Validation of Pronghorn Pressure Drop Correlations Against Pebble Bed Experiments

Abstract

The verification and validation of Pronghorn is imperative for predicting the fluid velocity, pressure, and temperature in advanced reactors, specifically high-temperature gas-cooled reactors. Pronghorn is a coarse-mesh, intermediate-fidelity, multidimensional thermal-hydraulic code developed by Idaho National Laboratory. The Pronghorn incompressible Navier-Stokes equations are validated by using the pressure drop measurements and axial velocity averaged from the particle image velocimetry data obtained at the engineering-scale pebble bed facility at Texas A&M University.

Pronghorn and STAR-CCM+ porous media models using the Handley, Kerntechnischer Ausschuss, and Carman correlations comparably estimate the pressure drop better than other functions with a maximum 3.34% average relative difference compared to the experimental measurements. The precise average pebble bed porosity estimation has a large impact on the pressure drop. The implementation of the volume-averaged porosity in several sectors, with each sector’s thickness larger than the representative elementary length, has the potential to improve pressure drop modeling or provide more detailed velocity profiles in nuclear reactors with high aspect ratios. The wall effects can be considered using this approach, applying the relatively higher volume-averaged porosity near walls. In addition, the pressure gradients and volume- or surface-averaged axial velocities from the realizable two-layer $k-\epsilon$ and shear stress transport $k-\omega$ models are in good agreement with the porous media simulations and particle image velocimetry data.

Keywords:

• Porous media modeling
• pebble bed reactors
• turbulence modeling

Nomenclature

$D$ =	=	bed diameter (m)
$D_h$ =	=	hydraulic diameter (m)
$d$ =	=	distance from the wall (m)
$d_p$ =	=	pebble diameter (m)
$F_s$ =	=	safety factor
$g$ =	=	gravitational acceleration vector (m/s2)
$H$ =	=	length of the porous medium (m)
$h$ =	=	mesh size or specific porous bed length (m)
$I$ =	=	turbulence intensity
$k$ =	=	turbulent kinetic energy (m2/s2)
$l$ =	=	turbulent length scale (m)
$l_{\epsilon }$ =	=	Wolfstein length scale (m)
$N$ =	=	total number of cells
$n$ =	=	refractive index
$P$ =	=	pressure (Pa)
$\bar{P}$ =	=	time-averaged pressure (Pa)
$\bar{p}$ =	=	formal order of accuracy
$\hat{p}$ =	=	observed order of accuracy
$R{e}_d$ =	=	wall-distance Reynolds number, $\sqrt{k}d/\nu$
$R{e}_h$ =	=	Reynolds number based on hydraulic diameter, ${\rho }_{f}uD/\mu$ for free flow region where $\epsilon$ = 1, or $2{\rho }_{f}u{d}_p\epsilon /(3(1-\epsilon )\mu )$ for porous bed
$R{e}_p$ =	=	particle Reynolds number, ${\rho }_f{u}_s{d}_p/\mu$
$R{e}^{\ast }$ =	=	modified Reynolds number, $R{e}_p/(1-\epsilon )$
$r$ =	=	Refinement ratio, radial distance from the center of the pebble bed (m), or fit point distance (mm)
$S_{ij}$ =	=	mean strain-rate tensor where $i$ and $j$ are the indices of vectors (1/s)
$t$ =	=	time (s) or $t$-value
$u$ =	=	interstitial/physical velocity (m/s)
$u_{ave}$ =	=	volume- or surface-averaged velocity (m/s)
$u_s$ =	=	superficial velocity, $\epsilon u$ (m/s)
$u_{\tau }$ =	=	friction velocity, $\sqrt{{\tau }_w/{\rho }_f}$ (m/s)
$\bar{u}$ =	=	time-averaged velocity vector (m/s)
$u^{\prime }$ =	=	fluctuating velocity (m/s)
$\overline{u_i^{\prime }{u}_j^{\prime }}$ =	=	Reynolds stress tensor where $i$ and $j$ are the indices of vectors (m2/s2)
$V_{spheres}$ =	=	total volume of spheres (m3)
$V_{total}$ =	=	total volume of the cylinder (m3)
$W$ =	=	interphase friction factor (1/s)
$W_{ij}$ =	=	mean rotation tensor where $i$ and $j$ are the indices of vectors (1/s)
$x$ =	=	coordinate vector (m)
Greek	=
${\delta }_{ij}$ =	=	Kronecker delta where $i$ and $j$ are the indices of vectors
$\mathrm{\Delta }{V}_i$ =	=	volume of the i-th cell (m3)
$\epsilon$ =	=	porosity or turbulent dissipation rate (m2/s3)
$\bar{\epsilon }$ =	=	average bed porosity
${\epsilon }_{ijk}$ =	=	permutation tensor where $i$, $j$, and $k$ are the indices of vectors
$\mu$ =	=	dynamic viscosity (Pa$\cdot$s)
${\mu }_t$ =	=	turbulent eddy viscosity (Pa$\cdot$s)
${\rho }_f$ =	=	fluid density (kg/m3)
${\tau }_w$ =	=	wall shear stress (Pa)
$\nu$ =	=	kinematic viscosity (m2/s)
${\nu }_t$ =	=	turbulent kinematic viscosity, ${\mu }_t/{\rho }_f$ (m2/s)
$\omega$ =	=	specific turbulent dissipation rate (1/s)
${\omega }_k$ =	=	angular velocity (1/s)

Acknowledgments

This research was partially funded by the Office of Nuclear Energy of the U.S. Department of Energy, NEAMS project, under contract DE-NE0008983. This research made use of the resources of the High Performance Computing Center at Idaho National Laboratory, which is supported by the Office of Nuclear Energy of the U.S. Department of Energy and the Nuclear Science User Facilities under contract DE-AC07-05ID14517. This paper has been authored by Battelle Energy Alliance, LLC under contract DE-AC07-05ID14517 with the U.S. Department of Energy. The U.S. government retains and the publisher, by accepting this paper for publication, acknowledges that the U.S. government retains a nonexclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. government purposes.

Disclosure Statement

No potential conflict of interest was reported by the authors.