the analytical predictions.

The radiation configuration factors for the two-dimensional geometry of the concentric cylindersare easily computed by the Hottel Crossed-String Method (Siegel and Howell, 1992). Thismethod compares the lengths of line segments between combinations of the respective edgesof two surfaces. The method is entirely geometric and does not involve the complicatedintegration procedure required of general three-dimensional problems. The computations forthis case are described in detail in Scientific Notebook 536E in the entry for March 20, 2006. The results of the calculations are summarized in Table 8-1. It is noted that the configurationfactors for a surface viewing itself (e.g., F1–1) are not zero for surfaces 1, 2, 3, and 4. Becausethe radiation module does not compute the self-viewing configuration factors, these values arenot listed here.

In the case of the three-dimensional rectangular box, there are only two general types ofconfigurations for the surfaces for all non-zero factors. First, there are six cases in which thesurfaces directly oppose each other. The radiation configuration factor for this configuration isgiven by Howell (1982)

( )( ) ( )( )

( )( )

( ) ( )

FXY

X Y

X YX Y X

Y

Y X Y

XX Y Y X

a b−

−

− − −

=

+ +

+ +

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+ ++

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ ++

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

− −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

2

1 1

11

1

11

2 2

2 2

0 5

2 0 5 1

2 0 5

2 0 5 1

2 0 51 1

π

ln tan

tan tan tan

.

.

.

.

.

(8-8)

8-6

( )( )

( )( ) ( )( )( )

( )( )( )

FA

AA

BB

B AB A

A B

A B

A A B

A A B

B B A

B B A

a b

W

−

− − −

′=

′

′′

⎛⎝⎜

⎞⎠⎟

+ ′′

⎛⎝⎜

⎞⎠⎟

− ′ + ′′ + ′

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

++ ′ + ′

+ ′ + ′

′ + ′ + ′

+ ′ ′ + ′

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

′ + ′ + ′

+ ′ ′ + ′

⎛

1

1 1 1

14

1 1

1

1

1

1

1

1 1 2 2 1

2 2 0 5

2 2

2 2

2 2 2

2 2 2

22 2 2

2 2 2

π

tan tan tan

ln

.

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

′H 2

Table 8-1. Configuration Factors, Fa-b, Two-Dimensional Cylinders, Exact SolutionSurface ‘b’

1 2 3 4 5 6 7 8

Surf

ace

‘a’

1 n/a 0.2254 0.1358 0.2347 0.2196 0.0404 0.000 0.04042 0.2254 n/a 0.2347 0.1358 0.0404 0.2195 0.0404 0.00003 0.1358 0.2347 n/a* 0.2347 0.0000 0.0404 0.2195 0.04044 0.2347 0.1358 0.2347 n/a 0.0404 0.0000 0.0404 0.21955 0.7274 0.1339 0.0000 0.1339 n/a 0.0000 0.0000 0.00006 0.1339 0.7274 0.1339 0.0000 0.0000 n/a 0.0000 0.00007 0.0000 0.1339 0.7274 0.1339 0.0000 0.0000 n/a 0.00008 0.1339 0.0000 0.1339 0.7274 0.0000 0.0000 0.0000 n/a

*n/a = not applicable

where

Fa!b — radiation configuration factor from surface ‘a’ to surface ‘b’L — distance between surfacesW — width of the surfacesH — height of the surfaces

AN — AL

BN — BL

The other general type of configuration for the enclosure is for orthogonal faces with a commonedge. All of the remaining 24 non-zero factors fall into this category. The configuration factorsare defined by Howell (1982) as

(8-9)

where

L — common edge of the two surfacesA — dimension of surface ‘a’ normal to common edgeB — dimension of surface ‘b’ normal to common edge

The configuration factors for this rectangular enclosure are summarized in Table 8-2.

