INL REPORT INL/EXT-18-44453 Unlimited Release Printed February 2018 Pronghorn Theory Manual April J. Novak Ling Zou John W. Peterson David Andrs Joe Kelly Rachel N. Slaybaugh Richard C. Martineau Hans D. Gougar Prepared by Idaho National Laboratory Idaho Falls, Idaho 83415 The Idaho National Laboratory is a multiprogram laboratory operated by Battelle Energy Alliance for the United States Department of Energy under DOE Idaho Operations Office. Contract DE-AC07-05ID14517. Approved for public release; further dissemination unlimited. Issued by the Idaho National Laboratory, operated for the United States Department of Energy by Battelle Energy Alliance. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. 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This report has been reproduced directly from the best available copy. NT O E F E TM N R E A R P G E Y •D • U N A I C T I E R D E S M T A ATES OF 2 INL/EXT-18-44453 Unlimited Release Printed February 2018 Pronghorn Theory Manual A.J. Novak∗, L. Zou†, J.W. Peterson†, D. Andrs†, J. Kelly×, R.N. Slaybaugh∗, R.C. Martineau†, and H.D. Gougar† ∗ University of California, Berkeley 3115 Etcheverry Hall Berkeley, CA 94708 ∗ Idaho National Laboratory Idaho Falls, ID 83402 × Nuclear Regulatory Commission Rockville, MD 20852 3 Contents 1 Introduction . 10 2 Physical Models. 11 2.1 Preliminary Considerations . 15 2.2 The “Math” Behind the Porous Media Equations . 16 2.2.1 Properties of Averages . 18 2.2.2 A Generic Transport Equation . 22 2.3 The Continuity Equation . 23 2.4 The Momentum Equation . 25 2.4.1 Porous Media Momentum Losses . 39 2.4.2 The Trans-Forchheimer Regime . 43 2.4.3 Anisotropic Porous Media . 46 2.5 The Fluid Energy Equation . 47 2.6 The Solid Energy Equation . 58 2.7 The Legacy Pronghorn Equations . 61 2.8 A Unified Approach to Solving the Navier-Stokes Equations . 61 2.9 Boundary Conditions . 66 2.9.1 The Continuity Equation . 67 2.9.2 The Momentum Equation . 67 2.9.3 The Energy Equation . 68 2.10 Nondimensionalization of the Governing Equations . 68 2.10.1 The Momentum Equation . 69 2.10.2 The Energy Equations . 69 2.10.3 Compressible Versus Incompressible Flows . 71 2.10.4 Definitions of Dimensionless Numbers . 74 2.10.5 Expected Values for Dimensionless Numbers. 76 2.11 Summary of Physical Models . 77 3 Kernels........................................................................................... 80 4 AuxKernels....................................................................................... 82 5 Interface Kernels . 83 6 Boundary Conditions . 84 6.1 The Conservative Equation Set . 85 6.2 The Primitive Equation Set . 86 7 Equations of State . 87 7.1 Barotropic Equation of State . 87 7.2 Stiffened Gas Equation of State . 87 7.3 Isentropic Stiffened Gas Equation of State . 88 7.4 Linear Equation of State . 88 7.5 Ideal Gas Equation of State . 88 7.5.1 Isentropic Relations . 89 8 Porosity, Coordination Number, and Tortuosity . 91 8.1 Porosity . 96 8.1.1 OscillatoryPorosityFunction ................................................ 96 8.1.2 ExponentialPorosityFunction ................................................ 96 8.1.3 AnnularExponentialPorosityFunction ......................................... 96 8.2 Coordination Number . 97 4 8.3 Tortuosity . 97 9 Friction Coefficients . 98 9.1 Pebble Bed Correlations . 98 9.1.1 Friction Coefficients Independent of the Wall . 99 9.1.1.1 ErgunDragCoefficients .............................................. 99 9.1.1.2 PiecewiseLinearDragCoefficients ....................................100 9.1.1.3 KTADragCoefficients ................................................101 9.1.2 Friction Coefficients Dependent on the Wall . 101 9.1.2.1 EisfeldDragCoefficients ............................................102 9.2 Pipe Flow Correlations . 102 9.2.1 ChurchillDragCoefficients ..................................................103 10 Heat Transfer Coefficients. 104 10.1 Pebble Bed Correlations . 104 10.1.1 Correlations Independent of Porosity Variation . 105 10.1.1.1 WakaoPebbleBedHTC ..................................................106 10.1.2 Correlations Dependent on Porosity Variation . 106 10.1.2.1 GnielinskiPebbleBedHTC .............................................106 10.1.2.2 KTAPebbleBedHTC ....................................................107 10.2 Wall-to-Fluid Correlations . ..