8.2.4 Test Results

FLOW-3D input files (prepin.*) were prepared for each of these scenarios. The input files usethe custom radiation module in which the required input specifications follow the instructions in

8-7

Table 8-2. Configuration Factors, Fa-b, Three-Dimensional Box, Exact SolutionSurface ‘b’

1 2 3 4 5 6Su

rfac

e ‘a

’ 1 n/a 0.0362 0.1673 0.1673 0.3146 0.31462 0.0362 n/a* 0.1673 0.1673 0.3146 0.31463 0.0837 0.0837 n/a 0.1653 0.3337 0.33374 0.0837 0.0837 0.1653 n/a 0.3337 0.33375 0.0787 0.0787 0.1669 0.1669 n/a 0.50906 0.0787 0.0787 0.1669 0.1669 0.5090 n/a


Table 8-3. Configuration Factors, Fa-b, Two-Dimensional Cylinders, FLOW-3D ResultsSurface ‘b’

1 2 3 4 5 6 7 8

Surf

ace

‘a’

1 n/a 0.2348 0.1307 0.2348 0.2199 0.0401 0.0000 0.04012 0.2348 n/a 0.2348 0.1307 0.0401 0.2199 0.0401 0.00003 0.1307 0.2348 n/a 0.2348 0.000 0.0401 0.2199 0.04014 0.2348 0.1307 0.2348 n/a 0.0401 0.000 0.0401 0.21995 0.7330 0.1340 0.0000 0.1340 n/a 0.0000 0.0000 0.00006 0.1340 0.7330 0.1340 0.0000 0.0000 n/a 0.0000 0.00007 0.0000 0.1340 0.7330 0.1340 0.0000 0.0000 n/a 0.00008 0.1340 0.0000 0.1340 0.7330 0.0000 0.0000 0.0000 n/a


Table 8-4. FLOW-3D Configuration Factor Errors, Two-Dimensional CylindersSurface ‘b’

1 2 3 4 5 6 7 8

Surf

ace

‘a’

1 — 4.19% !3.81% 0.08% 0.13% !0.82% — !0.82%2 4.19% — 0.08% !3.81% !0.82% 0.14% !0.82% —3 !3.81% 0.08% — 0.08% !0.82% 0.15% !0.82%4 0.08% !3.81% 0.08% — !0.82% — !0.82% 0.14%5 0.77% 0.11% — 0.11% — — — —6 0.10% 0.77% 0.10% — — — — —7 — 0.10% 0.77% 0.10% — — — —8 0.10% — 0.10% 0.77% — — — —

Scientific Notebook 536E (see entry for March 3, 2006) for using this module. The fluid flow andheat transfer are not of interest here; the solutions were executed for only one time step so thatthe radiation module initialization sequence would be executed. The radiation configurationfactors are computed during this initialization step.

A 74 × 74 grid of cells was used to simulate the concentric cylinder geometry. The results of theFLOW-3D computations for this case are summarized in Table 8-3. The variances between theexact values listed in Table 8-1 and the values computed by FLOW-3D are listed in Table 8-4. Itis shown that the errors are all less than 5 percent. Also, note that the configuration factor F1–2should be the same as F2–3. 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

8-8

slight error in the geometry specifications relative to grid line locations or round-off in thecomputations. Nonetheless, this is considered to be good agreement in light of the fact thatmost 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 of80 × 40 × 20 cells to discretize the air space in the enclosure. A single layer of cells was usedto define the six sides of the box. The results of the FLOW-3D computations for thethree-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 byFLOW-3D are listed in Table 8-6.

It is shown that the maximum error is just over 2 percent. Also, note that there is reasonablesymmetry to the factors. If the grid resolution is halved in all three directions, the errors in theconfiguration factors are approximately double those listed here. This case is described inScientific Notebook 536E.

Table 8-5. Configuration Factors, Fa-b, Three-Dimensional Box, FLOW-3D ResultsSurface ‘b’

1 2 3 4 5 6

Surf

ace

‘a’ 1 0.00000 0.03618 0.17069 0.17069 0.32133 0.32133

2 0.03618 0.00000 0.17069 0.17069 0.32133 0.321333 0.08535 0.08535 0.00000 0.16530 0.34043 0.340434 0.08535 0.08535 0.16530 0.00000 0.34043 0.340435 0.08033 0.08033 0.17022 0.17022 0.00000 0.509116 0.08033 0.08033 0.17022 0.17022 0.50911 0.00000

Table 8-6. FLOW-3D Configuration Factor Errors, Three-Dimensional BoxSurface ‘b’

1 2 3 4 5 6

Surf

ace

‘a’ 1 0.01% 2.02% 2.02% 2.14% 2.14%

2 0.01% 2.02% 2.02% 2.14% 2.14%3 2.02% 2.02% 0.02% 2.01% 2.01%4 2.02% 2.02% 0.02% 2.01% 2.01%5 2.14% 2.14% 2.02% 2.02% 0.02%6 2.14% 2.14% 2.02% 2.02% 0.02%

8-9

Figure 8-1. Schematic for Thermal Conduction and Radiation Between Opposing Surfaces {1 W/m-K = 0.578 BTU/ft-h-°F; 1 m = 3.28 ft;

T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

8-10

Figure 8-2. Temperature Distribution for One-Dimensional Radiation and Conduction Across an Air Gap

{1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

8-11

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

9-1

9 COMBINED HEAT TRANSFER TEST CASE

The test case described in this section validates the FLOW-3D code with the custommodules for moisture transport and radiation when all the pertinent modes of energytransfer—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 0.025 m [1 in] thick and has an internal heat generation rate such that theheat flux at the inner surface is 200 W/m2 [63.4 BTU/ft2-h]. The outer surface of this wall isadiabatic. The right vertical wall is 0.025 m [1 in] thick and its outer surface is held constant at300 K [80.3 °F]. The emissivity of both the left and right walls is 0.9. The vertical walls providefor the evaporation and condensation of water as needed under the existing temperature andconcentration 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 theother walls via this mode. These walls are furthermore assumed not to be a source or a sink forwater. The only interaction of these walls in the test case is to bound the flow and provide forviscous drag.

The acceleration due to gravity is assumed to be only 0.001 g so that the flow field for thesegeometric and thermal conditions will be laminar. The objective here is to compare the effectsof 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 scenariousing a empirical heat transfer correlation approach. This approach is based on the equationsdescribed below.

The Nusselt number correlation for natural convection in a two-dimensional square enclosuredescribed by Berkovsky and Polevikov (1977) is a widely used relationship for this case. Thenet mass transfer rate of water vapor through the enclosure will be estimated using the analogyof heat and mass transfer (Incropera and Dewitt, 1996). This is a common practice based onthe fundamentals of heat and mass transfer theory and similarity principles. Finally, theradiation heat transfer will be analyzed using the methods of Siegel and Howell (1992) for graydiffuse 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, kair = 0.026 W/(m2-K) [0.0046 BTU/h-ft2-°F]• Air/Vapor diffusivity, D = 2.6 × 10!5 m2/sec [1 ft2/hr]• Density, D = 1.169 kg/m3 (nominal value) [0.0728 lbs/ft3]• Wall thermal conductivity, kw = 1 W/(m2-K) [0.176 BTU/h-ft2-°F]

The density value listed above is used as the nominal density in the conservation of energyequation. In keeping with the moisture transport model, the incompressible ideal gas model is

9-2

used for this test case for the temperature and concentration dependent density that is used forthe momentum equation. The moisture model parameters pertinent to this case are

• Water heat of vaporization, ufg = 2,304,900 J/kg [993 BTU/lb]• Water vapor specific heat, Cvv = 1370 J/(kg*K) [0.328 BTU/(lb × R)]• Water liquid specific heat, Cvl = 4186 J/(kg*K) [1.00 BTU/(lb × R)]• Water vapor gas constant, Rv = 416 J/(kg*K) [0.100 BTU/(lb × R)]• Air gas constant, Ra = 289 J/(kg*K) [0.069 BTU/(lb × R)]

9.1.1 Test Input

FLOW-3D input files (prepin.*) were developed to model the idealized cases as follows:

1. Convection only2. Convection with radiation3. Convection with moisture transport4. Convection, radiation, and moisture transport

The convection and conduction aspects of the problem are handled by the standard portions ofthe FLOW-3D code. The moisture transport and thermal radiation processes are accomplishedby providing the user inputs to the customized portion of the code as described in ScientificNotebook 536E.


FLOW-3D was run with the input files described above until a steady-state condition wasachieved. An analysis using the empirical heat and mass transfer correlations was developedfor each of the four scenarios described above. The output of the heat transfer rates predictedby FLOW-3D is compared to the predictions of the engineering heat transfer analysis. Theaverage wall surface temperatures and heat transfer rates predicted by FLOW-3D arecompared to those resulting from the engineering analysis.


The acceptance criterion for this test case is that the local temperatures predicted by FLOW-3Dshall be within 20 percent (relative to the overall temperature difference between the twoisothermal surfaces) of the analytical predictions. Similarly, the overall heat transfer ratesshould be within 20 percent of the analytical prediction. This is a generally accepted criterion inlight of the approximate nature of the available empirical correlations for natural convection heatand mass transfer, especially in combination with radiation.

The analytical solutions for the four different scenarios investigated here are all described indetail in Scientific Notebook 536E. The analysis approach is outlined here.

The analysis seeks to determine the average temperatures of the vertical side walls and theheat transfer rates for the active heat transfer modes in each scenario. Consider a controlvolume enclosing the air in the box. The conservation of energy for this control volumedictates that

9-3

Q Q Q Q Qh c conv vap rad= = + + (9-1)

where

Qh — specified heat flux from the left hot surfaceQc — heat flux through the right cold wallQconv — heat flux via convection across the enclosureQvap — heat flux via phase change at the side walls (if present)Qrad — heat flux via radiation across enclosure (if present)

The heat flux via convection is computed from

( )Q h T Tconv s h s c= −, , (9-2)

where

h — heat transfer coefficientTs,h — hot wall surface temperature Ts,c — cold wall surface temperature

The heat transfer coefficient is estimated from the definition of the Nusselt number,

Nu hLkair

= (9-3)

where

L — enclosure dimension = 0.1 m [3.9 in]kair — air thermal conductivity = 0.026 W/(m2-K) [0.004 BTU/h-ft2-F]

The Nusselt number for a square enclosure is given by Berkovsky and Polevikov (1977) as

( )Nu Ra=+

⎛

⎝⎜⎜

⎞

⎠⎟⎟0 18

0 2

0 29

. Pr. Pr

.

(9-4)

where

Pr — air Prandtl number

Ra — Rayleigh number = ( )Ra a T T Ls,h s,c= −βν2

3 Pr

a — 0.001 g = acceleration due to gravity for this test case$ –2/(Ts,h + Ts,c) — approximate thermal expansion coefficient for airv — air kinematic viscosityPr — fluid Prandtl number

9-4

The heat flux via conduction through the right wall is

Q kT T

tc ws c C

w

=−,

(9-5)

where

kw — wall thermal conductivity = 1 W/(m2-K) [0.176 BTU/h-ft2-°F]tw — wall thickness = 0.025 m TC — cold wall outer surface temperature = 300 K [80.3 °F]

For the case when Qvap = Qrad = 0, Eqs. (9-1) and (9-5) form a nonlinear set of two equationswith two unknowns, Ts,h and Ts,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 viaradiation for this case is

( )Q F

FT Trad

h c

hh c

c

c

s,h s,c=+ −

−−

−

σ

εε

ε1 1

4 4

(9-6)

where

F — Stefan-Boltzmann constant = 5.67 × 10!8 W/(m2-K4) [1.17 × 10-9 BTU/(h-ft2-°R4)],h — ,c = surface emissivityFh!c — configuration factor = 0.414

When Qrad is included in Eq. (9-1), the equations still form a system of two equations in twounknowns, but the nonlinearity of the system is increased because of the T4 terms.

Finally, when moisture transport is considered, the heat flux due to evaporation andcondensation is approximated as

Q m hrad v vap= & (9-7)

where

hvap — heat of vaporization of water = 2304900 J/kg [993 BTU/lb] — mass flux of water [kg/(m2sec)]&mv

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 todefine the Sherwood number with the substitution of the Schmidt number for the Prandlt number

( )Sh h LD

ScSc

Ram

wa

= =+

⎛

⎝⎜⎜

⎞

⎠⎟⎟018

0 2

0 29

..

.

(9-8)

where

9-5

Sh — Sherwood numberhm — mass transfer coefficient (m/sec)Dwa — diffusion coefficient for air and water = 2.6 × 10!5 m2/sec [1.0 ft2/hr]Sc — Schmidt number

The water mass flux rate is then given by

( )&, ,m h c cv m w h w c= − (9-9)

where

cw,h — water vapor concentration at the hot wall surface (kg/m3)cw,c — water vapor concentration at the cold wall surface (kg/m3)

It is assumed here that the water vapor concentrations are those corresponding to air at100-percent humidity at the wall temperature.

So, adding Eq. (9-7) through Eq. (9-1) brings another term with the unknown temperatures intothe analysis.

The analysis results are summarized in Table 9-1 for each of the four different scenarios. Theheat transfer rates and the mass transfer rate are expressed in terms of power per unit depthand mass flow per unit depth.

Note the significant effect of the radiation and phase change on the hot wall surfacetemperature. 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, energytransfer by radiation or phase change modes accounts for several times more energy flow thanthe convection heat transfer.

9.1.4 Test Results

FLOW-3D input files (prepin.*) were prepared for each of these scenarios. The input files usethe custom radiation module and moisture transport module as required. The inputspecifications 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 ScientificNotebook 536E.

All of the simulations were conducted using a 20 × 30 grid mesh in the flow field. Five cellswere used through the thicknesses of the left and right walls, so the entire grid was30 × 30 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 theanalytical 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 fromthe empirical analysis. The basis for the temperature variance is the overall temperaturedifference between the hot and cold surfaces.

The FLOW-3D results agree with the analytical results for this case. All of the deviations areless than 5 percent of their basis values and meet the acceptance criterion for this test case.

9-6

Table 9-1. Analysis Results for Two-Dimensional Enclosure Heat and Mass Transfer

ScenarioTh, average

K*Tc, average

K

Heat Transfer RateMass

Transferkg/sec‡

Convection W/m†

RadiationW/m

PhaseChange

W/mTotalW/m

ConvectionOnly

650 305.0 20 n/a§ n/a 20 n/a

Convection +Radiation

362.1 305.0 2.7 17.3 n/a 20 n/a

Convection +MoistureTransport

342.5 305.0 1.7 n/a 18.3 20 7.9 × 10!6

Convection +Radiation +MoistureTransport

334.2 305.0 1.2 7.8 11.0 20 4.8 × 10!6

*T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]†1 W/m = 0.961 BTU/h-ft‡1 kg/sec = 2.2 lb/sec§n/a = not applicable

Table 9-2. FLOW-3D Results for Two-Dimensional Enclosure Heat and Mass Transfer

ScenarioTh, average

K*†Tc, average

K

Heat Transfer RateMass

Transferkg/sec§

Convection W/m‡

RadiationW/m

PhaseChange

W/mTotalW/m

ConvectionOnly

646.2(!1.1%)

304.5(0.1%)

20 n/a2 n/a 20 n/a

Convection +Radiation

362.7(1.1%)

304.5(!1.0%)

2.3(!2.0%)

17.7(2.0%)

n/a 20 n/a

Convection +MoistureTransport

340.9(!4.4%)

304.5(!1.4%)

1.8(0.2%)

n/a 18.2(!0.4%)

20 7.9 × 10!6

(0%)

Convection +Radiation +MoistureTransport

333.8(!1.5%)

304.5(!1.7%)

1.3(0.3%)

7.9(0.6%)

11.0(0%)

20.2(1.0%)

4.8 × 10!6

(0%)

*These values are the average of all surface cells where the temperature values correspond to the FLOW-3D cellcenter located half the cell width from the obstacle surface.†T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]‡1 W/m = 0.961 BTU/h-ft§1 kg/sec = 2.2 lb/sec2n/a = not applicable.

9-7

Figure 9-1. Schematic for Convection, Radiation, and Mass Transfer in a Two-Dimensional Enclosure

{1 W/m-K = 0.578 BTU/ft-h-°F; 1 W/m3 = 0.097 BTU/(hr-ft3); 1 m = 3.28 ft; T (°C) = (T (K) ! 273.15); T (°F) = [1.80 × (T (K) ! 273.15) + 32]}

9-8

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]

10-1

10 INDUSTRY EXPERIENCE

FLOW-3D is used widely in the casting industry because of its solid-liquid phase changecapabilities and in the aerospace industry for its free surface, surface tension (i.e., zero gravityconsiderations), and noninertial reference frame capabilities.

11-1

11 NOTES

None.

SOFTWARE VALIDATION REPORT FOR FLOW=3D ... VALIDATION REPORT FOR FLOW=3D@ VERSION 9.0 Prepared by S. Green C. Manepally Center for Nuclear Waste Regulatory Analyses San Antonio, Texas

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